Computing isogenies from modular equations in genus two
aa r X i v : . [ m a t h . AG ] O c t Computing isogenies from modular equationsin genus two
Jean Kieffer , Aurel Page , Damien Robert Université de Bordeaux, France Inria Bordeaux Sud-Ouest, France {jean.kieffer,aurel.page,damien.robert}@math.u-bordeaux.fr
Abstract
We present an algorithm solving the following problem: given twogenus curves over a field k with isogenous Jacobians, compute such anisogeny explicitly. This isogeny can be either an ℓ -isogeny or, in the realmultiplication case, an isogeny with cyclic kernel; we require that k havelarge enough characteristic and that the curves be sufficiently generic.Our algorithm uses modular equations for these isogeny types, and makesessential use of an explicit Kodaira–Spencer isomorphism in genus . We are interested in the following version of the isogeny problem: given twoisogenous abelian varieties, compute an isogeny between them explicitly.Let us start with some motivation. The isogeny problem in the case of ellip-tic curves was solved by Elkies [Elk98]. Given two ℓ -isogenous elliptic curves,where ℓ is a prime, his algorithm uses modular polynomials of level ℓ to computerational fractions defining this isogeny. Elkies’s algorithm is used to speed upSchoof’s point counting algorithm for elliptic curves over finite fields [Sch85]:replacing kernels of endomorphisms by kernels of isogenies yields smaller sub-groups of the elliptic curve, and therefore smaller polynomial computations,while giving the same amount of information on the Frobenius. This improve-ment is at the heart of the well-known SEA point counting algorithm [Sch95].The situation for point counting in genus 2 is different, as the existing com-plexity estimates and records only use kernels of endomorphisms [GKS11; GS12].One can therefore ask whether the idea of using isogenies generalizes. Modu-lar polynomials have now been computed in genus 2: the smallest ones areknown both for ℓ -isogenies [Mil15] and, in the real multiplication case, cyclic β -isogenies [MR17; Mar18]. This opened the way for Atkin-style methods in pointcounting [BGL+16], but isogeny computations remain the missing ingredient togeneralize Elkies’s method in genus . The object of this paper is precisely tofill this gap. 1e now present our main result in the case of ℓ -isogenies. For any field k , wedenote by A ( k ) the coarse moduli space of principally polarized abelian surfacesover k , and we denote by j = ( j , j , j ) the Igusa invariants as introduced byStreng (see §2.2). We also denote by Ψ ℓ,i for ≤ i ≤ the modular equations oflevel ℓ in Igusa invariants (see §2.6). Recall that if C is a hyperelliptic curve ofgenus over a field k , then its Jacobian Jac( C ) is a principally polarized abeliansurface which is birational to the symmetric square C , sym ; the points of C , sym over k are the Galois-invariant unordered pairs { P, Q } where P, Q ∈ C ( k ) . Theorem 1.1.
Let ℓ be a prime, and let k be a field such that char k = 0 or char k > ℓ + 7 . Let U ⊂ A ( k ) be the open set consisting of abelian surfaces A such that Aut ¯ k ( A ) ≃ {± } and j ( A ) = 0 . Assume that there is an algorithmto evaluate derivatives of modular equations of level ℓ at a given point of U × U over k using C eval ( ℓ ) operations in k .Let A, A ′ ∈ U , and let j ( A ) , j ( A ′ ) be their Igusa invariants. Assume that A and A ′ are ℓ -isogenous, and that the subvariety of A × A cut out by the modularequations Ψ ℓ,i for ≤ i ≤ is normal at ( j ( A ) , j ( A ′ )) . Then, given j ( A ) and j ( A ′ ) , we can compute1. a field extension k ′ /k of degree dividing ,2. hyperelliptic curve equations C , C ′ over k ′ whose Jacobians are isomorphicto A, A ′ respectively,3. a point P ∈ C ( k ′ ) ,4. rational fractions s, p, q, r ∈ k ′ ( u, v ) ,such that ( s, p, q, r ) equals the compositum C Jac( C ) Jac( C ′ ) C ′ , sym A Q [ Q − P ] ϕ ∼ m where ϕ is an ℓ -isogeny and m is the rational map given by { ( x , y ) , ( x , y ) } 7→ (cid:16) x + x , x x , y y , y − y x − x (cid:17) . The cost of the algorithm is O (cid:0) C eval ( ℓ ) (cid:1) + e O ( ℓ ) elementary operations and O (1) square roots in k ′ . In other words, given sufficiently generic genus curves C , C ′ whose Jaco-bians are ℓ -isogenous, obtained for instance by computing roots of modularequations of level ℓ , we compute rational fractions that determine an ℓ -isogenycompletely. We also obtain a similar result in the case of β -isogenies in thereal multiplication case: see Theorem 6.3. In a follow-up paper, the first au-thor will design evaluation algorithms for genus modular equations and theirderivatives, thereby obtaining estimates on C eval ( ℓ ) . Possible applications of ourresults to the point counting problem are a major goal for future work.2et us describe the outline of our algorithm in the case of ℓ -isogenies from ageometric point of view, in any dimension g . The central object is the map Φ ℓ = ( Φ ℓ, , Φ ℓ, ) : A g ( ℓ ) → A g × A g where A g ( ℓ ) denotes the stack of principally polarized abelian schemes of dimen-sion g with an ℓ -kernel, and A g denotes the stack of principally polarized abelianschemes of dimension g ; this map is given by ( A, K ) ( A, A/K ) . Both Φ ℓ, and Φ ℓ, are étale maps. Let ϕ : A → A ′ be an ℓ -isogeny, so that ( A, A ′ ) lies inthe image of Φ ℓ . Denote by T A ( A g ) the tangent space of A g at A , and denoteby T ( A ) the tangent space of A at its neutral point. Then there is a closerelation between two maps:• the deformation map D ( ϕ ) : T A ( A g ) → T A ′ ( A g ) defined as D ( ϕ ) := d Φ ℓ, ◦ d Φ ℓ, − ; • the tangent map dϕ : T ( A ) → T ( A ′ ) .This relation stems from a canonical isomorphism, called the Kodaira–Spencerisomorphism , between T A ( A g ) and Sym T ( A ) . Therefore, in any dimension g ,an isogeny algorithm could run as follows.1. Compute the deformation map by differentiating certain modular equa-tions giving a local model of A g ( ℓ ) and A g .2. Compute dϕ from the deformation map using an explicit version of theKodaira–Spencer isomorphism, that is, an explicit way to map a pair ( A, w ) where w is an element of Sym T ( A ) to the corresponding point of T A ( A g ) in the local model of A g .3. Finally, attempt to reconstruct ϕ itself by solving a differential system inthe formal group of A and performing a multivariate rational reconstruc-tion. In this last step, the characteristic of k should be large with respectto ℓ , hence the condition on the characteristic in Theorem 1.1. Otherwise,a standard solution is to use étaleness of the modular correspondence tolift the isogeny in characteristic , as in [JL06], and to control the precisionlosses when reconstructing the isogeny.In practice, working with stacks would involve adding an additional levelstructure and keeping track of automorphisms, which is not computationallyconvenient. Therefore, in order to make everything explicit in the case g = 2 ,we choose to replace the stack A by its coarse moduli scheme A . We even workup to birationality, by considering the birational map from A to A defined bythe three Igusa invariants ( j , j , j ) . These reductions have the drawback ofintroducing singularities; this is the reason for restricting to the open set U inTheorem 1.1. When the genericity conditions of Theorem 1.1 are not satisfied,one can still compute the isogeny by working at the level of stacks, or choosingother models, for instance when A or A ′ is a product of elliptic curves.3n the genus setting, the local model of A g ( ℓ ) that we use in Step 1 isgiven by modular equations in Igusa invariants; in order to compute the defor-mation map, it is enough to evaluate modular equations and their derivativesat ( A, A ′ ) . In Step 2, we choose to encode a basis of T ( A ) in the choice of ahyperelliptic curve equation. Then, the explicit Kodaira–Spencer isomorphismis simply an expression for certain Siegel modular forms, namely derivatives ofIgusa invariants, in terms of the coefficients of the curve (see Theorem 3.14). InStep 3, we take advantage of the fact that the curve C embeds in its Jacobianto compute with power series in one variable only.This paper is organized as follows. In Sections 2 and 3, we work over C : Sec-tion 2 is devoted to the necessary background on modular forms and isogenies,while Section 3 is devoted to the explicit Kodaira–Spencer isomorphism andthe computation of the tangent map. In Section 4, we call upon the languageof algebraic stacks to show that the calculations over C remain in fact validover any base. We present the computation of the isogeny from its tangentmap in Section 5, focusing on the large characteristic case which is sufficientfor applications to point counting, and we sum up the algorithm in Section 6.Finally, in Appendix A, we present variants in the algorithm in the case of realmultiplication by Q ( √ and compute an example of cyclic isogeny of degree . Acknowledgement.
The authors were supported by the ANR grant CIAO(French Agence Nationale de la Recherche).
We present the basic facts about Siegel and Hilbert modular only in the genus case. References for this section are [van08] for Siegel modular forms, and [Bru08]for Hilbert modular forms, where the general case is treated.We write × matrices in block notation using × blocks. We write m t for the transpose of a matrix m , and use the notations m − t = ( m − ) t , Diag( x, y ) = (cid:18) x y (cid:19) . The Siegel threefold.
Denote by H the set of complex symmetric × matrices with positive definite imaginary part. For every τ ∈ H , the quotient A ( τ ) = C / Λ( τ ) where Λ( τ ) = Z ⊕ τ Z is naturally endowed with the structure of a principally polarized abelian surfaceover C . A basis of differential forms on A ( τ ) is given by ω ( τ ) = (2 πi dz , πi dz ) z , z are the coordinates on C . Recall that the symplectic group Sp ( Z ) acts on H in the following way: ∀ γ = (cid:18) a bc d (cid:19) ∈ Sp ( Z ) , ∀ τ ∈ H , γτ = ( aτ + b )( cτ + d ) − . Proposition 2.1 ([BL04, Rem. 8.1.4]) . Let τ ∈ H , and let γ ∈ Sp ( Z ) withblocks a, b, c, d . Then there is an isomorphism η γ,τ : A ( τ ) → A ( γτ ) , z ( cτ + d ) − t z. Theorem 2.2 ([BL04, Prop. 8.1.3]) . Let A be a principally polarized abeliansurface over C . Then there exists τ ∈ H such that A is isomorphic to A ( τ ) ,and τ is uniquely determined up to action of Sp ( Z ) . The quotient space A ( C ) = Sp ( Z ) \ H is the set of complex points of thecoarse moduli space A alluded to in the introduction. Siegel modular forms.
Let ρ : GL ( C ) → GL( V ) be a finite-dimensionalholomorphic representation of GL ( C ) . We can assume that ρ is irreducible. A Siegel modular form of weight ρ is a holomorphic map f : H → V satisfyingthe transformation rule ∀ γ = (cid:18) a bc d (cid:19) ∈ Sp ( Z ) , ∀ τ ∈ H , f ( γτ ) = ρ ( cτ + d ) f ( τ ) . We say that f is scalar-valued if dim V = 1 , and vector-valued otherwise. A modular function is only required to be meromorphic instead of holomorphic.If A is a principally polarized abelian surface over C endowed with a basis ω of Ω ( A ) (the space of global differential forms on A ), and if if f is a Siegelmodular form of weight ρ , then it makes sense to evaluate f on the pair ( A, ω ) .We refer to §4 for a geometric interpretation of this fact. To compute thisquantity, choose τ ∈ H and an isomorphism η : A → A ( τ ) as in Theorem 2.2.Let r ∈ GL ( C ) be the matrix of the pullback map η ∗ : Ω ( A ( τ )) → Ω ( A ) inthe bases ω ( τ ) , ω . Then f ( A, ω ) = ρ ( r ) f ( τ ) . We can check using Proposition 2.1 that f ( A, ω ) does not depend on the choiceof τ and η . Classification of weights.
Finite-dimensional holomorphic representationsof GL ( C ) are well known. Let n ≥ be an integer. We denote by Sym n the n -th symmetric power of the standard representation of GL ( C ) on C .Explicitly, Sym n is a representation on the vector space C n [ x ] of polynomials ofdegree at most n , with Sym n (cid:18)(cid:18) a bc d (cid:19)(cid:19) W ( x ) = ( bx + d ) n W (cid:18) ax + cbx + d (cid:19) .
5e take ( x n , . . . , x, as the standard basis of C n [ x ] , so that we can write anendomorphism of C n [ x ] as a matrix; in particular we have Sym (cid:18) a bc d (cid:19) = a ab b ac ad + bc bdc cd d . Proposition 2.3.
The irreducible finite-dimensional holomorphic representa-tions of GL ( C ) are exactly the representations det k Sym n , for k ∈ Z and n ∈ N .Proof. Since SL ( C ) is a simply connected Lie group, there is an equivalencebetween holomorphic finite-dimensional representations of SL ( C ) and repre-sentations of its Lie algebra sl ( C ) [Bou72, Ch. III, §6.1, Th. 1]. By [Bou75,Ch. VIII, §1.3, Th. 1], irreducible representations of sl ( C ) are classified bytheir higher weight; on the Lie group side, this shows that the holomorphicfinite-dimensional irreducible representations of SL ( C ) are exactly the repre-sentations Sym n for n ∈ N . The case of GL ( C ) follows easily.The weight of a scalar-valued Siegel modular form f is of the form det k forsome k ∈ Z , and in fact k ≥ . We also say that f is a scalar-valued Siegelmodular form of weight k . Writing Sym n as a representation on C n [ x ] allowsus to multiply Siegel modular forms; hence, they naturally generate a graded C -algebra. Fourier expansions.
Let f be a Siegel modular form on H of any weight,with underlying vector space V . If s ∈ M ( Z ) is symmetric, then f ( τ + s ) = f ( τ ) for every τ ∈ H . Hence, if we write τ = (cid:18) τ τ τ τ (cid:19) and q j = exp(2 πiτ j ) for ≤ j ≤ , then f has a Fourier expansion of the form f ( τ ) = X n ,n ,n ∈ Z c f ( n , n , n ) q n q n q n . The Fourier coefficients c f ( n , n , n ) belong to V , and can be nonzero onlywhen n ≥ , n ≥ , and n ≤ n n . Note that n can still be negative.When computing with q -expansions, we consider them as elements of thepower series ring C ( q )[[ q , q ]] . Writing the beginning of a q -expansion meanscomputing modulo an ideal of the form (cid:0) q ν , q ν (cid:1) for some precision ν ≥ . Structure of scalar-valued forms.
The full graded C -algebra of Siegel mod-ular forms in genus 2 is not finitely generated [van08, §25], but the subalgebraof scalar-valued modular forms is. 6 heorem 2.4 ([Igu62; Igu67]) . The graded C -algebra of scalar-valued even-weight Siegel modular forms in genus is generated by four algebraically inde-pendent elements ψ , ψ , χ , and χ of respective weights , , , , and q -expansions ψ ( τ ) = 1 + 240( q + q )+ (cid:0) q + 13440 q + 30240 + 13340 q − + 240 q − (cid:1) q q + O (cid:0) q , q (cid:1) ,ψ ( τ ) = 1 − q + q )+ (cid:0) − q + 44352 q + 166320 + 44352 q − − q − (cid:1) q q + O (cid:0) q , q (cid:1) ,χ ( τ ) = (cid:0) q − q − (cid:1) q q + O ( q , q ) ,χ ( τ ) = (cid:0) q + 10 + q − (cid:1) q q + O (cid:0) q , q (cid:1) . The graded C -algebra of scalar-valued Siegel modular forms in genus is C [ ψ , ψ , χ , χ ] ⊕ χ C [ ψ , ψ , χ , χ ] where χ is a modular form of weight and q -expansion χ ( τ ) = q q ( q − q )( q − q − ) + O ( q , q ) . The q -expansions in Theorem 2.4 are easily computed from expressions interms of theta functions, and their coefficients are integers. We warn the readerthat different normalizations appear in the literature: our χ is − times themodular form χ appearing in Igusa’s papers, our χ is times Igusa’s χ ,and our χ is i times Igusa’s χ .The equality χ ( τ ) = 0 occurs exactly when A ( τ ) is isomorphic to a productof elliptic curves with the product polarization; otherwise, A ( τ ) is isomorphicto the Jacobian of a hyperelliptic curve.Following Streng [Str10, §2.1] and our choice of normalizations, we definethe Igusa invariants to be j = 2 − ψ ψ χ , j = 2 − ψ χ χ , j = 2 − ψ χ . They are Siegel modular functions of trivial weight, i.e. weight det . Proposition 2.5.
Igusa invariants define a birational map A ( C ) → C .Proof. By the theorem of Baily and Borel [BB66, Thm. 10.11], scalar-valuedSiegel modular forms of sufficiently high even weight realize a projective em-bedding of A ( C ) . Therefore, by Theorem 2.4, Igusa invariants generate thefunction field of A ( C ) . Remark 2.6.
Proposition 2.5 shows that generically, giving ( j , j , j ) in C uniquely specifies an isomorphism class of principally polarized abelian surfacesover C . This correspondence only holds on an open set: Igusa invariants are notdefined on products of elliptic curves, and do not represent a unique isomorphism7lass when ψ = 0 . If one wants to consider these points nonetheless, it is bestto make another choice of invariants: for instance one could use h = ψ ψ , h = χ ψ , h = χ ψ ψ which are generically well-defined on products of elliptic curves. See [Liu93,Thm. 1.V] for an interpretation of these invariants in terms of j ( E ) + j ( E ) and j ( E ) j ( E ) when evaluated on a product E × E . Examples of vector-valued forms.
Derivatives of Igusa invariants are mod-ular function themselves; as explained in the introduction, this property stemsfrom the existence of the Kodaira–Spencer isomorphism.
Proposition 2.7.
Let f be a Siegel modular function of trivial weight. Then dfdτ := ∂f∂τ x + ∂f∂τ x + ∂f∂τ is a Siegel modular function of weight Sym .Proof. Differentiate the relation f ( γτ ) = f ( τ ) with respect to τ .We will use another vector-valued modular form in the sequel. Example 2.8.
Following Ibukiyama [Ibu12], let E ⊂ R denote the lattice ofhalf-integer vectors v = ( v , . . . , v ) subject to the conditions X k =1 v k ∈ Z and ∀ ≤ k, l ≤ , v k − v l ∈ Z . Set a = (2 , , i, i, i, i, i, and b = (1 , − , i, i, , − , − i, i ) , where i = − . Define f , ( τ ) = 1111456000 X j =0 (cid:18) j (cid:19) Θ j ( τ ) x j where, using the notation h v, w i = X k =1 v k w k , Θ j ( τ ) = X v,v ′ ∈ E h v, a i j · h v ′ , a i − j · (cid:12)(cid:12)(cid:12)(cid:12) h v, a i h v ′ , a ih v, b i h v ′ , b i (cid:12)(cid:12)(cid:12)(cid:12) · exp (cid:16) iπ (cid:0) h v, v i τ + 2 h v, v ′ i τ + h v ′ , v ′ i τ (cid:1)(cid:17) . Then f , is a nonzero Siegel modular form of weight det Sym . This definitionprovides an explicit, but slow, method to compute the first coefficients of the q -expansion; using the expression of f , in terms of theta series [CFv17] would8e faster. We have f , ( τ ) = (cid:0) (4 q − q + 24 − q − + 4 q − ) q q + · · · (cid:1) x + (cid:0) (12 q − q + 24 q − − q − ) q q + · · · (cid:1) x + (cid:0) ( − q + 2 − q − ) q q + · · · (cid:1) x + (cid:0) ( − q + 2 q − ) q q + · · · (cid:1) x + (cid:0) ( − q + 2 − q − ) q q + · · · (cid:1) x + (cid:0) (12 q − q + 24 q − − q − ) q q + · · · (cid:1) x + (cid:0) (4 q − q + 24 − q − + 4 q − ) q q + · · · (cid:1) . In the context of Hilbert surfaces and abelian surfaces with real multiplication,we consistently use the following notation: K a real quadratic number field (embedded in R ) ∆ the discriminant of K , so that K = Q (cid:0) √ ∆ (cid:1) Z K the ring of integers in K Z ∨ K the trace dual of Z K , in other words Z ∨ K = 1 / √ ∆ Z K x x real conjugation in K Σ the embedding x ( x, x ) from K to R .Finally, we denote Γ K = SL (cid:0) Z K ⊕ Z ∨ K (cid:1) = (cid:26)(cid:18) a bc d (cid:19) ∈ SL ( K ) | a, d ∈ Z K , b ∈ (cid:0) Z ∨ K (cid:1) − , c ∈ Z ∨ K (cid:27) . A principally polarized abelian surface A over C has real multiplication by Z K if it is endowed with an embedding ι : Z K ֒ → End sym ( A ) , where End sym ( A ) denotes the set of endomorphisms of A that are invariantunder the Rosati involution. Hilbert surfaces.
Denote by H the complex upper half plane. For every t = ( t , t ) ∈ H , the quotient A K ( t ) = C / Λ K ( t ) where Λ K ( t ) = Σ (cid:0) Z ∨ K (cid:1) ⊕ Diag( t , t ) Σ (cid:0) Z K (cid:1) is naturally endowed with the structure of a principally polarized abelian surfaceover C , and has a real multiplication embedding ι K ( t ) given by multiplicationvia Σ . It is also endowed with the basis of differential forms ω K ( t ) = (2 πi dz , πi dz ) . σ of H given by σ (cid:0) ( t , t ) (cid:1) = ( t , t ) exchanges the twodifferential forms in the basis, and exchanges the real multiplication embeddingwith its conjugate.The embedding Σ induces a map Γ K ֒ → SL ( R ) . Through this embedding,the group Γ K acts on H by the usual action of SL ( R ) on H on each coordinate. Theorem 2.9 ([BL04, §9.2]) . Let ( A, ι ) be a principally polarized abelian surfaceover C with real multiplication by Z K . Then there exists t ∈ H such that ( A, ι ) is isomorphic to (cid:0) A K ( t ) , ι K ( t ) (cid:1) , and t is uniquely determined up to action of Γ K . The quotient H ( C ) = Γ K \ H is the set of complex points of an algebraicvariety H called a Hilbert surface . Hilbert modular forms.
Let k , k ∈ Z . A Hilbert modular form of weight ( k , k ) is a holomorphic function f : H → C satisfying the transformation rule ∀ γ = (cid:18) a bc d (cid:19) ∈ Γ K , ∀ t ∈ H , f ( γt ) = (cid:0) c t + d (cid:1) k (cid:0) c t + d (cid:1) k f ( t ) . We say that f is symmetric if f ◦ σ = f . If f is nonzero and symmetric, then itsweight ( k , k ) is automatically parallel , meaning k = k . A Hilbert modular function is only required to be meromorphic instead of holomorphic.All irreducible representations of GL ( C ) are 1-dimensional, so there isno need to consider vector-valued forms. The analogue of Proposition 2.7 forHilbert modular forms is the following. Proposition 2.10.
Let f be a Hilbert modular function of weight (0 , . Thenthe partial derivatives ∂f /∂t and ∂f /∂t are Hilbert modular functions ofweight (2 , and (0 , respectively.Proof. Differentiate the relation f ( γt ) = f ( t ) .Let ( A, ι ) be a principally polarized abelian surface over C with real multipli-cation by Z K . As in the Siegel case, we would like to evaluate Hilbert modularforms when a basis of differential forms on A is given; this is possible if we re-strict to bases of Ω ( A ) which behave well with respect to the real multiplicationembedding. Definition 2.11.
Let ω be a basis of Ω ( A ) . We say that ( A, ι, ω ) is Hilbert-normalized if for every α ∈ Z K , the matrix of ι ( α ) ∗ : Ω ( A ) → Ω ( A ) in thebasis ω is Diag( α, α ) .If ( A, ι, ω ) is Hilbert-normalized and f is a Hilbert modular form of weight ( k , k ) , then the quantity f ( A, ι, ω ) is computed as follows. Choose t ∈ H and an isomorphism η : ( A, ι ) → (cid:0) A K ( t ) , ι K ( t ) (cid:1) as in Theorem 2.9, and let r ∈ GL ( C ) be matrix of η ∗ in the bases ω ( t ) , ω . Then r is diagonal, r = Diag( r , r ) ,and f ( A, ι, ω ) = r k r k f ( t ) . .4 The Hilbert embedding Forgetting the real multiplication structure yields a map H ( C ) → A ( C ) fromthe Hilbert surface to the Siegel threefold. In fact, this forgetful map comesfrom a linear map H : H → H called the Hilbert embedding , which we now describe explicitly. Let ( e , e ) bethe Z -basis of Z K given by e = 1 and e = 1 − √ ∆2 if ∆ = 1 mod , e = √ ∆ otherwise.Set R = (cid:18) e e e e (cid:19) , and define H : H → H , t = ( t , t ) R t Diag( t , t ) R. Proposition 2.12.
For every t ∈ H , left multiplication by R t on C inducesan isomorphism A K ( t ) → A (cid:0) H ( t ) (cid:1) .Proof. By definition, Σ (cid:0) Z K (cid:1) = R Z , and since Z ∨ K is the trace dual of Z K , wehave Σ (cid:0) Z ∨ K (cid:1) = R − t Z . Then a direct computation shows that ∀ t ∈ H , Λ (cid:0) H ( t ) (cid:1) = R t Λ K ( t ) . The Hilbert embedding is compatible with the actions of the modular groups.
Proposition 2.13 ([LY11, Prop. 3.1]) .
1. Under H , the action of Γ K on H is transformed into the action of Sp ( Z ) on H by means of the morphism (cid:18) a bc d (cid:19) (cid:18) R t R − (cid:19) (cid:18) a ∗ b ∗ c ∗ d ∗ (cid:19) (cid:18) R − t R (cid:19) where we write x ∗ = Diag( x, x ) for x ∈ K .2. Define M σ = δ − δ − where δ = 1 if ∆ = 1 mod 4 , and δ = 0 otherwise. Then we have ∀ t ∈ H , H (cid:0) σ ( t ) (cid:1) = M σ H ( t ) . Moreover, pulling back a Siegel modular form via the Hilbert embeddinggives a Hilbert modular form. 11 roposition 2.14.
Let k ∈ Z , n ∈ N , and let f : H → C n [ x ] be a Siegelmodular form of weight ρ = det k Sym n . Define the functions g i : H → C for ≤ i ≤ n by ∀ t ∈ H , n X i =0 g i ( t ) x i = ρ ( R ) f (cid:0) H ( t ) (cid:1) . Then each g i is a Hilbert modular form of weight ( k + i, k + n − i ) .Proof. It is straightforward to check the transformation rule using Proposi-tion 2.13. The heart of the computation is that on diagonal matrices
Diag( r , r ) ,the representation det k Sym n splits: the coefficient before x i is multiplied by ( r r ) k r i r n − i . Corollary 2.15. If f is a scalar-valued Siegel modular form of weight det k ,then H ∗ f : t f (cid:0) H ( t ) (cid:1) is a symmetric Hilbert modular form of weight ( k, k ) .Proof. Since det( R ) k is a nonzero constant, by Proposition 2.14, the func-tion H ∗ f is a Hilbert modular form of weight ( k, k ) . Moreover det( M σ ) = 1 ,so H ∗ f is symmetric by Proposition 2.13.The image of the Hilbert embedding in A ( C ) is called a Humbert surface . Itcan be described by an equation in terms of Igusa invariants, which grows quicklyin size with the discriminant ∆ , but can be computed in small cases [Gru10]. Proposition 2.16.
Igusa invariants generate the field of symmetric Hilbertmodular functions of weight (0 , . They define a birational map from A ,K ( C ) to the closed subset of C cut out by the Humbert equation.Proof. The image of H in A ( C ) is not contained in the codimension subsetwhere Igusa invariants are not a local isomorphism to A .To ease notation, we also write j k for the pullback H ∗ j k , for each ≤ k ≤ . Let k be a field, and let A be a principally polarized abelian surface over k .Denote its dual by A ∨ and its principal polarization by π : A → A ∨ . Recallthat for every line bundle L on A , there is a morphism φ L : A → A ∨ defined by φ L ( x ) = T ∗ x L⊗ L − , where T x denotes translation by x on A . Finally, let NS( A ) denote the Néron–Severi group of A , consisting of line bundles up to algebraicequivalence. Theorem 2.17 ([Mil86a, Prop. 14.2]) . For every ξ ∈ End sym ( A ) , there is aunique symmetric line bundle L ξA such that φ L ξA = π ◦ ξ . This associationinduces an isomorphism of groups (cid:0) End sym ( A ) , + (cid:1) ≃ (cid:0) NS( A ) , ⊗ (cid:1) . Under this isomorphism, line bundles giving rise to polarizations correspond tototally positive elements in
End sym ( A ) .
12n this notation, L A is the line bundle associated with the principal polar-ization π . We will consider two different isogeny types that we now define. Definition 2.18.
Let k be a field.1. Let ℓ ∈ N be a prime, and let A, A ′ be principally polarized abelian surfacesover k . An isogeny ϕ : A → A ′ is called an ℓ -isogeny if ϕ ∗ L A ′ = L ℓA .
2. Let K be a real quadratic field, and let β ∈ Z K be a totally positive prime.Let ( A, ι ) and ( A ′ , ι ′ ) be principally polarized abelian surfaces over k withreal multiplication by Z K . An isogeny ϕ : A → A ′ is called a β -isogeny if ϕ ∗ L A ′ = L ι ( β ) A and ∀ α ∈ Z K , ϕ ◦ ι ( α ) = ι ′ ( α ) ◦ ϕ. For a generic principally polarized abelian surface, ℓ -isogenies are the sim-plest kind of isogenies that occur. They have degree ℓ . If we restrict to abeliansurfaces with real multiplication by Z K , then β -isogenies are smaller: their de-gree is only N K/ Q ( β ) [DJR+17, Prop. 2.1].Both ℓ - and β -isogenies are easily described over C . For t = ( t , t ) ∈ H ,write t/β := (cid:0) t /β, t /β (cid:1) . The following well-known statement is a consequence of Theorems 2.2 and 2.9,using the facts that the kernel of an ℓ -isogeny is a maximal isotropic subgroup ofthe ℓ -torsion, and the kernel of a β -isogeny is a cyclic subgroup of the β -torsion. Proposition 2.19.
1. For every τ ∈ H , the identity map on C induces an ℓ -isogeny A ( τ ) → A ( τ /ℓ ) . Let
A, A ′ be principally polarized abelian surfaces over C , and let ϕ : A → A ′ be an ℓ -isogeny. Then there exists τ ∈ H such that there is a commu-tative diagram A A ′ A ( τ ) A ( τ /ℓ ) . ϕ ∼ ∼ z z
2. For every t ∈ H , the identity map on C induces a β -isogeny (cid:0) A K ( t ) , ι K ( t ) (cid:1) → (cid:0) A K ( t/β ) , ι K ( t/β ) (cid:1) . et ( A, ι ) , ( A ′ , ι ′ ) be principally polarized abelian surfaces over C with realmultiplication by Z K , and let ϕ : ( A, ι ) → ( A ′ , ι ′ ) be a β -isogeny. Thenthere exists t ∈ H such that there is a commutative diagram ( A, ι ) ( A ′ , ι ′ ) (cid:0) A K ( t ) , ι K ( t ) (cid:1) (cid:0) A K ( t/β ) , ι K ( t/β ) (cid:1) . ϕ ∼ ∼ z z Modular equations encode the presence of an isogeny between principally po-larized abelian surfaces, as the classical modular polynomial does for ellip-tic curves. To define them, we use the fact that the extension of the field C (cid:0) j ( τ ) , j ( τ ) , j ( τ ) (cid:1) constructed by adjoining j ( τ /ℓ ) , j ( τ /ℓ ) , and j ( τ /ℓ ) isfinite and generated by j ( τ /ℓ ) . A similar statement holds for Igusa invariantsat t/β in the Hilbert case [MR17, Prop. 4.11]. Definition 2.20.
1. Let ℓ ∈ N be a prime. We call the Siegel modular equations of level ℓ thedata of the three polynomials Ψ ℓ, , Ψ ℓ, , Ψ ℓ, ∈ C ( J , J , J )[ J ′ ] defined asfollows:• Ψ ℓ, is the monic minimal polynomial of the function j ( τ /ℓ ) overthe field C (cid:0) j ( τ ) , j ( τ ) , j ( τ ) (cid:1) .• For i ∈ { , } , we have the following equality of meromorphic func-tions: j i ( τ /ℓ ) = Ψ ℓ,i (cid:0) j ( τ ) , j ( τ ) , j ( τ ) , j ( τ /ℓ ) (cid:1) .
2. Let K be a real quadratic field, and let β ∈ Z K be a totally positiveprime. We call the Hilbert modular equations of level β the data of thethree polynomials Ψ β, , Ψ β, , Ψ β, defined as follows:• Ψ β, is the monic minimal polynomial of the function j ( t/β ) overthe field C (cid:0) j ( t ) , j ( t ) , j ( t ) (cid:1) .• For i ∈ { , } , we have the following equality of meromorphic func-tions: j i ( t/β ) = Ψ β,i (cid:0) j ( t ) , j ( t ) , j ( t ) , j ( t/β ) (cid:1) . In the Hilbert case, since Igusa invariants are symmetric by Corollary 2.15,the modular equations encode β - and β -isogenies simulaneously [MR17, Ex. 4.17].It would be better to consider modular equations with non-symmetric invariants;however, we know of no good choice of such invariants in general.As explained in the introduction, modular equations really are equationsfor the image of a map defined at the level of algebraic stacks. As a conse-quence, they have coefficients in Q . Since Igusa invariants have poles on A H , modular equations in genus have denominators [MR17, Rem. 4.20].If we multiply by these denominators, then we may consider modular polyno-mials as elements of C [ J , J , J , J ′ , J ′ , J ′ ] that vanish on the Igusa invariantsof isogenous Jacobians: this is what we do in the sequel.From a practical point of view, modular equations in genus are very large.This is especially true for Siegel modular equations of level ℓ . The degree of Ψ ℓ, in J ′ is ℓ + ℓ + ℓ + 1 , and its degree in J , J , J has the same order ofmagnitude, not mentioning the height of the coefficients. The situation is lessdesperate for Hilbert modular equations of level β : the degree of Ψ β, in J ′ is N K/ Q ( β ) + 2 [MR17, Ex. 4.17]. Modular equations have been computed for ℓ = 2 and in the Siegel case, up to N ( β ) = 41 in the Hilbert case with K = Q ( √ using Gundlach invariants, and even up to N ( β ) = 97 for K = Q ( √ using theta constants as invariants [Mil]. C A nonsingular hyperelliptic equation C : v = E C ( u ) over C naturally en-codes a basis of differential forms ω ( C ) on the principally polarized abeliansurface Jac( C ) (§3.1). If f is a Siegel modular function, this gives rise to a map Cov( f ) : C 7→ f (cid:0) Jac( C ) , ω ( C ) (cid:1) . Then,
Cov( f ) is a covariant of the curve, and has an expression in terms ofthe coefficients. We give an algorithm to obtain this expression from the q -expansion of f (§3.2), and apply it to the derivatives of Igusa invariants (§3.3).The result is the explicit Kodaira–Spencer isomorphism. This allows us tocompute the deformation map and the tangent map of a given ℓ -isogeny over C (§3.4). Finally, we adapt these methods to the Hilbert case (§3.5). Let C be a nonsingular hyperelliptic equation of genus over C : C : v = E C ( u ) , with deg E C ∈ { , } . Then C is naturally endowed with the basis of differentialforms ω ( C ) = (cid:16) u duv , duv (cid:17) . Recall that the Jacobian
Jac( C ) is a principally polarized abelian surfaceover C [Mil86b, Thm. 1.1 and Summary 6.11]. Choose a base point P on C .This gives an embedding η P : C ֒ → Jac( C ) , Q [ Q − P ] . Proposition 3.1 ([Mil86b, Prop. 2.2]) . The map η ∗ P : Ω (cid:0) Jac( C ) (cid:1) → Ω ( C ) s an isomorphism and is independent of P . By Proposition 3.1, we can see ω ( C ) as a basis of differential forms on Jac( C ) .This basis depends on the particular hyperelliptic equation chosen. Lemma 3.2.
Let C be a genus hyperelliptic equation over C , and let r = (cid:18) a bc d (cid:19) ∈ GL ( C ) . Let E C ′ be the image of E C by det − Sym ( r ) , and let C ′ be the curve withequation y ′ = E C ′ ( x ′ ) . Let η : C ′ → C be the isomorphism defined by η ( x ′ , y ′ ) = (cid:18) ax ′ + cbx ′ + d , (det r ) y ′ ( bx ′ + d ) (cid:19) . Then the matrix of η ∗ : Ω ( C ) → Ω ( C ′ ) in the bases ω ( C ) , ω ( C ′ ) is r .Proof. Write ( x, y ) = η ( x ′ , y ′ ) . A simple calculation shows that dxy = ( bx ′ + d ) dx ′ y ′ and x dxy = ( ax ′ + c ) dx ′ y ′ , so the result follows. Corollary 3.3.
Let A be a principally polarized abelian surface over C that isnot a product of two elliptic curves, and let ω be a basis of Ω ( A ) . Then thereexists a unique hyperelliptic curve equation C of genus over C such that (cid:0) Jac( C ) , ω ( C ) (cid:1) ≃ ( A, ω ) . Proof.
By Torelli’s theorem, there is a curve equation C over C such that A is isomorphic to Jac( C ) . Then ω differs from ω ( C ) by a linear transformationin GL ( C ) . By Lemma 3.2, we can make a suitable change of variables to findthe correct C . It is unique because every isomorphism between hyperellipticcurves comes from such a matrix r . Definition 3.4.
The bases of differential forms chosen in §2 allows us to defineparticular curve equations attached to a point of H or H .1. Let τ ∈ H , and assume that χ ( τ ) = 0 . Then, by Corollary 3.3, thereexists a unique hyperelliptic equation C ( τ ) over C such that (cid:16) Jac (cid:0) C ( τ ) (cid:1) , ω (cid:0) C ( τ ) (cid:1)(cid:17) ≃ (cid:0) A ( τ ) , ω ( τ ) (cid:1) . We call C ( τ ) the standard curve attached to τ . We define the meromorphicfunctions a i ( τ ) for ≤ i ≤ to be the coefficients of C ( τ ) : C ( τ ) : y = X i =0 a i ( τ ) x i .
16. Let t ∈ H , and assume that χ ( H ( t )) = 0 , where H is the Hilbertembedding. Then, by Corollary 3.3, there exists a unique hyperellipticequation C K ( t ) over C such that (cid:16) Jac (cid:0) C K ( t ) (cid:1) , ω (cid:0) C K ( t ) (cid:1)(cid:17) ≃ (cid:0) A K ( t ) , ω K ( t ) (cid:1) . We call C K ( t ) the standard curve attached to t . Proposition 3.5.
The function τ
7→ C ( τ ) is a Siegel modular function ofweight det − Sym which has no poles on the open set { χ = 0 } .Proof. Over C , the Torelli map is biholomorphic, so this function is meromor-phic. By Corollary 3.3, it is defined everywhere on { χ = 0 } . CombiningProposition 2.1 with Lemma 3.2 shows the transformation rule.Finally, for t ∈ H , we can relate the standard curves C K ( t ) and C (cid:0) H ( t ) (cid:1) . Proposition 3.6.
For every t ∈ H , we have C K ( t ) = det − Sym ( R ) C (cid:0) H ( t ) (cid:1) . Proof.
Use Proposition 2.12 and Lemma 3.2. If f is a Siegel modular form, then we have a map Cov( f ) : C 7→ f (cid:0) Jac( C ) , ω ( C ) (cid:1) . We show that
Cov( f ) is a covariant of the curve equation. A recent referencefor covariants is Mestre’s article [Mes91]. Definition 3.7.
Denote by C [ x ] the space of polynomials of degree at most .Let ρ : GL ( C ) → GL( V ) be a finite-dimensional holomorphic representationof GL ( C ) . A covariant , or polynomial covariant , of weight ρ is a map C : C [ x ] → V which is polynomial in the coefficients, and such that the following transforma-tion rule holds: for every r ∈ GL ( C ) and W ∈ C [ x ] , C (cid:0) det − Sym ( r ) W (cid:1) = ρ ( r ) C ( W ) . If dim V ≥ , then C is said to be vector-valued , and otherwise scalar-valued . A fractional covariant is a map satisfying the same transformation rule which isonly required to have a fractional expression in terms of the coefficients.It is enough to consider covariants of weight det k Sym n for k ∈ Z , n ∈ N .What we call a vector-valued covariant of weight det k Sym n is in Mestre’s papera covariant of order n and degree k + n/ ; what we call a scalar-valued covariantof weight det k is in Mestre’s paper an invariant of degree k . The reason for thischange of terminology is the following.17 roposition 3.8. If f be a Siegel modular function of weight ρ , then Cov( f ) is a fractional covariant of weight ρ . Conversely, if F is a fractional covariantof weight ρ , then the meromorphic function τ F (cid:0) C ( τ ) (cid:1) is a Siegel modularfunction of weight ρ . These operations are inverse of each other.Proof. If f is a Siegel modular function, then Cov( f ) is well defined on a Zariskiopen set of C [ x ] and is algebraic, so must have a fractional expression in terms ofthe coefficients. We let the reader check the transformation rules (use Lemma 3.2and Proposition 3.5).Proposition 3.8 gives a bijection between Siegel modular functions and frac-tional covariants, but we need more. The following theorem establishes a re-lation between Siegel modular forms and polynomial covariants, and was firstproved in [CFv17, §4]. Theorem 3.9.
Let f be a holomorphic Siegel modular form. Then Cov( f ) is apolynomial covariant. Moreover, if f is a cusp form, then Cov( f /χ ) is also apolynomial covariant.Proof. The main difficulty is that nonsingular hyperelliptic equations only forma codimension subset of all degree polynomials: if f is a Siegel modular form,then the proof of Proposition 3.8 only shows that Cov( f ) is a polynomial dividedby some power of the discriminant. However, one can show that f extends tothe so-called toroidal compactification of A ( C ) , and this shows that Cov( f ) iswell defined on all curve equations with at most one node. Since this set hascodimension , the result follows.Unlike for Siegel modular forms, the graded C -algebra generated by polyno-mial covariants is finitely generated. Theorem 3.10 ([Cle72, p. 296]) . The graded C -algebra of covariants is gen-erated by elements defined over Q . The number of generators of weight det k Sym n is indicated in the following table: n \ k -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 150 1 1 1 1 12 1 1 1 1 1 14 1 1 1 1 16 1 1 1 28 1 1 110 112 1We only need to manipulate a small subset of these generators. Take ourscalar generators of even weight to be the Igusa–Clebsch invariants I , I , I , I ,in Mestre’s notation A ′ , B ′ , C ′ , D ′ [Mes91], and set I ′ := ( I I − I ) / . det by R , and denote by y , y , y the gen-erators of weights det Sym , det Sym , and det Sym respectively. Finally,the generator of weight det − Sym , denoted by X , is the degree 6 polynomialitself. Note that when computing these covariants as described in [Mes91, §1],the integers m and n on page 315 should be the orders of f and g , and nottheir degrees. To help the reader check their computations, we mention thatthe coefficient of a a in R is − − − . q -expansions to covariants We now explain how to compute the polynomial covariant associated with aSiegel modular form whose q -expansion is known up to a certain precision. Theworks of Igusa already provide the answer in the case of scalar covariants. Theorem 3.11.
We have ψ ) = I , ψ ) = I ′ , Cov( χ ) = I , Cov( χ ) = I I , − − Cov( χ ) = I R. Proof.
By [Igu62, p. 848], there exists a constant λ ∈ C × such that these re-lations hold up to a factor λ k , for k ∈ { , , , , } respectively. Notethat Igusa’s covariant E is − R . Then, Thomae’s formulæ ([Mum84,Thm. IIIa.8.1] and [Tho70, pp. 216–217]), which relate theta constants with thevalues of path integrals on the associated hyperelliptic curve, imply that λ =1 . Therefore, the Igusa invariants satisfy Cov( j ) = I I ′ I , Cov( j ) = I I I , Cov( j ) = I I . Let us compute the q -expansion of the standard curve C ( τ ) . Recall the Siegelmodular form f , of weight det Sym introduced in Example 2.8. Proposition 3.12.
We have
Cov( f , /χ ) = X . In other words, for every τ ∈ H such that χ ( τ ) = 0 , we have C ( τ ) = f , ( τ ) χ ( τ ) . Proof.
Since f , is a cusp form, by Theorem 3.9, Cov( f , /χ ) is a nonzeropolynomial covariant of weight det − Sym . By Theorem 3.10, this space ofcovariants is of dimension 1 and generated by X , so the relation holds up to afactor λ ∈ C × . This yields q -expansions for the coefficients a i ( τ ) of C ( τ ) up toa factor λ . Then, the relations from Theorem 3.11 imply λ = λ = λ = 1 ,hence λ = 1 . 19iven a Siegel modular form f of weight ρ whose q -expansion can be com-puted, the following algorithm recovers the expression of Cov( f ) as a polyno-mial. Algorithm 3.13.
1. Compute a basis B of the vector space of polynomialcovariants of weight ρ using Theorem 3.10.2. Choose a precision ν and compute the q -expansion of f modulo ( q ν , q ν ) .3. For every B ∈ B , compute the q -expansion of the Siegel modular function τ B (cid:0) C ( τ ) (cid:1) using Proposition 3.12.4. Do linear algebra; if the matrix does not have full rank, go back to step 2with a larger ν .Sturm-type bounds [BP17] provide a theoretical limit for the precision ν that we need to consider; for the examples given in this article, ν = 3 is enough.We now apply Algorithm 3.13 to derivatives of Igusa invariants. Recall fromProposition 2.7 that for ≤ k ≤ , the partial derivative dj k dτ := ∂j k ∂τ x + ∂j k ∂τ x + ∂j k ∂τ is a Siegel modular function of weight Sym . Theorem 3.14.
We have πi Cov (cid:16) dj dτ (cid:17) = 1 I (cid:16) I I y − I I y + 1352 I y + 465754 I I y − I y + 1366875 I y (cid:17) , πi Cov (cid:16) dj dτ (cid:17) = 1 I (cid:16) I I y + 900 I y + 40500 I I y (cid:17) , πi Cov (cid:16) dj dτ (cid:17) = 1 I (cid:16) I I y + 101250 I y (cid:17) . Proof.
Let ≤ k ≤ . The function χ j k has no poles on A ( C ) . Therefore,the Siegel modular function f k = χ dj k dτ is holomorphic on A ( C ) . Its q -expansion can be computed from the q -expansionof j k by formal differentiation. Since πi ∂∂τ i = q i ∂∂q i for ≤ i ≤ , we check that f k is a cusp form. Therefore, by Theorem 3.9, Cov( f k /χ ) is a polynomial covariant of weight det Sym . Looking at thetable in Theorem 3.10, we find that a basis of this space of covariants is givenby covariants of the form Iy where y ∈ { y , y , y } and I is a scalar-valuedcovariant of the appropriate even weight. Algorithm 3.13 succeeds with ν = 3 ;the computations were done using Pari/GP [The19].20 emark 3.15. Theorems 3.11 and 3.14 can be checked numerically. Comput-ing big period matrices of hyperelliptic curves [MN19] provides pairs (cid:0) τ, C ( τ ) (cid:1) with τ ∈ H . We can evaluate Igusa invariants at a given τ to high precisionusing their expression in terms of theta functions [Dup11]. Therefore we canalso evaluate their derivatives numerically with high precision and compute theassociated covariant using floating-point linear algebra. The computations weredone using the libraries hcperiods [Mol18] and cmh [ET14]; they provide a niceconsistency check to Theorem 3.14. Another consistency check is that we canrecover the relations from Theorem 3.11. Remark 3.16.
From Theorem 3.14, we can easily obtain similar formulæ forderivatives of other invariants, or even invariants for abelian surfaces with extrastructure such as theta constants. For instance, taking the invariants h , h , h defined in Remark 2.6, we obtain πi Cov (cid:16) dh dτ (cid:17) = 1 I (cid:16) − y I I + − y I I + 17018 y I I I + 1352 y I I + 1366875 y I I + 3462754 y I I I − y I I + − y I I − y I I − y I (cid:17) , πi Cov (cid:16) dh dτ (cid:17) = 1 I (cid:16) − y I I − y I I + 900 y I I (cid:17) , πi Cov (cid:16) dh dτ (cid:17) = 1 I (cid:16) − y I I I − y I I I + 270 y I I I + 1352 y I I + 1366875 y I I + 121500 y I I (cid:17) . Let C , C ′ be equations of genus hyperelliptic curves over C , let A, A ′ be theirJacobians, and let ϕ : A → A ′ be an ℓ -isogeny. The choice of curve equationsencodes a choice of bases of Ω ( A ) and Ω ( A ′ ) , or equivalently, by taking dualbases, a choice of bases of the tangent spaces T ( A ) and T ( A ′ ) . By an abuseof notation, we identify the tangent map dϕ : T ( A ) → T ( A ′ ) with its matrixwritten in these bases. Let us show how to compute dϕ from the data of thecurve equations and modular equations of level ℓ . Definition 3.17.
It is convenient to introduce matrix notations.• For τ ∈ H , we define D τ J ( τ ) = (cid:18) πi ∂j k ∂τ l ( τ ) (cid:19) ≤ k,l ≤ · . In other words, if we set v = (cid:18) (cid:19) , v = (cid:18) (cid:19) , v = (cid:18) (cid:19) , l -th column of D τ J ( τ ) contains (up to πi ) the derivatives ofIgusa invariants at τ in the direction v l . We can check that for r ∈ GL ( C ) , the l -th column of D τ J ( τ ) Sym ( r ) contains the derivatives ofIgusa invariants at τ in the direction r v l r t .Let ( A, ω ) be a principally polarized abelian surface over C with a basis ofdifferential forms, let η : A → A ( τ ) be an isomorphism for some τ ∈ H ,and let r be the matrix of η ∗ in the bases ω ( τ ) , ω . Then the fact thatderivatives of Igusa invariants have weight Sym translates as D τ J ( A, ω ) = D τ J ( τ ) Sym ( r t ) . We denote by
C 7→ D τ J ( C ) the associated fractional covariant; Theorem 3.14 expresses the entries ofthis matrix up to a constant in terms of the coefficients of C .• Consider the Siegel modular equations Ψ ℓ, , Ψ ℓ, , Ψ ℓ, of level ℓ as elementsof the ring Q [ J , J , J , J ′ , J ′ , J ′ ] . We define D Ψ ℓ,L = (cid:18) ∂ Ψ ℓ,n ∂J k (cid:19) ≤ n,k ≤ and D Ψ ℓ,R = (cid:18) ∂ Ψ ℓ,n ∂J ′ k (cid:19) ≤ n,k ≤ . Definition 3.18.
Let ϕ be an ℓ -isogeny as above, write j as a shorthand for theIgusa invariants ( j , j , j ) of A , and j ′ for the invariants ( j ′ , j ′ , j ′ ) of A ′ . Wesay that the isogeny ϕ is generic if the × matrices D Ψ ℓ,L ( j, j ′ ) , D Ψ ℓ,R ( j, j ′ ) , D τ J ( C ) and D τ J ( C ′ ) are invertible. In this case, we define the deformationmatrix D ( ϕ ) of ϕ as D ( ϕ ) = − D τ J ( C ′ ) − · D Ψ ℓ,R ( j, j ′ ) − · D Ψ ℓ,L ( j, j ′ ) · D τ J ( C ) . In Section 4, we will interpret D ( ϕ ) as the matrix of the deformation map inthe bases of T A ( A ) and T A ′ ( A ) associated with ω ( C ) , ω ( C ′ ) via the Kodaira–Spencer isomorphism. Let us relate the deformation matrix D ( ϕ ) with thetangent matrix dϕ . Proposition 3.19.
Assume that ϕ is generic. Then we have Sym ( dϕ ) = ℓ D ( ϕ ) . Proof.
By Proposition 2.19, we can find τ ∈ H and isomorphisms η, η ′ suchthat there is a commutative diagram A A ′ A ( τ ) A ( τ /ℓ ) . ϕη η ′ z z r be the matrix of η ∗ in the bases ω ( τ ) , ω ( C ) , and define r ′ similarly. Thenwe have dϕ = r ′ t r − t . By the definition of modular equations, we have Ψ ℓ,n (cid:0) j ( τ ) , j ( τ ) , j ( τ ) , j ( τ /ℓ ) , j ( τ /ℓ ) , j ( τ /ℓ ) (cid:1) = 0 for ≤ n ≤ . We differentiate with respect to τ , τ , τ and obtain D Ψ ℓ,L ( j, j ′ ) · D τ J ( τ ) + 1 ℓ D Ψ ℓ,R ( j, j ′ ) · D τ J ( τ /ℓ ) = 0 . We rewrite this relation as − ℓ D Ψ ℓ,L ( j, j ′ ) · D τ J ( C ) · Sym ( r t ) = D Ψ ℓ,R ( j, j ′ ) · D τ J ( C ′ ) · Sym ( r ′ t ) , and the result follows.Once we compute D ( ϕ ) , the matrix dϕ itself is easily computed up to sign. We now explain how to recover the tangent matrix in the Hilbert case, in thesame spirit as the Siegel case. An important difference is that we have to restrictto Hilbert-normalized bases of differential forms (recall Definition 2.11), so notall curve equations will do. For the moment, assume that we have a β -isogeny ϕ : ( A, ι ) → ( A ′ , ι ′ ) between abelian surfaces with real multiplication by Z K , andwe are given curve equations C , C ′ such that the associated bases ω ( C ) and ω ( C ′ ) are Hilbert-normalized. We address the question of constructing C , C ′ in §3.6. Definition 3.20. • For t ∈ H , we define D t J ( t ) = (cid:18) πi ∂j k ∂t l ( t ) (cid:19) ≤ k ≤ , ≤ l ≤ . If C is a curve equation such that ω ( C ) is Hilbert-normalized, we denoteby D t J ( C ) the value of this modular form on C .• We define the × matrices D Ψ β,L and D Ψ β,R in the case of Hilbertmodular equations of level β as in Definition 3.17.• Write j as a shorthand for the Igusa invariants ( j , j , j ) of A , and j ′ forthe invariants ( j ′ , j ′ , j ′ ) of A ′ . We say that the isogeny ϕ is generic ifthe denominators of modular equations do not vanish at j and the × matrices D Ψ β,L ( j, j ′ ) · D t J ( C ) and D Ψ β,R ( j, j ′ ) · D t J ( C ′ ) have rank . 23 emma 3.21. Let ( A, ι, ω ) be Hilbert-normalized, and let t ∈ H such that thereis an isomorphism η : ( A, ι ) → ( A K ( t ) , ι K ( t )) . Let r be the matrix of η ∗ in thebases ω K ( t ) , ω . Then we have D t J ( A, ω ) = D t J ( t ) · r . Proof.
By Proposition 2.10, derivatives of Igusa with respect to t and t areHilbert modular functions of weight (2 , and (0 , respectively. Proposition 3.22.
Let ( A, ι, ω ) be Hilbert-normalized. Then we have D t J ( A, ω ) = D τ J ( A, ω ) · T where T = . Proof.
Let t, η, r as in Lemma 3.21, and write τ = H ( t ) . By the expression ofthe Hilbert embedding, D t J ( t ) contains the derivatives of Igusa invariants at τ in the directions πi R t (cid:18) (cid:19) R and πi R t (cid:18) (cid:19) R. Hence we have D t J ( t ) = D τ J ( τ ) · Sym ( R t ) · T. By Proposition 2.12, we have an isomorphism ζ : A K ( t ) → A ( τ ) such that thematrix of ζ ∗ in the bases ω ( τ ) , ω K ( t ) is R . Therefore D t J ( A, ω ) = D t J ( t ) r , D τ J ( A, ω ) = D τ J ( τ ) Sym (( rR ) t ) . The result follows.It is natural that the matrix R defining the Hilbert embedding does notappear in Proposition 3.22: evaluating derivatives of Igusa invariants on ( A, ω ) has an intrinsic meaning in terms of the Kodaira–Spencer isomorphism, and thechoice of Hilbert embedding does not matter. Proposition 3.23.
Let ϕ : A → A ′ be a β -isogeny and let C , C ′ be Hilbert-normalized curve equations as above. Then the tangent matrix dϕ is diagonal,and we have D Ψ β,L ( j, j ′ ) · D t J ( C ) = − D Ψ β,R ( j, j ′ ) · D t J ( C ′ ) · Diag(1 /β, /β ) · ( dϕ ) . Proof.
By Proposition 2.19, we can find t ∈ H and isomorphisms η, η ′ suchthat there is a commutative diagram (cid:0) A, ι (cid:1) (cid:0) A ′ , ι ′ (cid:1)(cid:0) A K ( t ) , ι K ( t ) (cid:1) (cid:0) A K ( t/β ) , ι K ( t/β ) (cid:1) . ϕη η ′ z z r be the matrix of η ∗ in the bases ω K ( t ) , ω , and define r ′ similarly; they arediagonal. We have dϕ = r ′ t r − t = r ′ r − . Differentiating the modular equations,we obtain D Ψ β,L ( j, j ′ ) · D t J ( t ) + D Ψ β,R ( j, j ′ ) · D t J ( t/β ) · Diag(1 /β, /β ) = 0 . By Lemma 3.21, we have D t J ( t ) = D t J ( C ) · r , D t J ( t/β ) = D t J ( C ′ ) · r ′ and the result follows.This relation allows us to compute ( dϕ ) from derivatives of modular equa-tions when ϕ is generic. In contrast with the Siegel case, the knowledge of ( dϕ ) does not allow us to recover the diagonal matrix dϕ up to sign, as we have toperform two uncorrelated root extractions: we obtain two possible candidates. Let ( A, ι ) is an abelian surface over C with real multiplication by Z K . Given theIgusa invariants ( j , j , j ) of A , we would like to construct a curve equation C such that ( A, ι, ω ( C )) is Hilbert-normalized. Our method is to compute a firstcurve equation using Mestre’s algorithm [Mes91], and then look for a suitablehomographic change of variables. However, we are missing some information, asthe two pairs ( A, ι ) and ( A, ι ) , where ι denotes the real conjugate of ι , have thesame Igusa invariants. The best we can hope for is to compute an equation C such that either ( A, ι, ω ( C )) or ( A, ι, ω ( C )) is Hilbert-normalized. In this case, wesay that C is potentially Hilbert-normalized. This uncertainty is a consequenceof our using symmetric invariants on the Hilbert surface.
Proposition 3.24.
Let C be a hyperelliptic curve equation of genus over C such that Jac( C ) has real multiplication by Z K . Denote its Igusa invariants by ( j , j , j ) . Then the curve C is potentially Hilbert-normalized if and only if thetwo columns of the × matrix D τ J ( C ) · T where T = define tangent vectors to the Humbert surface at ( j , j , j ) .Proof. Let t ∈ H such that there is an isomorphism η : Jac( C ) → A K ( t ) , andwrite τ = H ( t ) . Let r ∈ GL ( C ) be the matrix of η ∗ in the bases ω K ( t ) , ω . Thenthe columns of D τ J ( C ) · T contain, up to πi , the derivatives of Igusa invariantsat τ in the directions R t r (cid:18) (cid:19) r t R and R t r (cid:18) (cid:19) r t R. These directions are tangent to the Humbert surface if and only if r is is eitherdiagonal or anti-diagonal. 25ssume that the equation of the Humbert surface for K in terms of Igusainvariants is given: this precomputation depends only on K . Given Igusa in-variants ( j , j , j ) on the Humbert surface, the algorithm to reconstruct a po-tentially Hilbert-normalized curve equation runs as follows. Algorithm 3.25.
1. Construct any curve equation C with Igusa invariants ( j , j , j ) using Mestre’s algorithm.2. Find r ∈ GL ( C ) such that the two columns of the matrix D τ J ( C ) · Sym ( r t ) · T are tangent to the Humbert surface at ( j , j , j ) .3. Output det − Sym ( r ) C .In step 2, if a, b, c, d denote the entries of r , we only have to solve a quadraticequation in a, c , and a quadratic equation in b, d . Therefore Algorithm 3.25 costs O K (1) square roots and field operations.In practice, when computing a β -isogeny ϕ : A → A ′ in the Hilbert case,we are only given the Igusa invariants of A and A ′ . Constructing potentiallyHilbert-normalized curves is then equivalent to making a choice of real multipli-cation embedding for each abelian surface. If these embeddings are incompatiblevia ϕ , we obtain antidiagonal matrices when computing the tangent matrix; inthis case, we apply the change of variables x /x on one of the curve equa-tions to make them compatible. Even if they are compatible, ϕ will be either a β - or a β -isogeny depending on the choices of real multiplication embeddings.Therefore we really obtain four candidates for the tangent matrix, among whichonly one is correct. In this section, we use the language of algebraic stacks to show how to computethe deformation map of a given isogeny ϕ , and to show its relation with thetangent map dϕ , for abelian schemes of any dimension over any base.We start by recalling well-known and general facts about separated Deligne–Mumford stacks and their coarse moduli spaces (§4.1). Then we recall theproperties of several moduli stacks for principally polarized abelian schemesof dimension g , namely A g (abelian schemes with no extra structure), A g,n (abelian schemes with a level n structure), A g ( ℓ ) (abelian schemes endowedwith the kernel of an ℓ -isogeny), and their coarse moduli schemes A g , A g,n , A g ( ℓ ) (§4.2). In particular, we have a map at the level of algebraic stacks, Φ ℓ = ( Φ ℓ, , Φ ℓ, ) : A g ( ℓ ) → A g × A g sending ( A, K ) to ( A, A/K ) such that both Φ ℓ, and Φ ℓ, are étale. Therefore,for an ℓ -isogeny ϕ seen as a point of A g ( ℓ ) , the deformation map D ( ϕ ) = d Φ ℓ, ( ϕ ) ◦ d Φ ℓ, ( ϕ ) −
26s well-defined at the level of stacks. However, the induced maps at the level ofcoarse spaces, ( Φ ℓ, , Φ ℓ, ) : A g ( ℓ ) → A g × A g are not étale everywhere, so that we can only recover the deformation map onan open set of the coarse spaces (see Corollary 4.10). In the genus case, whenwe work with the modular polynomials Ψ ℓ,i from Section 2.6, this phenomenonworsens; still, we can give precise conditions on the isogeny that ensure genericityin the sense of Definition 3.18 (see Proposition 4.13). We also extend theseresults to the Hilbert case.After that, we give the general relation between the tangent map and thedeformation map of a given ℓ - or β -isogeny (§4.3). Finally we show that in di-mension , the relations between modular forms and covariants given in Propo-sition 3.12 hold over Z and not only over C (§4.4). This allows us to give anexplicit version of the Kodaira–Spencer isomorphism over any base (§4.5), thatwe could use for instance to construct explicit families of abelian varieties withreal multiplication.In summary, this section explains the relationship between the fine modulispace A g ( ℓ ) and its coarse moduli space A g ( ℓ ) , and the geometric meaning ofthe genericity conditions of Theorem 1.1; moreover it gives a purely algebraic,rather than analytic, interpretation of the results of Section 3. Another wayto extend the results of Section 3 over any base would be to lift the isogeny tocharacteristic zero (in the case of fields), then interpolate between fibers usingrigidity; however, we find that the moduli-theoretic approach is superior as itprovides more geometric insight. In this paper, we always assume stacks to be of finite type over a Noetherianbase scheme. Let X be a separated Deligne–Mumford stack over S ; we recallthat an Artin stack is Deligne–Mumford if and only if its diagonal is unramified[The18, Tag 06N3]. Here we summarize well-known results on the geometryof X and its coarse moduli space.By a point x of X , we mean a point of the underlying topological space | X | ,and we implicitly take a representative Spec k → X of x . For any scheme T ,a T -point of X is a morphism T → X . We denote by I X the inertia stackof X , and if x is a point of X , we denote by I x the pullback of I X to x ;this pullback is simply the space Aut( x ) of automorphisms, or stabilizers, of x .Since we assume X separated, I x is in fact finite. The stabilizer I x does notdepend on the representative chosen since I x is the pullback of the residual gerbe G x → k ( ξ ) at x through Spec k → Spec k ( ξ ) : see [LM00, Ch. 11], [The18, Tag06ML]. We identify open substacks of X with the underlying open topologicalspaces of | X | [The18, Tag 06FJ].We recall that a map f : X → Y is representable if and only if the inducedmap I X → X × Y I Y is a monomorphism [The18, Tag 04YY]. Also, if f is unramified, then its diagonal is étale by [The18, Tag 0CIS] and [Ryd11];27ence the map I X → X × Y I Y is étale. Therefore, if f is representable andunramified, then the map I X → X × Y I Y is an open immersion.A coarse moduli space X of X is an algebraic space X endowed with a map π : X → X such that π is categorical and induces a bijection π : X ( k ) → X ( k ) for every algebraically closed field k . We also use the following terminology from[MFK94] (see also [KM97, Def. 1.8] and [Ryd13, Defs. 2.2 and 6.1]): a map q : X → Z is topological if q is a universal homeomorphism, and geometric if itis topological and furthermore O Z → q ∗ O X is an isomorphism. A GC quotient is a geometric quotient that is also (uniformly) categorical; in particular, itsimage is a coarse moduli space ([KM97, Def. 1.8] and [Ryd13, Def. 3.17 andRem. 3.18]).
Theorem 4.1.
Let X → S be a separated Deligne–Mumford stack.(i) (Keel–Mori). There exists a coarse moduli space π : X → X , where X isof finite type over S . The map π is a GC quotient, is proper, quasi-finiteand separated; moreover the construction is stable under flat base change.(ii) Let x ∈ X ( k ) be a point, and let I x be the stabilizer of any point in X above x . Then étale-locally around x , X is a quotient stack by I x and X is a geometric quotient by I x . More precisely, there is an affine scheme U ,an étale morphism U → X whose image contains x , and a finite morphism V → U with an action of I x on V such that X U := X × X U = [ V /I x ] isan I x -quotient stack, and U = V /I x .Proof. Theorem 4.1.(i) is valid for Artin stacks with finite inertia; the originalproof is in [KM97], and reformulations of the proof using the language of stacksrather than groupoids are given in [Con05], [Ryd13] and [The18, Tag 0DUK].Since X is a separated Deligne–Mumford stack, its inertia I X is finite, so theKeel–Mori theorem applies.For Theorem 4.1.(ii), see [AV02, Lem. 2.2.3] which shows that X is locallya quotient, and [Ols06, Thm. 2.12] which shows that we can take the quotientto be a quotient by I x . If V = Spec R , then V /I x is the affine scheme Spec R I x .The fact that U = (Spec R ) /I x then follows easily from the theory of quotientsof affine schemes: see for instance [Ryd13, §4] or [DR73, §I.8.2.2]. See also[The18, Tag 0DU0] for extensions of this result in the case of quasi-DM stacks,and [AHR19; AHR20] for a far reaching generalization.By Zarsiki’s main theorem, the coarse moduli space X is characterized bythe fact that π : X → X is proper and quasi-finite, and O X ≃ π ⋆ O X on theétale site [Con05, §1].The formation of coarse moduli spaces is not stable under base change ingeneral. This causes problems when reducing coarse moduli spaces, defined forinstance over Z , modulo a prime p , as the morphism Spec F p → Spec Z is notflat. Coarse moduli spaces have better properties in the case of tame stacks.The stack X is said to be tame [AOV08] if the map π : X → X is cohomo-logically affine; in particular it is a good moduli space in the sense of [Alp13]. Afinite fppf group scheme G/S is linearly reductive if BG → S is tame ([MFK94],28AOV08, Def. 2.4], [Alp13, Def. 12.1]). In [AOV08], it is shown that G/S islinearly reductive if and only if its geometric fibers are geometrically reductive,if and only if its geometric fibers are locally (in the fppf topology) a split exten-sion of a constant tame group by a group of multiplicative type. If x ∈ X ( k ) isa geometric point of X , we say that x is a tame point of X if x has a linearlyreductive stabilizer. Theorem 4.2.
Let X → S be a separated Deligne–Mumford stack, and let π : X → X be its coarse moduli space.(i) If every geometric point of X is tame, then X is tame. If X is tame, thenthe formation of its coarse space commutes with arbitrary base change.(ii) More generally, if x ∈ X ( k ) is tame, then there is an open tame sub-stack U of X containing x . Furthermore, the image of U in X is Cohen–Macaulay.(iii) The map π : X → X is always an adequate moduli space in the senseof [Alp14]. In particular, if T → S is a morphism of algebraic spaces, X T denotes the base change of X to T and X T denotes the coarse mod-uli space of X T , then the natural map X T → X × S T is an universalhomeomorphism.Proof. Theorems 4.2.(i) and 4.2.(ii) are proved in the case of Artin stacks withfinite inertia in [AOV08]. The openness of tame points is the main result of thispaper [AOV08, Thm. 3.2, Prop. 3.6]. Since we restrict to separated Deligne–Mumford stacks, it also follows from Theorem 4.1.(ii). Formation of the coarsemoduli space commutes with pullbacks in the tame case by [AOV08, Cor. 3.3].If x is a tame point of X , then by the local structure theorem, étale-locallyaround x , there is an open substack of the form U = [ V /I x ] , and I x is linearlyreductive. By the Hochster–Roberts theorem [MFK94, Appendix 1.E], the affinescheme V /I x is Cohen–Macaulay. Being Cohen–Macaulay is a local notion forthe étale topology, so the image of U in X is also Cohen–Macaulay.Finally, Theorem 4.2.(iii) is proved in [Alp14], which shows that the coarsemoduli space of an Artin stack with finite inertia is always an adequate modulispace. The natural map X T → X × S T is then an adequate homeomorphismin the sense of [Alp14], and in particular is a universal homeomorphism [Alp14,Main Theorem]. Corollary 4.3.
Let X be a separated Deligne–Mumford stack.(i) The set U of points x such that I x is trivial is an open substack of X (which may be empty), and π : U → π ( U ) is an isomorphism.(ii) Let x ∈ X ( k ) be a point, and let b O X ,x be the strict Hensel ring of X at x . Then b O X,x = b O I x X ,x . (1) In particular, if X is a normal, then its coarse moduli space is normal.Proof. These two statements are immediate consequences of Theorem 4.1.(ii).For Corollary 4.3.(ii), see also [DR73, §I.8.2.1] which states that the kernel of29he action of I x acting on b O X ,x is exactly the set of automorphisms of x thatcan be extended to Spec b O X ,x → X .Finally, we know when an étale map between algebraic stacks induces anétale map on their coarse moduli spaces. Theorem 4.4 (Luna’s fundamental lemma) . Let f : X → Y be a representableand unramified morphism of separated Deligne–Mumford stacks. Then the setof points where f is stabilizer preserving , meaning that the monomorphism oninertia I x → I f ( x ) induced by f is an isomorphism, is an open substack U of X .The morphism I U → I Y × Y U induced by f is an isomorphism.If f is étale and U = X , that is if f is stabilizer preserving at every point,then the induced map on coarse spaces f : X → Y is étale, and even stronglyétale; in other words X = X × Y Y .Proof. The fact that U is open is [The18, Tag 0DUA], [Ryd13, Prop. 3.5].Since X and Y are separated Deligne–Mumford stacks, the induced map isétale by Corollary 4.3.(ii).The general case of Artin stacks with finite inertia is treated in [Ryd13,Prop. 6.5 and Thm. 6.10]. In this reference, stabilizer preserving is called fixedpoint reflecting , but we prefer to use the terminology of the Stacks project[The18, Tag 0DU6]. The fact that f is strongly étale comes from the cartesiandiagram in [Ryd13, Thm. 6.10]. See also [AHR19, Thm. 3.14] where this isproved in a more general setting. Remark 4.5. If f : X → Y is proper (resp. finite), then the induced map f : X → Y is proper (resp. finite), because the maps from X and Y to theircoarse moduli spaces are proper quasi-finite [The18, Tag 02LS], [Gro64, EGAIV.8.11.1]). Remark 4.6. If x is a tame smooth k -point of X , then by Luna’s étale slicetheorem ([Lun73], [AHR20, Thm 1.1 and Thm 2.1], [AHR19, Thm 19.4]), theétale local structure of Theorem 4.1.(ii) takes a particularly nice form. Indeed,taking an étale local presentation X U = [ V /I x ] as in Theorem 4.1.(ii), then(possibly after an étale extension of k and after shrinking V ) there is a stronglyétale morphism [ V /I x ] → [ T x X /I x ] which sends x to , where I x acts via itsnatural linear action on T x X . In particular, étale locally around x the map π : X → X is given by [ T x X /I x ] → T x X /I x . In this section, we apply the general results gathered in §4.1 to the case ofmoduli spaces of abelian schemes. This allows us to investigate the propertiesof the map Φ ℓ on coarse moduli spaces in the Siegel case, and its analogue Φ β in the Hilbert case. 30 .2.1 Siegel stacks Recall that we denote by A g the moduli stack of principally polarized abelianvarieties, and by A g,n the moduli stack of principally polarized abelian varietieswith a level n symplectic structure; here we mean a level ( Z /n Z ) g structureas in [FC90] rather than a ( Z /n Z ) g × µ gn structure as in [Mum71; de 93], sothat A g,n is defined over Z [1 /n ] rather than over Z . Both A g and A g,n areseparated Deligne–Mumford stacks, and moreover A g,n is smooth over Z [1 /n ] with φ ( n ) geometrically irreducible fibers [FC90].We denote by A g , A g,n their corresponding coarse moduli spaces. By Mum-ford’s Geometric Invariant Theory [MFK94], they are quasi-projective schemes.We can extend A g,n over Z by taking the normalization of A g in A g,n / Z [1 /n ] ,as in [Mum71; DR73; de 93]. Over C , the analytification of A g is the Siegelspace H g / Sp g ( Z ) seen as an orbifold.If n ≥ , then the inertia of the stack A g,n is trivial. Therefore A g,n isisomorphic to A g,n by Corollary 4.3.(i), and A g,n is smooth over Z [1 /n ] . Thisshows in particular that there is a p such that A g is tame at every abelianvariety defined over a field of characteristic p ≥ p .If n ≤ , then the generic automorphism group on A g,n is µ . We canrigidify A g,n by µ in such a way that A g,n → [ A g,n /µ ] is a µ -gerbe [AOV08,Appendix A]. The map A g,n → A g,n factors through [ A g,n /µ ] , so the coarsemoduli space of [ A g,n /µ ] is still A g,n . By Theorem 4.1.(ii) or Theorem 4.2.(ii),there exists an affine étale open scheme U above A g,n whose image is dense andcontains all points with only generic automorphisms. Then [ A g,n /µ ] → A g,n becomes an isomorphism over U by Corollary 4.3.(i). Since [ A g,n /µ ] is smooth,the image of U in A g,n is also smooth by étale descent.We now proceed to construct the moduli stack A g ( ℓ ) parametrizing ℓ -isogenies.If Γ is a level subgroup of Sp g ( b Z ) , and n is an integer such that the level sub-group Γ( n ) is contained in Γ , we define A g, Γ / Z [1 /n ] as the quotient stack [ A g,n / e Γ] where e Γ is the image of Γ in Sp g ( Z /n Z ) . A T -point of [ A g,n / e Γ] corresponds toan abelian scheme A/T which is étale-locally endowed with a level n structuremodulo the action of e Γ [DR73, §IV.3.1]. The maps A g,n → A g, Γ and A g, Γ → A g are finite, étale, and representable [DR73, §IV.2, §IV.3]. We can extend A g, Γ to Z by normalization, as we did for A g,n . We can check as in [DR73, §IV.3.6]that the definition does not depend on the integer n such that Γ( n ) ⊂ Γ .We apply this construction to Γ = Γ ( ℓ ) , the standard level subgroup en-coding ℓ -isogenies, and we denote by A g ( ℓ ) := A g, Γ ( ℓ ) the resulting stack. Thestack A g ( ℓ ) is smooth over Z [1 /ℓ ] . We denote by Φ ℓ = ( Φ ℓ, , Φ ℓ, ) : A g ( ℓ ) → A g × A g the map ( A, K ) ( A, A/K ) . Proposition 4.7.
1. The maps Φ ℓ, and Φ ℓ, are finite, étale and repre-sentable.2. Let x ∈ A g ( ℓ )( k ) be a point represented by ( A, K ) , and let K ′ ⊂ A/K bethe kernel of the contragredient isogeny. Then Φ ℓ, is stabilizer preserving t x if and only if all automorphisms of A stabilize K , and Φ ℓ, is stabilizerpreserving at x if and only if all automorphisms of A/K stabilize K ′ .Proof. The automorphisms of x in A g ( ℓ ) are exactly the automorphisms of A stabilizing K . In particular Φ ℓ, induces a monomorphism of the automorphismgroups, so is representable; it is stabilizer preserving if and only if all automor-phisms of A stabilize K .If α is an automorphism of ( A, K ) , then α descends to A ′ = A/K , so Φ ℓ, is representable as well. An automorphism of A ′ comes from an automorphismof A if and only if it stabilizes K ′ , hence the condition for Φ ℓ, to be stabilizerpreserving.Finally, the map Φ ℓ, is finite étale because it is of the form A g, Γ → A g for Γ = Γ ( ℓ ) . Denote by π : X g → A g the universal abelian scheme, and by π ℓ : X g ( ℓ ) → A g ( ℓ ) the universal abelian scheme with a Γ ( ℓ ) -level structure.Then the universal isogeny f : X g ( ℓ ) → X g × A g A g ( ℓ ) is separable over Z [1 /ℓ ] .If we let s : A g → X g and s ℓ : A g ( ℓ ) → X g ( ℓ ) be the zero sections, then wehave Φ ℓ, = Φ ℓ, ◦ π × A g A g ( ℓ ) ◦ f ◦ s ℓ . Therefore Φ ℓ, : A g ( ℓ ) → A g is finite étale as well.The map Φ ℓ induces a map Φ ℓ : A g ( ℓ ) → A g on the coarse moduli spaces.This map is not injective, but the same reasoning as in [DR73, §VI.6] showsthat it is generically radicial, and even a birational isomorphism. The opensubscheme U of A g ( ℓ ) where Φ ℓ is an embedding is dense in every fiber ofcharacteristic p ∤ ℓ . Proposition 4.8.
Let Ψ denote the schematic image of Φ ℓ . Then A g ( ℓ ) isthe normalization of Ψ . If x lies in the image, then Φ ℓ : A g ( ℓ ) → Ψ inducesa local isomorphism around x if and only if x is normal in Ψ .Proof. The map A g ( ℓ ) → Ψ is separated quasi-finite, and birational by the dis-cussion above. Since A g ( ℓ ) is normal by Corollary 4.3.(ii), we deduce that A g ( ℓ ) is the normalization of Ψ by Zariski’s main theorem [Gro64, Cor. IV.8.12.11].If Φ ℓ induces a local isomorphism at x , then x is normal since A g ( ℓ ) isnormal. In fact it suffices to ask that Φ ℓ : A g ( ℓ ) → Ψ is étale at x , becausenormality is a local notion in the smooth topology [The18, Tag 034F]. Theconverse also follows from Zariski’s main theorem [Gro64, Cor. IV.8.12.10 andCor. IV.8.12.12]: there exists an open neighborhood U of x in Ψ such thatthe map Φ − ℓ ( U ) → U is an isomorphism.If x is a point of A g ( ℓ ) or A g , we abuse notation by also calling x its reductionto the associated coarse moduli space. Proposition 4.9.
Let x be a k -point of A g ( ℓ ) .1. Assume that Φ ℓ, is stabilizer preserving at x . Then: • The map Φ ℓ, is strongly étale at x , and the point x is smoothin A g ( ℓ ) if and only if Φ ℓ, ( x ) is smooth in A g . The point x = Φ ℓ ( x ) is normal in Ψ if and only if the projection p : Ψ → A g is étale at x . • If Φ ℓ, ( x ) is represented by an abelian variety A defined over k , thenthe isogeny ϕ : A → A ′ representing x is also defined over k .2. Assume that Φ ℓ, ( x ) only has generic automorphisms. Then Φ ℓ, is sta-bilizer preserving at x , the point x is smooth in A g ( ℓ ) , and the map A g ( ℓ ) → A g ( ℓ ) (resp. A g → A g ) is étale at x (resp. at Φ ℓ, ( x ) ).Proof. The first part of Item 1 comes from Theorem 4.4: in this case, themap Φ ℓ, is étale at x , and Φ ℓ, is étale-locally around x the pullback of Φ ℓ, by the map A g ( ℓ ) → A g ( ℓ ) .For the second part, we know that Φ ℓ, = p ◦ Φ ℓ is étale at x , and we haveseen in Proposition 4.8 that Φ ℓ is étale at x if and only if x is normal in Ψ .Therefore x is normal in Ψ if and only if p is étale at x .The final part of Item 1 comes from [DR73, §VI.3.1]. Indeed, if ( A, K ) represents x over k , the obstruction for ( A, K ) to descend over k is given byan element in H (Spec k, Aut( x )) in the sense of Giraud. But this obstructionvanishes since Φ ℓ, ( x ) is represented by A/k , and the automorphism groups of x and Φ ℓ, ( x ) are equal. The set of isomorphism classes over k is then canonicallygiven by H (Spec k, Aut( x )) .If y = Φ ℓ, ( x ) only has generic automorphisms, then x too, so Φ ℓ, is stabi-lizer preserving at x . The rigidification A g → [ A g /µ ] is étale (it is a µ -gerbe)and [ A g /µ ] → A g is an isomorphism above y by Corollary 4.3.(i). Therefore A g → A g is étale at y , and y is smooth in A g . By the same reasoning, the map A g ( ℓ ) → A g ( ℓ ) is étale at x .Proposition 4.9 also holds for Φ ℓ, in place of Φ ℓ, . Corollary 4.10.
Let x be a k -point of A g ( ℓ ) such that both Φ ℓ, ( x ) and Φ ℓ, ( x ) only have generic automorphisms. Then x is a smooth k -point of A g ( ℓ ) , thepoints Φ ℓ, ( x ) and Φ ℓ, ( x ) are both smooth k -points of A g , and we have acommutative diagram T Φ ℓ, ( x ) ( A g ) T x ( A g ( ℓ )) T Φ ℓ, ( x ) ( A g ) T Φ ℓ, ( x ) ( A g ) T x ( A g ( ℓ )) T Φ ℓ, ( x ) ( A g ) d Φ ℓ, d Φ ℓ, d Φ ℓ, d Φ ℓ, where the vertical arrows are isomorphisms induced by the maps A g ( ℓ ) → A g ( ℓ ) and A g → A g . In particular, the deformation map of the isogeny ϕ represent-ing x is given by D ( ϕ ) = d Φ ℓ, ( x ) ◦ d Φ ℓ, − ( x ) .Furthermore, let Ψ ⊂ A g × A g be the image of Φ ℓ , denote by p , p : Ψ → A g the two projections, and let x = Φ ℓ ( x ) . If Ψ is normal at x , then thedeformation map D ( ϕ ) is given by dp ( x ) ◦ dp ( x ) − . roof. For the first part, apply Proposition 4.9 for both Φ ℓ, and Φ ℓ, . For thesecond part, if Ψ is normal at y , then Φ ℓ : A g ( ℓ ) → Ψ is an isomorphismaround x by Proposition 4.8. Remark 4.11.
Let x be a k -point of A g ( ℓ ) such that both Φ ℓ, and Φ ℓ, arestabilizer preserving at x . Let y = Φ ℓ, ( x ) , y = Φ ℓ, ( x ) , and let y ′ , y ′ belifts of y , y to A g . Let G = I x be the common automorphism group of theseobjects. Even if G contains non-generic automorphisms, strong étaleness stillallows us to compute the deformation map by looking at the coarse spaces, asfollows.Indeed, suppose that x is smooth in A g ( ℓ ) (equivalently, by Proposition 4.9, y , or y , is smooth in A g ). Then, the same reasoning as in Corollary 4.10holds for x , except that in the commutative diagram the vertical maps are notisomorphisms, since the maps to the coarse moduli spaces are not étale at x andits images. From strong étaleness, the maps on the bottom are isomorphisms,and it remains to explain how to recover the maps on the top from them.Let B be the completed local ring of A g at y ′ . Then by Corollary 4.3.(ii),the completed local ring of A g at y is B G . Therefore, given m = g ( g + 1) / uniformizers u ′ , . . . , u ′ m of A g at y ′ , we obtain g ( g + 1) / uniformizers of A g at y as G -invariant polynomials in u ′ , . . . , u ′ m . Knowing these polynomials andproceeding in the same way at y allows us to recover the deformation map atthe level of stacks up to an action of non-generic elements of G , which amountsto changing the lifts y ′ and y ′ .In practice, it may be more convenient to work at the level of stacks to re-cover the deformation map directly, rather than using G -invariants uniformizerson A g . Algorithmically, the choice depends on the degree of the field extensionone has to take to add enough level structure to rigidify the stack. For instance,if g = 2 and k is a finite field, we only need an extension of degree at most toget the -torsion, whereas over a number field this could take an extension ofdegree up to . Remark 4.12.
Let k be a field. Then Proposition 4.9 and Corollary 4.10 alsoapply to the map A g ( ℓ ) ( k ) → A ( k ) g × A ( k ) g , where A g ( ℓ ) ( k ) and A ( k ) g are thecoarse moduli space of A g ( ℓ ) ⊗ k and A g ⊗ k respectively. In practice this doesnot change the results much, since at points x with generic automorphisms, weknow that A ( k ) g is isomorphic to A g ⊗ k locally around x by Theorem 4.2.(ii).Moreover, if the characteristic of k is large enough, then all points above k aretame, so A ( k ) g = A g ⊗ k by Theorem 4.2.(i).Now assume that we are in the situation of Remark 4.11, with x a k -pointof A g ( ℓ ) such that both Φ ℓ, and Φ ℓ, are stabilizer preserving at x . Assumefurthermore that x is a tame point, and that the characteristic of k is p . Let x = Φ ℓ ( x ) . If Φ ℓ is étale at x , or equivalently x is normal in Ψ , then Φ ℓ isétale above lifts in characteristic of x . The converse is also true: if x is notnormal, then it must come from a singular point in characteristic zero. Indeed,normality is equivalent to the conditions S and R ; since Ψ ⊗ k is reduced, so34s S and R , it suffices to check normality at lifts of characteristic zero. Thisgeneralizes the remark of [Sch95, p. 248]. In the case g = 2 , the structure of the coarse moduli space A and the possibleautomorphism groups have been worked out explicitly.Recall that the Jacobian locus, denoted by M , is the open locus in A consisting of Jacobians of hyperelliptic curves. Igusa showed in [Igu60] that M = Proj[ J , J , J , J , J ] / ( J J − J − J ) ( J ) , and that there is only one singular point of M over Z , given by the hyperellipticcurve C : y = x − , which corresponds to the point J = J = J = J = 0 .Over C , in [Igu62], Igusa shows that A has also in its singular locus twoprojective lines which represent products of elliptic curves, one of which beingisomorphic to y = x − or to y = x − . Finally, the structure of A over Z is described in [Igu79], but the singular locus is not determined.The possible (reduced) groups of automorphisms of genus curves over analgebraic closure are determined in [Igu60, §VIII]; see also [Liu93, §4.1]. Werestrict to a characteristic different from . Define C : y = x − and C : y = x − x . Then every curve C not isomorphic to C or C satisfies C ) ∈ { , , } . In characteristic different from , we have Aut C = Z / Z and C ∈ { , } . In characteristic , Aut C is an extension of PGL ( F ) ,which has cardinality , by Z / Z . In particular we see that in characteristic and p > all curves have a tame automorphism group.From [Igu60; Str10; GL12], the covariants I , I , I ′ , I are defined over Z .They are zero modulo , and I , I , I ′ are all polynomials in J modulo .Therefore the Igusa invariants j , j , j have bad reduction modulo and donot generate the function field of M modulo . Over Z [1 / however, they arebirational invariants, and determine an isomorphism from U = { I = 0 } ⊂ M to { j = 0 } ⊂ A . Every point with I = 0 maps to ( j , j , j ) = (0 , , .The modular polynomials Ψ ℓ,i from §2.6 are equations for the image Ψ ⊂ A g × A g of Φ ℓ intersected with U × U in A × A via j , j , j . Proposition 4.13.
Let Ψ denote the normalization of the variety cut out bythe modular polynomials Ψ ℓ,i . Let ϕ : A → A ′ be an ℓ -isogeny over a field k ofcaracteristic p > or zero , and let x be the k -point of Ψ corresponding to ϕ .Assume that A and A ′ are Jacobians with no extra automorphisms and that A, A ′ ∈ U . Then the deformation map D ( ϕ ) of ϕ is given by dp ( x ) ◦ dp ( x ) − ,where p , p denotes the projections Ψ ⊗ k → A k .Proof. By assumption, the Igusa invariants induce an isomorphism between thetangent spaces T A ( A g ) and T j ( A ) ( A k ) , and similarly for A ′ . Since A and A ′ haveno extra automorphisms, Φ ℓ, and Φ ℓ, are automatically stabilizer preserving.The normalization Ψ is isomorphic to the preimage of U × U in the coarsemoduli space A g ( ℓ ) by the discussion before Proposition 4.9. Since ϕ is a tame35oint, by Theorem 4.2.(ii), Ψ ⊗ k is still the coarse moduli space of A g, Γ ( ℓ ) ⊗ k locally around ϕ , so we conclude by Proposition 4.9. Remark 4.14.
We summarize different incarnations of the deformation map.• At the level of stacks, the two projections Φ ℓ, , Φ ℓ, : A g ( ℓ ) → A g arealways étale and we can always compute the deformation map at anisogeny ϕ as d Φ ℓ, ( ϕ ) ◦ d Φ ℓ, ( ϕ ) − .• At the level of the coarse moduli space A g ( ℓ ) , we can still compute thedeformation map at the points where Φ ℓ, and Φ ℓ, are stabilizer preserv-ing. If this is not the case, we must add a level structure that kills theautomorphisms that do not stabilize the kernel of the isogeny.• We may then replace A g ( ℓ ) by its birational image in A g . We recoverthe deformation map at points x ∈ A g where there is a local isomorphism Φ − ℓ ( U ) → U for some open set U containing x . If this is not the case,we may instead recover A g ( ℓ ) from its birational image by computing thenormalization. It is usually enough to compute the normalization onceand for all over Z , since by Theorem 4.2 the formation of A g ( ℓ ) commuteswith arbitrary base change at tame points.• Finally, when g = 2 , we can use the birational morphism from A to A given by the three Igusa invariants. Modular polynomials are usually givenin this form. With Streng’s version of Igusa invariants, they can be usedas long as I = 0 , i.e. j = 0 . Otherwise, one has to compute the modularpolynomials for another set of invariants which are defined at A and A ′ .As we go down the list, modular equations become algorithmically moretractable, at the expense of introducing more exceptions; but if we find such anexception, we can always spend more computation time if needed in order torecover the deformation map. We now briefly describe Hilbert–Blumenthal stacks, and refer to [Rap78; Cha90]for more details. Let K be a real number field of dimension g , and let Z K be itsmaximal order. We say that an abelian scheme A → S has real multiplicationby Z K (or, for short, is RM) if it is endowed with a morphism ι : Z K → End( A ) such that Lie( A ) is a locally free Z K ⊗ O S -module of rank . This last conditioncan be checked on geometric fibers [Rap78, Rem. 1.2] and is automatic on fibersof characteristic zero [Rap78, Prop. 1.4].We let H g be the stack of principally polarized abelian schemes with realmultiplication by Z K . It is algebraic and smooth of relative dimension g over Spec Z [Rap78, Thm. 1.14]. Moreover, H g is connected and its generic fiber isgeometrically connected [Rap78, Thm. 1.28]. Forgetting the real multiplicationembedding ι yields a map H g → A g , called the Hilbert embedding , which is an
Isom( Z K , Z K ) ≃ Gal( K ) -gerbe over its image, the Humbert stack . We described36he analytification of H g and the Hilbert embedding in Section 2. The map from H g → A g is finite by [Gro64, EGA IV.15.5.9], [DR73, Lem 1.19] (or by lookingat the compactifications of [Rap78], [FC90]).One can define the stack H g,n → Z [1 /n ] of RM abelian schemes with alevel n structure in the usual way. The map H g,n → H g is étale over Z [1 /n ] [Rap78, Thm. 1.22], its generic fiber is connected, and geometrically has φ ( n ) components defined over Q ( ζ n ) [Rap78, Thm. 1.28]. If β is a totally positiveprime of Z K , this allows us to construct, in a similar fashion to A g ( ℓ ) , thestack H g ( β ) = H g, Γ ( β ) of RM abelian schemes endowed with a subgroup K which is maximal isotropic for the β -pairing. We have a map Φ β = ( Φ β, , Φ β, ) : H g ( β ) → H g × H g given by forgetting the extra structure and taking the isogeny respectively. Thecondition on β ensures that Φ β, sends H g ( β ) to H g .The methods of Section 4.2.1 also apply to compute the Hilbert deformationmap. We have the following analogue of Corollary 4.10, with a similar proof. Proposition 4.15.
Let x be a k -point of H g ( β ) such that Φ β, ( x ) and Φ β, ( x ) only have generic automorphisms. Then x maps to a smooth point of the coarsemoduli space H g ( β ) , both Φ β, ( x ) and Φ β, ( x ) map to smooth points of thecoarse moduli space H g , and we have a commutative diagram T Φ β, ( x ) ( H g ) T x ( H g ( β )) T Φ β, ( x ) ( H g ) T Φ β, ( x ) ( H g ) T x ( H g ( β )) T Φ β, ( x ) ( H g ) d Φ β, d Φ β, d Φ β, d Φ β, where the vertical arrows are isomorphisms induced by the maps A g ( ℓ ) → A g ( ℓ ) and A g → A g , and Φ β,i is the map induced by Φ β,i at the level of coarse spaces.In particular, the deformation map of the isogeny ϕ representing x is given by D ( ϕ ) = d Φ β, ( x ) ◦ d Φ β, − ( x ) . Corollary 4.16.
Let x be a k -point of H g ( β ) such that both x = Φ β, ( x ) and x = Φ β, ( x ) only have generic automorphisms. Assume furthermore that ( x , x ) does not lie in Φ β ( H g ( β )) : this means that the corresponding abelianvarieties are β -isogenous but not β -isogenous.Let Ψ β ⊂ H g × H g be the image of Φ β . Let Ψ β,β ⊂ A g × A g be the imageof Ψ β , and let y = ( y , y ) the image of ( x , x ) by the forgetful morphism H g × H g → A g × A g . Denote by p , p : Ψ β,β → A g the two projections. If Ψ β,β is normal at y , then the deformation map D ( ϕ ) is given by dp ( y ) ◦ dp ( y ) − .Proof. The map H g → A g is finite étale, and under our assumptions the maps H g → H g and A g → A g are étale at x and x (resp. at their images y , y in A g ). Therefore the map H g × H g → A g × A g is étale at x ′ = ( x , x ) .Furthermore the pullback of Ψ β,β by H g × H g → A g × A g is Ψ β ∪ Ψ β ⊂ H g × H g , so the map Ψ β ∪ Ψ β → Ψ β,β is étale at x ′ . Since Φ β is finite, its image37 β ⊂ H g × H g is closed. By our assumption on x , there is an open subschemecontaining x which does not intersect Ψ β , so the map Ψ β → Ψ β,β is étale at x ′ .In particular, Ψ β is normal at x ′ if and only if Ψ β,β is normal at x . The sameproof as in Corollary 4.10 shows that the projections maps Ψ β → H g are étaleat x ′ , and can be used to compute the deformation matrix. Since H g → A g is étale at x and x , the projections p and p are also étale at y , and can beused to compute the deformation matrix as well. In this section, we present the Kodaira–Spencer isomorphism, which for a prin-cipally polarized abelian variety A identifies T A ( A g ) with Sym ( T ( A )) . Thisyields a relation between the deformation and tangent maps of a given ℓ -isogeny(Proposition 4.19). We also present an analogous result in the Hilbert case. The Kodaira-Spencer morphism was first introduced in [KS58]; we refer to[FC90, §III.9] and [And17, §1.3] for more details.Let p : A → S be a proper abelian scheme, and assume for simplicity that S is smooth. Then, using the Gauss-Manin connection ∇ : R p ∗ Ω A/S → R p ∗ Ω A/S ⊗ Ω S , one can define the Kodaira–Spencer morphism κ : T S → R p ∗ T A/S , where T A/S is the dual of Ω A/S .Recall that
Lie S A = p ∗ T A/S is the dual of p ∗ Ω A/S , and is canonically iden-tified with s ∗ T A S where s : S → A is the zero section [MvE12, Prop. 3.15]. Bythe projection formula [FGI+05, Thm. 8.3.2], [The18, Tag 0943], we have R p ∗ T A/S = Lie S ( A ) ⊗ O S R p ∗ O A . Moreover, R p ∗ O A is naturally isomorphic to Lie S ( A ∨ ) , where A ∨ → S is thedual of A . Therefore, we can also write the Kodaira–Spencer map as κ : T S → R p ∗ T A/S ≃ Lie S ( A ) ⊗ O S Lie S ( A ∨ ) . The Kodaira-Spencer map κ is invariant by duality. A polarization A → A ∨ induces another version of the Kodaira–Spencer map: κ : T S → Sym Lie S ( A ) = Hom Sym (Ω A/S , Ω ∨ A ∨ /S ) = Hom Sym (cid:0)
Lie S ( A ) ∨ , Lie S ( A ∨ ) (cid:1) . If we apply this construction to the universal abelian scheme X g → A g (orrather, the pullback of X g to an étale presentation S of A g ), the Kodaira–Spencer map is an isomorphism [And17, §2.1.1]. Its analytification can be de-scribed explicitly. 38 roposition 4.17. Let V be the trivial vector bundle C g on H g , identified withthe tangent space at of the universal abelian variety A ( τ ) over H g . Then thepullback of the Kodaira–Spencer map κ : T A g → Sym Lie S X g by H g → A g an is an isomorphism T H g ≃ Sym V given by κ (cid:16) δ jk πi ∂∂τ jk (cid:17) = 1(2 πi ) ∂∂z j ⊗ ∂∂z k . for each ≤ j, k ≤ g , where δ jk is the Kronecker symbol.Proof. This is [And17, §2.2]. The identification can be derived by looking atthe deformation of a section s of the line bundle on X g giving the principalpolarization. On H g × C g → H g , we can take the theta function θ as a section,and its deformation along τ is given by the heat equation [Cv00, p. 9]: πi (1 + δ jk ) ∂θ∂τ jk = ∂ θ∂z j ∂z k . When identifying the tangent space at τ with the symmetric matrices, theaction of Sym at a matrix U on the tangent space is given by M M U M t .It is then easy to check that this action is indeed compatible with the actionof Sp g ( Z ) on τ and U . From Proposition 4.17, we recover that derivatives ofSiegel modular invariants have weight Sym in the sense of §2; moreover thebasis of differential forms ω ( τ ) from §2.1 and the matrix D τ J defined in §3.4are correctly normalized.To sum up, if x : Spec k → A g is a point represented by a principally polar-ized abelian variety A/k , we have a canonical isomorphism T x A g ≃ Sym ( T ( A )) . Definition 4.18.
Let k be a field of characteristic distinct from ℓ , let ϕ : A → A ′ be an ℓ -isogeny representing a point of A g ( ℓ )( k ) , and fix bases of T ( A ) and T ( B ) as k -vector spaces. We call the matrix of the tangent map dϕ inthese bases the tangent matrix of ϕ .By functoriality, this choice of bases induces bases of T A ( A g ) and T A ′ ( A g ) over k . We call the matrix of the deformation map D ( ϕ ) in these bases the deformation matrix of ϕ .We still denote these matrices by dϕ and D ( ϕ ) , but this abuse of notationshould cause no confusion.We can now extend the relation that we gave in Proposition 3.19 betweenthe tangent and deformation matrices, as follows. Proposition 4.19.
Let ϕ be as in Definition 4.18, and let dϕ (resp. D ( ϕ ) ) beits tangent (resp. deformation) matrix. Then we have Sym ( dϕ ) = ℓ D ( ϕ ) .Proof. It suffices to prove it for the universal ℓ -isogeny ϕ : X g ( ℓ ) → X g × A g A g ( ℓ ) over Z [1 /ℓ ] . All line bundles involved in the relation we have to prove are locallyfree on smooth stacks, so are flat over Z ; therefore, since Z → C is injective,39t suffices to prove the relation over C . By rigidity [MFK94, Prop. 6.1 andThm. 6.14], it suffices to prove the relation on each fiber.Hence we may assume that ϕ : A → A ′ is an ℓ -isogeny over C . We can find τ ∈ H g such that A is isomorphic to C g / ( Z g + τ Z g ) and A ′ is isomorphic to C g / ( Z g + τ /ℓ Z g ) , with ϕ induced by the identity on C g . Then, the deformationmap at ϕ is given by τ → τ /ℓ , so the result follows. In the Hilbert case, the Kodaira–Spencer isomorphism is as follows.
Proposition 4.20.
Let A → S be an abelian scheme in H g . Then we havecanonical isomorphisms T A ( H g ) ≃ Hom Z K ⊗O S (Lie( A ) ∨ , Lie( A ∨ )) = Lie( A ∨ ) ⊗ Z K ⊗O S Lie( A ) ⊗ Z K Z ∨ K . Proof.
Combine [Rap78, Prop. 1.6] with [Rap78, Prop. 1.9].Proposition 4.20 shows that for Hilbert–Blumenthal stacks, the deformationmap is actually represented by an element of Z K ⊗ O S rather than a matrixin O S . The action of the Hilbert embedding on tangent spaces is also easy todescribe. Proposition 4.21.
Let A be a k -point of H g . Then the map T A ( H g ) → T A ( A g ) induced by the forgetful functor fits in the commutative diagram T A ( H g ) T A ( A g )Hom Z K ⊗O k (Lie( A ) ∨ , Lie( A ∨ )) Hom Sym (Lie( A ) ∨ , Lie( A ∨ )) . where the vertical arrows are the Kodaira–Spencer isomorphisms.Proof. The bottom arrow is well-defined:
Lie( A ) is a projective Z K ⊗O k -sheaf ofrank , so its image in Hom O k (Lie( A ) ∨ , Lie( A ∨ )) obtained by forgetting the Z K -structure is automatically symmetric. We omit the proof of commutativity.Combining Proposition 4.21 with the analytic description of the Kodaira–Spencer in the Siegel case (Proposition 4.17) and the analytic description ofthe forgetful map (§2.4), we obtain the following analytic description of theKodaira–Spencer isomorphism in the Hilbert case. Corollary 4.22.
The pullback of κ : T H g → Sym Lie S X g by H g → H g an isgiven by κ (cid:16) πi ∂∂t j (cid:17) = 1(2 πi ) ∂∂z j ⊗ ∂∂z j for every ≤ j ≤ g . ,the part of T A ( A ) that comes from the Hilbert space corresponds to the spanof dz ⊗ dz and dz ⊗ dz .We obtain the analogue of Proposition 4.19 in the Hilbert case by a similarproof; in this statement, we see D ( ϕ ) as an element of a Z K ⊗ O S -module. Proposition 4.23.
Let ϕ : A → A ′ be a β -isogeny. Then Sym ( dϕ ) = β D ( ϕ ) . Remark 4.24.
We give an algebraic interpretation of the notion of Hilbert-normalized bases from §2.3, and the reduction to diagonal matrices that weused in §3.5 to compute the tangent matrix in the Hilbert case.Let k be a field, and let A be an abelian variety representing a k -point of H g .Then Lie( A ) is a free Z K ⊗ k -module of rank , and any choice of basis v inducesan isomorphism with Z K ⊗ k itself. Provided that char k ∤ Discr( K ) , and up totaking an étale extension of k , we may assume that k splits Z K : Z K ⊗ k = ⊕ gi =1 k σ i where k σ i ≃ k has a Z K -module structure induced by the i -th embedding σ i : Z K → k . We fixed such a trivialization in §2.3 in the case k = C . Then, v in-duces a basis of Lie( A ) as a k -vector space on which Z K acts diagonally, in otherwords a Hilbert-normalized basis of Lie( A ) . With such choices of trivializations,the deformation map as given by a g × g matrix in the basis ( v ⊗ v , . . . , v g ⊗ v g ) of the tangent spaces to H g .Let us discuss, as a generalization of §3.6, the construction of Hilbert-normalized basis when only the Humbert equation is given. Assume that k splits Z K and fix a trivialization; let ( v , . . . , v g ) be a Hilbert-normalized basisof Lie( A ) , let ( w , . . . , w g ) be another k -basis and let M be the base-changematrix. Then w ⊗ w , . . . , w g ⊗ w g are tangent to the Humbert variety if andonly if they are in the image of the map Hom Z K ⊗O k (Lie( A ) ∨ , Lie( A ∨ )) → Hom
Sym (Lie( A ) ∨ , Lie( A ∨ )) . Via the trivialization, the left hand side is isomorphic to ⊕ gi =1 Hom k ( k σ i , k σ i ) .So w ⊗ w , . . . , w g ⊗ w g are tangent to the Humbert variety if and only if M is diagonal up to a permutation. When Gal( K/ Q ) is not the full symmetricgroup S g , this is not enough in general to ensure that the basis ( w , . . . , w g ) ispotentially Hilbert-normalized. This issue does not appear in genus .As a final remark, assume that ϕ : A → A ′ is an isogeny compatible with thereal multiplication, and assume that we are given bases of Lie( A ) and Lie( A ′ ) as Z K ⊗ k -modules (which we assume is étale for simplicity). Then, knowing Sym ( dϕ ) , the number of possibilities for dϕ is s where s is the number ofconnected components of the étale algebra Z K ⊗ k . For instance, if g = 2 and k = F p there are or possibilities according to whether p is inert or splitin Z K . 41 .4 Modular forms and covariants In this section, we give an algebraic interpretation of modular forms and covari-ants over Z , as well as a completely algebraic proof of Theorem 3.9. This yieldsan explicit version of the Kodaira–Spencer isomorphism in the model of A g given by Igusa invariants over Z [1 / and not only over C .Let π : X g → A g be the universal abelian variety. The vector bundle h = π ∗ Ω X g / A g over A g , which is dual to Lie X g / A g , is called the Hodge bundle . If ρ is a rep-resentation of GL g , a Siegel modular form of weight ρ is a section of ρ ( h ) ; inparticular, a scalar-valued modular form of weight k is a section of Λ g h ⊗ k . Inother words, a Siegel modular form f can be seen as a map ( A, ω ) f ( A, ω ) where A is a point of A g and ω is a basis of differential forms on A , with thefollowing property: if η : A → A ′ is an isomorphism, and r ∈ GL g is the matrixof η ∗ in the bases ω ′ , ω , then f ( A ′ , ω ) = ρ ( r ) f ( A, ω ′ ) . The link with classicalmodular forms over C is the following: if τ ∈ H g , then we define f ( τ ) = f (cid:0) C g / ( Z g + τ Z g ) , (2 πi dz , . . . , πi dz g ) (cid:1) . This choice of basis is made so that the q -expansion principle holds [FC90,p. 141]. We already used it to define f ( A, ω ) over C in §2.1. The canonical linebundle h = Λ g h is ample, so modular forms give local coordinates on A g .The link between modular forms and covariants comes from the Torelli mor-phism τ g : M g → A g where M g denotes the moduli stack of smooth curves of genus g . Let C g → M g denote the universal curve; then the pullback τ ∗ g h of the Hodge bundle by theTorelli morphism is π ∗ Ω C g / M g , with both having canonical action by GL g . Inother words a Siegel modular form of weight ρ induces a Teichmuller modularform of weight ρ .Now assume that g = 2 . Over Z [1 / , the moduli stack M is identified withthe moduli stack of nondegenerate binary forms of degree . Let V = Z x ⊕ Z y ,let X = det − V ⊗ Sym V , and let U be the open locus determined by the dis-criminant. Then U → M is naturally identified with the Hodge frame bundleon M : in other words, U is the moduli space of genus hyperelliptic curves π : C → S endowed with a rigidification O ⊕ S ≃ π ∗ Ω C/S . In this identifica-tion, we send the binary form f ( x, y ) to the curve v = f ( u, with a basisof differential forms given by ( u du/v, du/v ) [CFv17, §4]. The natural actionof GL on the Hodge bundle corresponds to the action of GL on U that wedescribe in Section 3.2. This shows why a Siegel modular form of weight ρ pullsback to a fractional covariant of weight ρ , at least over Z [1 / . In fact, onecan show as in Theorem 3.9, by considering suitable compactifications, that a42iegel modular form pulls back to a polynomial covariant over any ring R inwhich is invertible. Using Igusa’s universal form [Igu60, §2], one can also usebinary forms of degree to describe the moduli stack of genus curves even incharacteristic two. Proposition 4.25.
The equality
Cov( f , ) = Cov( χ ) X from Proposition 3.12holds over Z .Proof. By the q -expansion principle, f , is defined over Z [1 / , / , / , / ;the covariants I and X are defined over Z [1 / since they have integral co-efficients. Checking the value of Cov( χ ) X on Igusa’s universal hyperellipticcurve as in [Igu60, §3] shows that this covariant is even defined over Z . Since theHodge bundle is without torsion, it is enough to check equality over C , which isthe content of Proposition 3.12.This suggests another, entirely algebraic proof of Proposition 3.12. By di-mension considerations, we have Cov( f , ) = λ Cov( χ ) X for some λ ∈ Q × .We have seen above that Cov( χ ) X is defined over Z and primitive; therefore,if we can show that the Fourier coefficients of f , are integers with gcd , wewill have λ = ± . In order to obtain λ = 1 , we can use Thomae’s formula onone curve, perform a certified numerical evaluation over C , or study degenera-tions from hyperelliptic curves to elliptic curves using the formula from [Liu93,Thm. 1.II].As a consequence of Proposition 4.25, the identification of derivatives ofIgusa invariants as explicit covariants (Theorem 3.14) still holds over Z [1 / .For the algebraic interpretation of Hilbert modular forms as sections of theHodge bungle on H g , the Koecher principle and the q -expansion principle forHilbert modular forms, we refer to [Cha90, §4] and [Rap78, Thm. 6.7]. We cancheck that the relation between derivatives of Igusa invariants on the Hilbert andSiegel sides (Proposition 3.22) and the characterization of potentially Hilbert-normalized curves (Proposition 3.24) are still valid over Z [1 / . In this section, we work over a field k of characteristic different from and ;this restriction is not essential and comes from our choice of invariants. Wehave seen that derivatives of Igusa invariants are defined over Z [1 / , and hencemake sense over k . We keep the matrix notations from §3.4. Proposition 4.26.
Let U be the open set of A over k consisting of abeliansurfaces A such that Aut( A ) = {± } and j ( A ) = 0 . Let ϕ : A → A ′ be an ℓ -isogeny over k . Assume that A, A ′ lie in U , and denote their Igusa invariantsby j, j ′ . Assume further that the subvariety of A × A cut out by modularequations is normal at ( j ( A ) , j ( A ′ )) . Let C , C ′ be hyperelliptic equations over k whose Jacobians are isomorphic to A, A ′ respectively. Then . The isogeny ϕ is generic in the sense of Definition 3.18, in other wordsthe × matrices D Ψ ℓ,L ( j, j ′ ) , D Ψ ℓ,R ( j, j ′ ) , D τ J ( C ) and D τ J ( C ′ ) are invertible.2. Let dϕ be the tangent matrix of ϕ with respect to C , C ′ . Then Sym ( dϕ ) = − ℓD τ J ( C ′ ) − · D Ψ ℓ,R ( j, j ′ ) − · D Ψ ℓ,L ( j, j ′ ) · D τ J ( C ) . Proof.
By Corollary 4.10, both A and A ′ are smooth points of A g , and thedeformation map D ( ϕ ) is d Φ ℓ, ( ϕ ) ◦ d Φ ℓ, ( ϕ ) − . Since A has generic auto-morphisms, A is not a product of elliptic curves; moreover j ( A ) = 0 , sothe birational map ( j , j , j ) : A g → A is well-defined and étale at A . Themap A g → A g is also étale at A , so the Igusa invariants are local uniformizersaround A in A g . This shows that ϕ is generic in the sense of Definition 3.18.We obtain the expression of Sym ( dϕ ) by Proposition 4.19.If A lies in the open set U defined in Proposition 4.26 and C is a hyperellipticequation for A , then giving an element of T A ( A g ) is equivalent to giving one ofthe following:1. A deformation C ǫ of C over k [ ǫ ] / ( ǫ ) ,2. The Igusa invariants j ( C ǫ ) , j ( C ǫ ) , j ( C ǫ ) in k [ ǫ ] / ( ǫ ) ,3. If ( w , w ) = ( x dx/y, dx/y ) is the canonical basis of differential formson C , a vector v = αw + βw w + γw in Sym Ω ( C ) .Switching from one representation to another can be done at the cost of O (1) operations in k using the formulæ for Igusa invariants, the expression of theirderivatives as a covariant, and linear algebra.In the Hilbert case, it is more difficult to ensure genericity in the sense ofDefinition 3.20 because the Hilbert embedding H g → A g comes into play. Weassume that k splits Z K , and fix a trivialization of Z K ⊗ k . Proposition 4.27.
Let
A, A ′ be abelian varieties representing k -points of H g ,and let C , C ′ be hyperelliptic equations over k whose Jacobians are isomorphic to A, A ′ respectively; assume that C , C ′ are Hilbert-normalized and that there existsa β -isogeny ϕ : A → A ′ . Then we have D Ψ β,L ( j, j ′ ) · D t J ( C ) = − D Ψ β,R ( j, j ′ ) · D t J ( C ′ ) · Diag(1 /β, /β ) · ( dϕ ) . Proof.
This comes from the relation between the deformation and tangent ma-trices (Proposition 4.23).The equality in Proposition 4.27 only allows to compute ( dϕ ) when ϕ isgeneric. Even in this case, we get several possible candidates for dϕ up to sign.The discussion of Remark 4.24 shows that Proposition 3.24, which gives an al-gorithm to construct potentially Hilbert-normalized curve equations in genus ,is still valid over k . 44 emark 4.28. The × matrices D t J ( C ) and D t J ( C ′ ) have rank two when theIgusa invariants contain uniformizers of H g at A and A ′ by [Gro64, p. IV.17.11.3].Given the relation between derivatives of Igusa invariants on the Hilbert andSiegel sides (Proposition 3.24, which is valid over k by Proposition 4.21), thiswill be the case at soon as the images of A and A ′ in A g lie in the open set U from Proposition 4.26.Assume that generators of the ring of Hilbert modular forms are known, andthe expression of Igusa invariants in terms of these generators is given. Sincemodular forms realize a projective embedding of H g , one can compute from thisdata an open set V in H g where the Igusa invariants contain local uniformizers.Then, if A lies in V and Aut( A ) ≃ {± } , the Igusa invariants will contain localuniformizers of H g , hence D t J ( C ) will have rank .In the Hilbert case, if Igusa invariants contain local uniformizers of H g at A and if C is a Hilbert-normalized curve equation for A , then giving an elementof T A ( H g ) is equivalent to giving1. A deformation C ǫ of C over k [ ǫ ] / ( ǫ ) with real multiplication by Z K ,2. Igusa invariants j ( C ǫ ) , j ( C ǫ ) , j ( C ǫ ) in k [ ǫ ] / ( ǫ ) lying on the Humbertsurface (if j ( C ) = 0 ),3. If ( w , w ) = ( x dx/y, dx/y ) is the canonical basis of differential formson C , a vector v = αw + γw in Sym Ω ( C ) .Switching from one representation to another can be done at the cost of O (1) operations in k . Assume that we are given the tangent map dϕ of a separable isogeny ϕ : A → A ′ of principally polarized abelian varieties of dimension g defined over a field k .In general, the task of computing ϕ explicitly may be specified as follows: givenmodels of A and A ′ , that is given very ample line bundles L A and L A ′ on A and A ′ and a choice of global sections ( a i ) (resp. ( a ′ j ) ) which give a projectiveembedding of A (resp. A ′ ), express the functions ϕ ∗ a ′ j on A as rational fractionsin terms the coordinates ( a i ) .One method to determine ϕ given dϕ is to work over the formal groups of A and A ′ . Let x , . . . , x g be uniformizers at A , and let y , . . . , y g be uniformizersat A ′ . Knowing the map dϕ allows us to express the differential form ϕ ∗ dy j in term of the differential forms dx i on A , so the functions ϕ ∗ a ′ j satisfy a dif-ferential system. A possible strategy to solve this differential system is to use amultivariate Newton algorithm, possibly over an extension of the formal group.If this algorithm is successful, we recover the functions ϕ ∗ a ′ j as power series in k [[ x , . . . , x g ]] up to some precision. The next step is to use multivariate rational45econstruction to obtain ϕ as a rational map. In order for the rational recon-struction algorithm to succeed, the power series precision must be large enoughwith respect to the degrees of the result in the variables ( a i ) . These degreescan be estimated from the intersection degree of ϕ ∗ L A ′ and L A , or alternativelyfrom the intersection degree of ϕ ∗ L A and L A ′ .This strategy to compute ϕ is not new: the idea of using a differentialequation to compute isogenies in genus appears in [Elk98], and [BMS+08]uses a Newton algorithm to solve this differential equation. To the best of ourknowledge, the first article to extend these ideas to genus is [CE15]. Themethod is further extended to compute endomorphisms of Jacobians over anumber field in [CMS+19]. In [CMS+19, §6], the endomorphism is representedas a divisorial correspondence; the interpolation of this divisor is done a bitdifferently, via linear algebra on Riemann–Roch spaces.A necessary condition for the whole method to work is that ϕ be completelydetermined by its tangent map. In general, this will be the case when char k is large with respect to the degree of ϕ . For instance, we have the followingstatement in the case of ℓ -isogenies. Lemma 5.1.
Let A and A ′ be two principally polarized abelian varieties over afield k , and M : T ( A ) → T ( A ′ ) a linear map. Assume that that char k = 0 or char k > N . Then there is at most one ℓ -isogeny ϕ : A → A ′ with ℓ ≤ N suchthat dϕ = M .Proof. Let ϕ and ϕ be two such isogenies. Then ϕ = ϕ + ψ where ψ isinseparable. If char k = 0 , this implies ψ = 0 and hence ϕ = ϕ . Otherwise,write p = char k and denote by ϕ the contragredient isogeny. Then if ψ = 0 ,we have ψψ = ϕ ϕ + ϕ ϕ − ϕ ϕ − ϕ ϕ . But ψψ is equal to p m for some m ≥ , and ϕ ϕ = ℓ , ϕ ϕ = ℓ with ℓ , ℓ ≤ N by hypothesis. Therefore we obtain p m ≤ N + 2 √ N √ N = 4 N .In practice, Newton iterations will fail to reach sufficiently high power seriesprecision if char k is too small, hence the bound given in Theorem 1.1.In the rest of this section, we carry out this strategy in detail when A, A ′ arethe Jacobians of genus hyperelliptic curves C , C ′ . Concretely, we are given thematrix of dϕ in the bases of T ( A ) and T ( A ′ ) that are dual to ω ( C ) and ω ( C ′ ) respectively (see §3.1). In this case, a nice simplification occurs: the isogeny ϕ is completely determined by the compositum C Jac( C ) Jac( C ′ ) C ′ , sym A Q [ Q − P ] ϕ ∼ m where P is any point on C , and m is the rational map given by { ( x , y ) , ( x , y ) } 7→ (cid:16) x + x , x x , y y , y − y x − x (cid:17) . s, p, q, r ∈ k ( u, v ) thatwe call the rational representation of ϕ at the base point P . We choose a uni-formizer z of C around P and perform the Newton iterations and rational re-construction over the univariate power series ring k [[ z ]] .We explain how we choose the base point P and solve the differential systemin Section 5.2. One difficulty is that the differential system we obtain is singular(Lemma 5.6), so we need to use the geometry of the curves (Proposition 5.4)to find the first few terms in the series before switching to Newton iterations(Proposition 5.8). In Section 5.3, we estimate the degrees of the rational frac-tions that we want to compute and present the rational reconstruction step. We keep the notation used in §5.1. Write the curve equations C , C ′ and thetangent matrix as C : v = E C ( u ) , C ′ : y = E C ′ ( x ) , dϕ = (cid:18) m , m , m , m , (cid:19) . We assume that ϕ is separable, so that dϕ is invertible. If P is a base point on C , we denote by ϕ P the associated map C → C ′ , sym . Step 1: choice of base point and power series.
Let P be a point on C which is not at a point at infinity; up to enlarging k , we assume that P ∈ C ( k ) .Since ϕ P ( P ) is zero in Jac( C ′ ) , we have ϕ P ( P ) = (cid:8) Q, i ( Q ) (cid:9) for some Q ∈ C ′ , where i denotes the hyperelliptic involution. We say that ϕ P is of Weierstrass type if Q is a Weierstrass point of C ′ , and of generic type otherwise. If z is a local uniformizer of C at P , and R is an étale extensionof k [[ z ]] , we define a local lift of ϕ P at P with coefficients in R to be a tuple e ϕ P = ( x , x , y , y ) ∈ R such that we have a commutative diagram Spec R C ′ Spec k [[ z ]] C C ′ , sym . ( x ,y ) , ( x ,y ) ϕ P If the power series x , x , y , y define a local lift of ϕ P , then they satisfy thedifferential system ( S ) given by x dx y + x dx y = ( m , u + m , ) duvdx y + dx y = ( m , u + m , ) duvy = E C ′ ( x ) y = E C ′ ( x ) , ( S )47here we consider the coordinates u, v on C as elements of k [[ z ]] , and the letter d denotes derivation with respect to z .When solving ( S ), we want ϕ P to be of generic type. Proposition 5.4 showshow to choose P to enforce this condition; in order to prove it, we first studythe existence of local lifts for arbitrary base points. Lemma 5.2.
Let z be a uniformizer of C at P . Then there is a quadraticextension k ′ /k such that a local lift of ϕ P at P with coefficients in R = k ′ [[ √ z ]] exists. Moreover, if ϕ P is of generic type, or if P is a Weierstrass point of C ,then the same statement holds with R = k ′ [[ z ]] .Proof. First assume that ϕ P is of generic type. Since the unordered pair (cid:8) Q, i ( Q ) (cid:9) is Galois-invariant, there is a quadratic extension k ′ /k such that Q is defined over k ′ . The map C ′ → C ′ , sym is étale at (cid:0) Q, i ( Q ) (cid:1) , so induces anisomorphism of completed local rings. Therefore a local lift exists over k ′ [[ z ]] .Second, assume that ϕ P is of Weierstrass type. The map Spec k [[ z ]] →C ′ , sym defines a k (( z )) -point of C ′ , sym , and there exists a preimage of this pointdefined over an extension K/k (( z )) of degree . Let R be the integral closure of k [[ z ]] in K . Then R is contained in k ′ [[ √ z ]] for some quadratic extension k ′ /k [The18, Tag 09E8]. By the valuative criterion of properness, our K -point of C ′ extends to an R -point uniquely, so a local lift exists over R .Finally, assume that ϕ P is of Weierstrass type and that P is a Weierstrasspoint of C . Let ( x , x , y , y ) be a local lift of ϕ P over k ′ [[ √ z ]] . The completedlocal ring of the Kummer line of C at P is k [[ z ]] , and the unordered pair { x , x } is defined on the Kummer line; by the same argument as above, x and x aredefined over k ′ [[ z ]] . The system ( S ) can be written as (cid:18) /y /y (cid:19) = (cid:18) x x ′ x x ′ x ′ x ′ (cid:19) − (cid:18) R ( z ) R ( z ) (cid:19) for some series R , R ∈ k [[ z ]] , hence y and y are defined over k ′ [[ z ]] as well.Consider the tangent space T ( Q,i ( Q )) C ′ of C ′ at (cid:0) Q, i ( Q ) (cid:1) . It decomposesas T ( Q,i ( Q )) C ′ = T Q C ′ ⊕ T i ( Q ) C ′ ≃ ( T Q C ′ ) where the last map is given by the hyperelliptic involution on the second term. Lemma 5.3.
Assume that a local lift e ϕ P of ϕ P to k ′ [[ z ]] exists. Then thetangent vector d e ϕ P /dz at z = 0 cannot be of the form ( v, v ) where v ∈ T Q C ′ .Proof. Assume the contrary. The direction ( v, v ) is contracted to zero in theJacobian, so every differential form on the Jacobian is pulled back to zero via ϕ P .This is a contradiction because ϕ ∗ is nonzero. Proposition 5.4.
The point Q is uniquely determined by the property that, upto a scalar factor, ϕ ∗ ω ′ Q = ω P where ω P (resp. ω ′ Q ) is a nonzero differential form on C (resp. C ′ ) vanishingat P (resp. Q ). roof. First, assume that a local lift e ϕ P exists over k ′ [[ z ]] . By Lemma 5.3, thetangent vector d e ϕ P /dz at z = 0 is of the form ( v + w, w ) for some v, w ∈ T Q C ′ such that v = 0 . Let ω ′ be the unique nonzero differential form pulled backto ω P by ϕ . Then ω ′ must vanish on ( v, , in other words ω ′ must vanish at Q .Second, assume that no such lift exists. By Lemma 5.2, Q is a Weierstrasspoint on C ′ , and P is not a Weierstrass point on C . After a change of variables,we may assume that Q is not at infinity. Write P = ( u , v ) with v = 0 , and Q = ( x , . We have to show that x = m , u + m , m , u + m , . Let ( x , y , x , y ) be a lift over k ′ [[ √ z ]] as in Lemma 5.2, and look at thedifferential system ( S ). Write the lift as y = v √ z + t z + O ( z / ) , y = v √ z + t z + O ( z / ) . Then the relation y = E C ′ ( x ) forces x , x to have no term in √ z , so that x = x + w z + O ( z / ) , x = x + w z + O ( z / ) . Using the relation dx/y = 2 dy/E ′C ′ ( x ) , we have x dy E ′C ′ ( x ) + 2 x dy E ′C ′ ( x ) = ( m , u + m , ) duv , dy E ′C ′ ( x ) + 2 dy E ′C ′ ( x ) = ( m , u + m , ) duv . Inspection of the ( √ z ) − term gives the relation v = − v . Write e = E ′C ′ ( x ) .Then the constant term of the series on the left hand side are respectively x (cid:16) t e + t e (cid:17) and (cid:16) t e + t e (cid:17) . The differential forms on the right hand side do not vanish simultaneously at P ,therefore m , u + m , must be nonzero. Taking the quotient of the two linesgives the result.Using Proposition 5.4, we choose a base point P on C such that ϕ P is ofgeneric type. By Lemma 5.2, a local lift e ϕ P = ( x , x , y , y ) of ϕ P existsover k ′ [[ z ]] , where k ′ is a quadratic extension of k . Step 2: initialization.
Now we explain how to compute the power series x , x , y , y up to O ( z ) . We can compute the point Q = ( x , y ) using Propo-sition 5.4. Write x = x + v z + O ( z ) , x = x + v z + O ( z ) . y , y up to O ( z ) in terms of v , v respectively. Let u (resp. d ) be the constant term of the power series u (resp. du/v ). Then ( S ) gives v + v = y x ( m , u + m , ) d = y ( m , u + m , ) d . (2)Combining the two lines, we also obtain ( x − x ) dx y + ( x − x ) dx y = R, where R = r z + O ( z ) has no constant term. At order 1, this yields v + v = y r . (3)Equalities (2) and (3) yield a quadratic equation satisfied by v , v . This givesthe values of v and v in a quadratic extension k ′ /k . Step 3: Newton iterations.
Assume that the series x , x , y , y are knownup to O ( z n ) for some n ≥ . The system ( S ) is satisfied up to O ( z n − ) forthe first two lines, and O ( z n ) for the last two lines. We attempt to double theprecision, and write x = x ( z ) + δx ( z ) + O ( z n ) , etc.where x is the polynomial of degree at most n − that has been computed.The series δx i and δy i start at the term z n . From now on, we also denote by x ′ the derivative of a power series x with respect to z . Proposition 5.5.
The power series δx , δx satisfy a linear differential equa-tion of the first order M ( z ) (cid:18) δx ′ δx ′ (cid:19) + N ( z ) (cid:18) δx δx (cid:19) = R ( z ) + O ( z n − ) ( E n ) where M, N, R ∈ M (cid:0) k ′ [[ z ]] (cid:1) have explicit expressions in terms of x , x , y , y , u , v , E C and E C ′ . In particular, M ( z ) = (cid:18) x /y x /y /y /y (cid:19) and, writing e = E ′C ′ ( x ) , the constant term of N is v y − x v y e v y − x v y e − v y e − v y e . Proof.
Linearize the system ( S ). We omit the calculations.50n order to solve ( S ) in quasi-linear time in the precision, it is enough tosolve equation ( E n ) in quasi-linear time in n . One difficulty here, that doesnot appear in similar works [CE15; CMS+19], is that the matrix M is notinvertible in k ′ [[ z ]] . Still, we can adapt the generic divide-and-conquer algorithmfrom [BCG+17, §13.2]. Lemma 5.6.
The determinant det M ( z ) = x − x y y has valuation one.Proof. We know that y and y have constant term ± y = 0 . The polynomi-als x and x have the same constant term x , but they do not coincide atorder : if they did, then so would y and y because of the curve equation,contradicting Lemma 5.3.By Lemma 5.6, we can find I ∈ M (cid:0) k [[ z ]] (cid:1) such that IM = (cid:18) z z (cid:19) . Lemma 5.7.
Let κ ≥ , and assume that char k > κ + 1 . Let A = IN . Thenthe matrix A + κ has an invertible constant term.Proof. By Lemma 5.6, the leading term of det( M ) is λz for some nonzero λ ∈ k ′ .Using Proposition 5.5, we compute that the constant term of det( A + κ ) is λ κ ( κ + 1) . We omit the calculations. Proposition 5.8.
Let ≤ ν ≤ n − , and assume that char k > ν . Thenwe can solve ( E n ) to compute δx and δx up to precision O ( z ν ) using e O ( ν ) operations in k ′ .Proof. Write θ = (cid:18) δx δx (cid:19) . Multiplying ( E n ) by I , we obtain the equation zθ ′ + ( A + κ ) θ = B + O ( z d ) , where d = 2 n − , κ = 0 . We show that θ can be computed from this kind of equation up to O ( z d ) usinga divide-and-conquer strategy. If d > , write θ = θ + z d θ where d = ⌊ d/ ⌋ .Then we have zθ ′ + ( A + κ ) θ = B + O ( z d ) for some other series B . By induction, we can recover θ up to O ( z d ) . Then zθ ′ + ( A + κ + d ) θ = E + O ( z d − d ) where E has an expression in terms of θ . This is enough to recover θ upto O ( z n − − d ) , so we can recover θ up to O ( z n − ) . We initialize the inductionwith the case d = 1 , where we have to solve for the constant term in ( A + κ ) θ = B. θ starts at z , the values of κ that occur are , . . . , ν − when computingthe solution of ( S ) up to precision O ( z ν ) . By Lemma 5.7, the constant term of A + κ is invertible. This concludes the induction, and the result follows fromstandard lemmas in computer algebra [BCG+17, Lem. 1.12]. Proposition 5.9.
Let ν ≥ , and assume that char k > ν . Then we can computethe lift e ϕ P up to precision O ( z ν ) within e O ( ν ) operations in k ′ .Proof. This is a consequence of Proposition 5.8 and [BCG+17, Lem. 1.12].
Finally, we want to recover the rational representation ( s, p, q, r ) of ϕ at P fromits power series expansion e ϕ P at some finite precision. First, we estimate thedegrees of the rational fractions we want to compute; second, we present thereconstruction algorithm. Degree estimates.
The degrees of s, p, q, r as morphisms from C to P canbe computed as intersection numbers of divisors on Jac( C ′ ) , namely ϕ P ( C ) andthe polar divisors of s , p , q and r as functions on Jac( C ′ ) . They are alreadyknown in the case of an ℓ -isogeny. Proposition 5.10 ([CE15, §6.1]) . Let ϕ : Jac( C ) → Jac( C ′ ) be an ℓ -isogeny,and let P ∈ C ( k ) . Let ( s, p, q, r ) be the rational representation of ϕ at the basepoint P . Then the degrees of s , p , q and r as morphisms from C to P are ℓ , ℓ , ℓ , and ℓ respectively. Now assume that
Jac( C ) and Jac( C ′ ) have real multiplication by Z K givenby embeddings ι, ι ′ , and that ϕ : (cid:0) Jac( C ) , ι (cid:1) → (cid:0) Jac( C ′ ) , ι ′ (cid:1) is a β -isogeny. Denote the theta divisors on Jac( C ) and Jac( C ′ ) by Θ and Θ ′ respectively, and denote by η P : C →
Jac( C ) the map Q [ Q − P ] . Then η P ( C ) is algebraically equivalent to Θ . Lemma 5.11.
The polar divisors of s, p, q, r as rational functions on
Jac( C ′ ) are algebraically equivalent to ′ , ′ , ′ and ′ respectively.Proof. See [CE15, §6.1]. For instance, s = x + x has a pole of order alongeach of the two divisors (cid:8) ( ∞ ± , Q ) | Q ∈ C (cid:9) , where ∞ ± are the two points atinfinity on C , assuming that we choose a degree 6 hyperelliptic model for C ′ .Each of these divisors is algebraically equivalent to Θ ′ . The proof for p , q , and r is similar.Recall that divisor classes on Jac( C ′ ) are in bijective correspondence withisomorphism classes of line bundles. By Theorem 2.17, if ( A, ι ) is a principallypolarized abelian surface with real multiplication by Z K , then there is a bijection α
7→ L ι ( α ) A between Z K and the Néron–Severi group of A .52 emma 5.12. Let ϕ be a β -isogeny as above. Then the divisor ϕ P ( C ) is alge-braically equivalent to the divisor corresponding to the line bundle L ι ′ ( β )Jac( C ′ ) .Proof. By Theorem 2.17, there exists an α ∈ Z K such that the divisor ϕ P ( C ) corresponds to the line bundle L ι ′ ( α )Jac( C ′ ) up to algebraic equivalence. Look at thepullback ϕ ∗ (cid:0) ϕ P ( C ) (cid:1) as a divisor on Jac( C ) : by definition, we have ϕ ∗ (cid:0) ϕ P ( C ) (cid:1) = X x ∈ ker ϕ (cid:0) x + η P ( C ) (cid:1) and therefore, up to algebraic equivalence, ϕ ∗ (cid:0) ϕ P ( C ) (cid:1) = ( ϕ )Θ = N K/ Q ( β )Θ . Since ϕ is a β -isogeny, by Definition 2.18, the pullback ϕ ∗ Θ ′ corresponds to L ι ( β )Jac( C ) up to algebraic equivalence. Therefore, for every γ ∈ Z K , ϕ ∗ L ι ′ ( γ )Jac( C ′ ) = L ι ( γβ )Jac( C ) . By Theorem 2.17 applied on
Jac( C ) , we have αβ = N K/ Q ( β ) , so α = β .The next step is to compute the intersection degree of Θ ′ and the divisorcorresponding to L ι ( α )Jac( C ′ ) on Jac( C ′ ) , for every α ∈ Z K . Proposition 5.13 ([Kan19, Rem. 16]) . Let ( A, ι ) be a principally polarizedabelian surface with real multiplication by Z K , and let Θ be its theta divisor.Then the quadratic form D ( D · Θ) − D · D ) on NS( A ) corresponds via the isomorphism of Theorem 2.17 to the quadraticform on Z K given by α K/ Q ( α ) −
12 Tr K/ Q ( α ) . Corollary 5.14.
Let ( A, ι ) be a principally polarized abelian surface with realmultiplication by Z K , and let Θ be its theta divisor. Then for every α ∈ Z K , wehave (cid:0) L ι ( α ) A · Θ (cid:1) = Tr K/ Q ( α ) . Proof.
Write α = a + b √ ∆ . By Proposition 5.13, we can compute (cid:0) L ι ( α ) A · Θ (cid:1) − (cid:0) L ι ( α ) A · L ι ( α ) A (cid:1) = 2 Tr( α ) −
12 Tr( α ) = 4 b ∆ . On the other hand, the Riemann–Roch theorem [Mil86a, Thm. 11.1] gives (cid:0) L ι ( α ) A · L ι ( α ) A (cid:1) = 2 χ (cid:0) L ι ( α ) A (cid:1) = 2 p deg ι ( α ) = 2( a − b ∆) . The result follows. 53 roposition 5.15.
Let ϕ be a β -isogeny as above, and let ( s, p, q, r ) be therational representation of ϕ at P . Then, considered as morphisms from C to P ,the respective degrees of s , p , q , and r are β ) , β ) , β ) and β ) .Proof. The degrees of s, p, q, r can be computed as the intersection of the po-lar divisors from Lemma 5.11 and the divisor ϕ P ( C ) . By Lemma 5.12, theline bundle associated with ϕ P ( C ) , up to algebraic equivalence, is L β Jac( C ′ ) . Itsintersection number with Θ ′ is nonnegative, hence by Corollary 5.14, we have (cid:0) ϕ P ( C ) · Θ ′ (cid:1) = Tr K/ Q ( β ) = Tr K/ Q ( β ) . The result follows by Lemma 5.11.
Rational reconstruction.
Now we present the rational reconstruction algo-rithm, and compute the power series precision that is precisely needed.
Lemma 5.16.
Let s : C → P be a morphism of degree d .1. If s is invariant under the hyperelliptic involution i , then we can write s ( u, v ) = X ( u ) where the degree of X is bounded by d/ .2. In general, let X , Y be the rational fractions such that s ( u, v ) = X ( u ) + v Y ( u ) . Then the degrees of X and Y are bounded by d and d + 3 respectively.Proof. For item , use the fact that the function u itself has degree . Foritem , write s ( u, v ) + s ( u, − v ) = 2 X ( u ) , s ( u, v ) − s ( u, − v ) v = 2 Y ( u ) . The degrees of these morphisms are bounded by d and d + 6 respectively. Proposition 5.17.
Let e ϕ P and e ϕ i ( P ) be local lifts of ϕ P at P and i ( P ) in theuniformizers z and i ( z ) . Let ν = 8 ℓ + 7 in the Siegel case, and ν = 4 Tr K/ Q ( β ) +7 in the Hilbert case. Then, given e ϕ P and e ϕ i ( P ) at precision O ( z ν ) , we cancompute the rational representation of ϕ at P within e O ( ν ) operations in k ′ .Proof. It is enough to recover the rational fractions s and p ; afterwards, q and r can be deduced from the equation of C ′ .First, assume that P is a Weierstrass point of C . Then s , p are invariant underthe hyperelliptic involution. Therefore we have to recover univariate rationalfractions in u of degree d ≤ ℓ (resp. d ≤ Tr( β ) ). This can be done in quasi-lineartime from their power series expansion up to precision O ( u d +1 ) [BCG+17, §7.1].Since u has valuation in z , we need to compute e ϕ P at precision O ( z d +1 ) .Second, assume that P is not a Weierstrass point of C . Then the seriesdefining s ( u, − v ) and p ( u, − v ) are given by e ϕ i ( P ) . We now have to computerational fractions of degree d ≤ ℓ + 3 (resp. d ≤ β ) + 3 ) in u . Since u hasvaluation in z , this can be done in quasi-linear time if e ϕ P and e ϕ i ( P ) are knownup to precision O ( z d +1 ) . 54 Summary of the algorithm
In this final section, we summarize the isogeny algorithm and prove Theorem 1.1.We also state the analogous result in the case of β -isogenies (Theorem 6.3). Algorithm 6.1.
Let j, j ′ the Igusa invariants of principally polarized abelianvarieties A, A ′ over k . Assume that A, A ′ are ℓ -isogenous (the Siegel case), orthat A, A ′ have real multiplication by Z K and are β -isogenous (the Hilbert case).1. Use Mestre’s algorithm [Mes91] to construct curve equations C , C ′ whoseJacobians are isomorphic to A, A ′ . In the Hilbert case, use Algorithm 3.25to ensure that C , C ′ are potentially Hilbert-normalized.2. Compute at most candidates for the tangent matrix of the isogeny ϕ using Proposition 4.26 in the Siegel case, or Proposition 4.27 in the Hilbertcase. Run the rest of the algorithm for all the candidates; in general, onlyone will produce meaningful results.3. Choose a base point P on C such that ϕ P is of generic type, and com-pute the power series e ϕ P and e ϕ i ( P ) up to precision O (cid:0) z ℓ +7 (cid:1) , respec-tively O (cid:0) z β )+7 (cid:1) using Proposition 5.9.4. Recover the rational representation of ϕ at P using Proposition 5.17.We recall the statement of Theorem 1.1 from the introduction. Theorem 6.2.
Let ℓ be a prime, and let k be a field such that char k = 0 or char k > ℓ + 7 . Let U ⊂ A ( k ) be the open set consisting of abelian surfaces A such that Aut( A ) ≃ {± } and j ( A ) = 0 . Assume that there is an algorithm toevaluate derivatives of modular equations of level ℓ at a given point of U × U over k using C eval ( ℓ ) operations in k .Let A, A ′ ∈ U , and let j ( A ) , j ( A ′ ) be their Igusa invariants. Assume that A and A ′ are ℓ -isogenous over k , and that the subvariety of A × A cut out bythe modular equations Ψ ℓ,i for ≤ i ≤ is normal at ( j ( A ) , j ( A ′ )) . Then,given j ( A ) and j ( A ′ ) , Algorithm 6.1 succeeds and returns1. a field extension k ′ /k of degree dividing ,2. hyperelliptic curve equations C , C ′ over k ′ whose Jacobians are isomorphicto A, A ′ respectively,3. a point P ∈ C ( k ′ ) ,4. the rational representation ( s, p, q, r ) ∈ k ′ ( u, v ) of an ℓ -isogeny ϕ : Jac( C ) → Jac( C ′ ) at P .The cost of Algorithm 6.1 in the Siegel case is O (cid:0) C eval ( ℓ ) (cid:1) + e O ( ℓ ) elementaryoperations and O (1) square roots in k ′ . roof. Mestre’s algorithm returns curve equations C , C ′ defined over extensionsof k of degree at most , and costs O (1) operations in k and O (1) squareroots. Under our hypotheses, Proposition 4.26 applies and allows us to recover Sym ( dϕ ) using O ( C eval ( ℓ )) + O (1) operations in k . We recover dϕ up to signusing O (1) square roots and elementary operations; since ϕ is defined over k ,extending the base field is not necessary. We choose the base point P on C such that ϕ P is of generic type using Proposition 5.4, perhaps taking anotherextension of degree . By Proposition 5.9, we can compute the local lifts e ϕ P and e ϕ i ( P ) up to precision ℓ +7 within e O ( ℓ ) field operations; this is where we usethe hypothesis on char k . Finally, we recover the rational representation at P using a further e O ( ℓ ) field operations by Proposition 5.17. The result is definedover an extension of k of degree dividing .We conclude with the analogue of Theorem 6.2 in the Hilbert case. Theorem 6.3.
Let K be a real quadratic field, and let β ∈ Z K be a totallypositive prime. Let k be a field such that char k = 0 or char k > K/ Q ( β ) + 7 .Assume that there is an algorithm to evaluate derivatives of modular equationsof level β at a given point ( j, j ′ ) over k using C eval ( β ) operations in k .Let A, A ′ be principally polarized abelian surfaces over k with real multipli-cation by Z K whose Igusa invariants j ( A ) , j ( A ′ ) are well defined, and assumethat there exists a β -isogeny ϕ : A → A ′ defined over k which is generic in thesense of Definition 3.18. Then, given j ( A ) and j ( A ′ ) , Algorithm 6.1 succeedsand returns1. a field extension k ′ /k of degree dividing ,2. hyperelliptic curve equations C , C ′ over k ′ whose Jacobians are isomorphicto A, A ′ respectively,3. a point P ∈ C ( k ′ ) ,4. at most possible values for the rational representation ( s, p, q, r ) ∈ k ′ ( u, v ) of a β - or β -isogeny ϕ : Jac( C ) → Jac( C ′ ) at P .The cost of Algorithm 6.1 in the Hilbert case is O (cid:0) C eval ( β ) (cid:1) + e O (cid:0) Tr K/ Q ( β ) (cid:1) + O K (1) elementary operations and O (1) square roots in k ′ ; the implied constants, O K (1) excepted, are independent of K . Note that C eval ( β ) also depends on K . We expect that the algorithm returnsonly one answer for the rational representation of ϕ at P ; if the algorithmoutputs several answers, we could implements tests for correctness, but theymight be more expensive than the isogeny algorithm itself. Proof.
We use Algorithm 3.25 to construct the curve equations C , C ′ . By Re-mark 4.24, we obtain potentially Hilbert-normalized curves, and each of themis defined over an extension of k of degree at most . This requires O K (1) ele-mentary operations and O (1) square roots in k . We may assume that C , C ′ are56ilbert-normalized for some choice of real multiplication embeddings that arecompatible via ϕ , which becomes either a β - or a β -isogeny.Under our hypotheses, Proposition 4.27 applies, so we recover two possiblevalues for ( dϕ ) within O ( C eval ( β )) + O (1) operations in k , and hence possiblevalues for dϕ , using O (1) square roots. We can now make a change of variablesto the (not necessarily Hilbert-normalized) curves output by Mestre’s algorithm,so that each curve is defined over an extension of k of degree . The end of thealgorithm is similar to the Siegel case: we take an extension of degree to findthe base point, then try to compute the rational representation for each valueof dϕ using e O (Tr K/ Q ( β )) operations in k . For the correct value of dϕ , rationalreconstruction will succeed and output fractions of the correct degrees. References [AOV08] D. Abramovich, M. Olsson, and A. Vistoli. “Tame stacks in posi-tive characteristic”. In:
Ann. Inst. Fourier (Grenoble)
J. Amer. Math. Soc.
Ann. Inst.Fourier (Grenoble)
Algebr. Geom. .[AHR20] J. Alper, J. Hall, and D. Rydh. “A Luna étale slice theorem foralgebraic stacks”. In:
Ann. Math.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)
Ann. of Math. (2)
84 (1966), pp. 442–528.[BGL+16] S. Ballentine, A. Guillevic, E. Lorenzo García, C. Martindale, M.Massierer, B. Smith, and J. Top. “Isogenies for point countingon genus two hyperelliptic curves with maximal real multiplica-tion”. In:
Algebraic Geometry for Coding Theory and Cryptography .Vol. 9. Los Angeles: Springer, Feb. 2016, pp. 63–94.[BL04] C. Birkenhake and H. Lange.
Complex abelian varieties . Second.Springer-Verlag, Berlin, 2004.57BMS+08] A. Bostan, F. Morain, B. Salvy, and É. Schost. “Fast algorithmsfor computing isogenies between elliptic curves”. In:
Math. Comp.
Algorithmes efficaces en calcul formel . Printed byCreateSpace, 2017.[Bou72] N. Bourbaki.
Groupes et algèbres de Lie. Chapitres II et III . Ac-tualités Scientifiques et Industrielles 1349. Hermann, Paris, 1972.[Bou75] N. Bourbaki.
Groupes et algèbres de Lie. Chapitres VII et VIII .Actualités Scientifiques et Industrielles 1364. Hermann, Paris, 1975.[Bru08] J. H. Bruinier. “Hilbert modular forms and their applications”. In:
The 1-2-3 of modular forms . Universitext. Springer, Berlin, 2008,pp. 105–179.[BP17] J. I. Burgos Gil and A. Pacetti. “Hecke and Sturm bounds forHilbert modular forms over real quadratic fields”. In:
Math. Comp.
Ann. of Math. (2)
Surveysin differential geometry . Vol. 7. Surv. Differ. Geom. Int. Press,Somerville, MA, 2000, pp. 61–81.[Cle72] A. Clebsch.
Theorie der binären algebraischen Formen . B. G.Teubner, Leipzig, 1872.[CFv17] F. Cléry, C. Faber, and G. van der Geer. “Covariants of binarysextics and vector-valued Siegel modular forms of genus two”. In:
Math. Ann.
Math. Comp.
Lond. Math. Soc. J. Comput. Math.
Math. Ann.
Modular functions of one variable, II (Proc. Inter-nat. Summer School, Univ. Antwerp, Antwerp, 1972) . 1973, 143–316. Lecture Notes in Math., Vol. 349.58DJR+17] A. Dudeanu, D. Jetchev, D. Robert, and M. Vuille. “Cyclic isogeniesfor abelian varieties with real multiplication”. 2017.[Dup11] R. Dupont. “Fast evaluation of modular functions using Newtoniterations and the AGM”. In:
Math. Comp.
Computational perspectives onnumber theory (Chicago, IL, 1995) . Vol. 7. AMS/IP Stud. Adv.Math. Amer. Math. Soc., Providence, RI, 1998, pp. 21–76.[ET14] A. Enge and E. Thomé.
CMH: Computation of Igusa class polyno-mials . 2014.[FC90] G. Faltings and C.-L. Chai.
Degeneration of abelian varieties .Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 22. Springer-Verlag, Berlin, 1990.[FGI+05] B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure, andA. Vistoli.
Fundamental algebraic geometry . Mathematical Surveysand Monographs 123. American Mathematical Society, Providence,RI, 2005.[GKS11] P. Gaudry, D. Kohel, and B. Smith. “Counting points on genus 2curves with real multiplication”. In:
ASIACRYPT 2011 . Vol. 7073.Lecture Notes in Computer Science. Seoul, South Korea: Springer,2011, pp. 504–519.[GS12] P. Gaudry and É. Schost. “Genus 2 point counting over primefields”. In:
J. Symb. Comput.
IMRN
Inst.Hautes Études Sci. Publ. Math.
20 (1964), p. 259.[Gru10] D. Gruenewald. “Computing Humbert surfaces and applications”.In:
Arithmetic, geometry, cryptography and coding theory 2009 .Vol. 521. Contemp. Math. Amer. Math. Soc., Providence, RI,2010, pp. 59–69.[Ibu12] T. Ibukiyama. “Vector-valued Siegel modular forms of symmetrictensor weight of small degrees”. In:
Comment. Math. Univ. St.Pauli
Amer. J.Math.
84 (1962), pp. 175–200.[Igu67] J.-I. Igusa. “Modular forms and projective invariants”. In:
Amer.J. Math.
89 (1967), pp. 817–855.[Igu79] J.-I. Igusa. “On the ring of modular forms of degree two over Z ”.In: Amer. J. Math.
Ann.of Math. (2)
72 (1960), pp. 612–649.[JL06] A. Joux and R. Lercier. “Counting points on elliptic curves inmedium characteristic”. 2006.[Kan19] E. Kani. “Elliptic subcovers of a curve of genus 2. I. The isogenydefect”. In:
Ann. Math. Qué.
Ann. of Math.(2)
Ann. of Math. (2)
67 (1958), pp. 328–401.[LM00] G. Laumon and L. Moret-Bailly.
Champs algébriques . Ergebnisseder Mathematik und ihrer Grenzgebiete (3) 39. Springer-Verlag,Berlin, 2000.[LY11] K. Lauter and T. Yang. “Computing genus 2 curves from invariantson the Hilbert moduli space”. In:
J. Number Theory et leur schéma de modules”.In: Math. Ann.
Sur les groupes algébriques . 1973,81–105. Bull. Soc. Math. France, Paris, Mémoire 33.[Mar18] C. Martindale. “Isogeny graphs, modular polynomials, and applica-tions”. PhD thesis. Universiteit Leiden and Université de Bordeaux,2018.[Mes91] J.-F. Mestre. “Construction de courbes de genre à partir de leursmodules”. In: Effective methods in algebraic geometry (Castiglion-cello, 1990) . Vol. 94. Progr. Math. Birkhäuser, Boston, 1991,pp. 313–334.[Mil] E. Milio.
Modular polynomials . https://members.loria.fr/EMilio/modular-polynomials.[Mil15] E. Milio. “A quasi-linear time algorithm for computing modularpolynomials in dimension 2”. In:
LMS J. Comput. Math.
18 (2015),pp. 603–632.[MR17] E. Milio and D. Robert. “Modular polynomials on Hilbert surfaces”.Sept. 2017.[Mil86a] J. S. Milne. “Abelian varieties”. In:
Arithmetic geometry (Storrs,Conn., 1984) . Springer, New York, 1986, pp. 103–150.[Mil86b] J. S. Milne. “Jacobian varieties”. In:
Arithmetic geometry (Storrs,Conn., 1984) . Springer, New York, 1986, pp. 167–212.[Mol18] P. Molin.
Hcperiods: Period matrices and Abel-Jacobi maps of hy-perelliptic and superperelliptic curves . 2018.60MN19] P. Molin and C. Neurohr. “Computing period matrices and theAbel-Jacobi map of superelliptic curves”. In:
Math. Comp.
Abelian varieties .2012.[MFK94] D. Mumford, J. Fogarty, and F. Kirwan.
Geometric invariant the-ory . Third edition. Ergebnisse der Mathematik und ihrer Grenzge-biete (2) 34. Springer-Verlag, Berlin, 1994.[Mum71] D. Mumford. “The structure of the moduli spaces of curves andabelian varieties”. In:
Actes du Congrès International des Mathé-maticiens (Nice, 1970), Tome 1 . 1971, pp. 457–465.[Mum84] D. Mumford.
Tata lectures on theta. II . Progr. Math. 43. Birkhäuser,Boston, 1984.[Nag83] S. Nagaoka. “On the ring of Hilbert modular forms over Z ”. In: J.Math. Soc. Japan
Hom -stacks and restriction of scalars”. In:
DukeMath. J.
Compositio Math.
Math. Z.
J.Algebraic Geom. p ”. In: Math. Comp.
J. Théorie Nr. Bordx.
Pari/GP version 2.11.0 . Univ. Bordeaux, 2019.[The18] The Stacks project authors. “The Stacks Project”. 2018.[Tho70] J. Thomae. “Beitrag zur Bestimmung von ϑ (0 , , . . . durch dieKlassenmoduln algebraischer Functionen”. In: J. Reine Angew.Math.
71 (1870), pp. 201–222.[van08] G. van der Geer. “Siegel modular forms and their applications”. In:
The 1-2-3 of modular forms . Universitext. Springer, Berlin, 2008,pp. 181–245. 61
The case K = Q ( √ We present a variant of our algorithm in the case of principally polarized abelianvarieties with real multiplication by Z K where K = Q ( √ . In this case, thestructure of the ring of Hilbert modular form is well known, and the Humbertsurface is rational: its function field can be generated by only two elementscalled Gundlach invariants . Having only two coordinates reduces the size ofmodular equations.We work over C , but the methods of §4 show that the computations arevalid in general. We illustrate our algorithm with an example of cyclic isogenyof degree over a finite field. A.1 Hilbert modular forms for K = Q ( √ We keep the notation used to describe the Hilbert embedding (§2.4). Hilbertmodular forms have Fourier expansions in terms of w = exp (cid:0) πi ( e t + e t ) (cid:1) and w = exp (cid:0) πi ( e t + e t ) (cid:1) . We use this notation and the term w -expansions to avoid confusion with expan-sions of Siegel modular forms. Apart from the constant term, a term in w a w b can only appear when ae + be is a totally positive element of Z K . Since e = 1 and e has negative norm, for a given a , only finitely many b ’s appear. Thereforewe can consider truncations of w -expansions as elements of C ( w )[[ w ]] moduloan ideal of the form ( w ν ) . Theorem A.1 ([Nag83]) . The graded C -algebra of symmetric Hilbert modularforms of even parallel weight for K = Q ( √ is generated by three elements G , F , F of respective weights , and , with w -expansions G ( t ) = 1 + (120 w + 120) w + (cid:0) w + 600 w + 720 w + 600 + 120 w − (cid:1) w + O ( w ) ,F ( t ) = ( w + 1) w + (cid:0) w + 20 w − w + 20 + w − (cid:1) w + O ( w ) ,F ( t ) = ( w − w + 1) w + O ( w ) . The
Gundlach invariants for K = Q ( √ are g = G F and g = G F F . Recall that we denote by σ the involution ( t , t ) ( t , t ) of H ( C ) Proposition A.2.
The Gundlach invariants define a birational map H ( C ) /σ → C . Proof.
This is a consequence of the theorem of Baily and Borel [BB66, Thm. 10.11]and Theorem A.1. 62y Proposition 2.14, the pullbacks of the Siegel modular forms ψ , ψ , χ and χ via the Hilbert embedding H are symmetric Hilbert modular forms ofeven weight, so they have expressions in terms of G , F , F . These expressionscan be computed using linear algebra on Fourier expansions [MR17, Prop. 2.12]:in our case, the Hilbert embedding is defined by e = 1 , e = (1 − √ / , so q = w , q = w , q = w w . As a corollary, we obtain the expression for the pullback of Igusa invariants.
Proposition A.3 ([MR17, Cor. 2.14]) . In the case K = Q ( √ , we have H ∗ j = 8 g (cid:18) g g − (cid:19) ,H ∗ j = 12 g (cid:18) g g − (cid:19) ,H ∗ j = 18 g (cid:18) g g − (cid:19) (cid:18) g g + 2 g g − (cid:19) . Let β ∈ Z K be a totally positive prime. We call the Hilbert modular equationsof level β in Gundlach invariants the data of the two polynomials Ψ β, , Ψ β, ∈ C ( G , G )[ G ′ ] defined as follows:• Ψ β, is the univariate minimal polynomial of the function g ( t/β ) over thefield C (cid:0) g ( t ) , g ( t ) (cid:1) .• We have the following equality of meromorphic functions on H ( C ) : g ( t/β ) = Ψ β, (cid:0) g ( t ) , g ( t ) , g ( t/β ) (cid:1) . Modular equations using Gundlach invariants for K = Q ( √ also have denom-inators. They have been computed up to N K/ Q ( β ) = 41 [Mil]. A.2 Variants in the isogeny algorithm
Constructing potentially Hilbert-normalized curves.
We give anothermethod to reconstruct such curves using the pullback of the modular form f , from Example 2.8 as a Hilbert modular form. Let H : H → H be the Hilbertembedding from §2.4. Proposition A.4.
Define the functions b i ( t ) for ≤ i ≤ on H by ∀ t ∈ H , det Sym ( R ) f , (cid:0) H ( t ) (cid:1) = X i =0 b i ( t ) x i . hen b and b are identically zero, and b = 4 F F ,b b = 3625 F F − F G ,b b = − F F + 15 F G ,b (cid:0) b b + b b (cid:1) = 123 F F − F F G + 288125 F F G − F . Proof.
By Proposition 2.14, each coefficient b i is a Hilbert modular form ofweight (8 + i, − i ) . We can check using the action of M σ that σ exchanges b i and b − i . From the Siegel q -expansion for f , , we can compute the w -expansionsof the b i ’s; then, we use linear algebra to identify symmetric combinations ofthe b i ’s of parallel even weight in terms of the generators G , F , F .By Propositions 3.6 and 3.12, the standard curve C K ( t ) attached to t ∈ H is proportional to the curve y = P b i ( t ) x i . The algorithm to compute apotentially Hilbert-normalized curve C from its Igusa invariants ( j , j , j ) runsas follows. Algorithm A.5.
1. Compute Gundlach invariants ( g , g ) mapping to theIgusa invariants ( j , j , j ) via H using Proposition A.3, and computevalues for the generators G , F , F giving these invariants.2. Compute b , b b , etc. using Proposition A.4.3. Recover values for the coefficients: choose any square root for b ; chooseany value for b , which gives b ; finally, solve a quadratic equation tofind b and b .We can always choose values G , F , F such that b is a square in k ;then, the output is defined over a quadratic extension of k . Even if arbitrarychoices are made during Algorithm A.5, the output will be potentially Hilbert-normalized. Computing the tangent matrix.
Consider Ψ β, and Ψ β, as elements ofthe ring Q ( G , G )[ G ′ , G ′ ] . Define the × matrices D Ψ β,L = (cid:18) ∂ Ψ n ∂G k (cid:19) ≤ n,k ≤ and D Ψ β,R = (cid:18) ∂ Ψ n ∂G ′ k (cid:19) ≤ n,k ≤ . Then we have an analogue of Proposition 4.27, where we replace derivatives ofIgusa invariants in Proposition 3.19 by derivatives of Gundlach invariants. Therelation between these derivatives is given by Proposition A.3. This time, usingthe formalism of §4, we can prove that all × matrices will be invertible ifthe abelian varieties A, A ′ have only Z × K as automorphisms, have g = 0 , andif the modular equations in Gundlach invariants cut out a normal subvarietyof A × A at ( g ( A ) , g ( A ′ )) . 64 .3 An example of cyclic isogeny We illustrate our algorithm in the Hilbert case with K = Q ( √ by computinga β -isogeny between Jacobians with real multiplication by Z K , where β = 3 + 1 + √ ∈ Z K , N K/ Q ( β ) = 11 , Tr K/ Q ( β ) = 7 . We work over the prime finite field k = F , whose characteristic is largeenough for our purposes. We choose a trivialization of Z K ⊗ k , in other wordsa square root of in k , so that β = 26213 .Consider the Gundlach invariants ( g , g ) = (cid:0) , (cid:1) , ( g ′ , g ′ ) = (cid:0) , (cid:1) . In order to reconstruct a Hilbert-normalized curve, we apply Algorithm A.5.We obtain the curve equations C : v = 13425 u + 34724 u + 102 u + 54150 u + 11111 C ′ : y = 47601 x + 35850 x + 40476 x + 24699 x + 40502 . The derivatives of Gundlach invariants are given by D t G ( C ) = (cid:18) (cid:19) , D t G ( C ′ ) = (cid:18) (cid:19) . Computing derivatives of the modular equations as in Proposition 3.19, we findthat the isogeny is compatible with the real multiplication embeddings for which C , C ′ are Hilbert-normalized. We do not known whether ϕ is a β - or a β -isogeny,so we have four candidates for the tangent matrix up to sign: dϕ β, ± = (cid:18) α + 19466 00 ± (53318 α + 26659) (cid:19) ,dϕ β, ± = (cid:18) α + 53481 00 ± (11076 α + 5538) (cid:19) where α + α + 2 = 0 . We see that the isogeny is only defined over a quadraticextension of k .The curve C has a rational Weierstrass point (cid:0) , (cid:1) . We can bring it to (0 , , so that C is of the standard form C : v = 33461 u + 7399 u + 16387 u + 34825 u + 14713 u + u. This multiplies the tangent matrix on the right by (cid:18) (cid:19) . Choose P = (0 , as a base point on C , and z = √ u as a uniformizer; itis a Weierstrass point, and we check that ϕ P is of generic type. We solve the65ifferential system up to precision O ( z ) , or any higher precision. It turns outthat the correct tangent matrix is dϕ β, + as the other series do not come fromrational fractions of the prescribed degree. We obtain s ( u ) = 50255 u + 40618 u + 17196 u + 9527 u + 22804 u + 49419 u + 11726 u + 40883 u + 22913 u + 41828 u + 18069 u + 14612 u + 7238 ,p ( u ) = 35444 u + 9569 u + 52568 u + 3347 u + 9325 u + 32206 u + 7231 u + 40883 u + 22913 u + 41828 u + 18069 u + 14612 u + 7238 . The degrees agree with Proposition 5.15. The isogeny is k -rational at the levelof Kummer surfaces, but not on the Jacobians themselves: α appears on thenumerator of r ( u, v ))