Degree and height estimates for modular equations on PEL Shimura varieties
aa r X i v : . [ m a t h . AG ] M a r Degree and height estimates for modularequations on PEL Shimura varieties
Jean Kieffer
Université de Bordeaux [email protected]
Abstract
We define modular equations in the setting of PEL Shimura varieties asequations describing Hecke correspondences, and prove degree and heightbounds for them. This extends known results about classical modularpolynomials. In particular, we obtain tight degree bounds for modularequations of Siegel and Hilbert type for abelian surfaces. One step consistsin proving tight bounds on the heights of rational fractions over numberfields in terms of the heights of their evaluations; the results we obtainare of independent interest.
Modular equations encode the presence of isogenies between polarized abelianvarieties. The classical modular polynomial Φ ℓ is an example: this bivariatepolynomial vanishes on the j -invariants of ℓ -isogenous elliptic curves [9, §11.C],and can be used to detect and compute such isogenies. Analogues of Φ ℓ for prin-cipally polarized abelian surfaces, called Siegel and Hilbert modular polynomialsin dimension , have recently been defined and computed [18, 19, 15].In this paper, we define modular equations in the general setting of PELShimura varieties of finite level; these varieties are moduli spaces for abelianvarieties with Polarization, Endomorphisms, and Level structure. Let H δ be aHecke correspondence of degree d ( δ ) on such a Shimura variety. In the modularinterpretation, H δ parametrizes isogenies of a certain degree ℓ ( δ ) . Given con-nected components S , S ′ of the Shimura variety and a choice of invariants onthem, we define modular equations Ψ δ,m as a set of polynomials describing H δ on S × S ′ . The coefficients of Ψ δ,m are rational fractions defined over a numberfield, and we prove that their degrees and heights are governed by d ( δ ) and ℓ ( δ ) . Theorem 1.1.
Let S , S ′ be connected components of a simple PEL Shimuravariety of type (A) or (C) of finite level, and choose invariants on them. Thenthere exists explicit constants C , C depending on this setting such that thefollowing holds. Let H δ be a Hecke correspondence of degree d ( δ ) describingisogenies of degree ℓ ( δ ) between abelian varieties with PEL structure, and assumethat H ( δ ) intersects S × S ′ nontrivially. Let F be a rational fraction occurringas a coefficient of one of the modular equations Ψ δ,m . Then1. The total degree of F is bounded by C d ( δ ) . The same estimate holds ifwe require all the fractions F to share a common denominator.2. The height of F is bounded by C d ( δ ) log ℓ ( δ ) . Φ ℓ , which has coefficients in Z , is symmetric, and has degree ℓ + 1 inboth variables. The height h (Φ ℓ ) is then the maximum value of log | c | , where c ranges over the coefficients of Φ ℓ . We have h (Φ ℓ ) ∼ ℓ log( ℓ ) as ℓ grows [8], andexplicit bounds can be given [5]. In particular Theorem 1.1 seems optimal up tothe value of the constants. For abelian surfaces, even degree bounds for modularequations were unknown, and the degree bounds we obtain match exactly withexperimental data.From the algorithmic point of view, Theorem 1.1 provides complexity boundsfor algorithms involving modular equations. For instance, classical modularpolynomials are used for instance in the SEA algorithm to count points onelliptic curves [26], and in multi-modular methods to compute class polynomialsof imaginary quadratic fields [28]; being able to compute isogenies also hasapplications in elliptic curve cryptography. General modular equations are ofsimilar interest.The strategy to prove part 1 of Theorem 1.1 is to exhibit a particular modularform that behaves as the denominator of Ψ δ,m , and to control its weight; then, weshow that rewriting quotients of modular forms in terms of invariants transformsbounded weight into bounded degree. The proof of part 2 is inspired by previousworks on Φ ℓ [24]. We prove height bounds on evaluations of modular equationsat certain points using well-known results on the Faltings height of isogenousabelian surfaces. To obtain bounds on coefficients of modular equations, weneed to bound the height of a rational fraction from the heights of its valuesat certain points. Since the bounds obtained from a direct analysis of theinterpolation algorithm are inefficient in our setting, it is crucial to use moreevaluation points than the required number for the interpolation problem tohave a unique solution. We prove the following result in this direction. Theorem 1.2.
Let L be a number field of degree d L . Then there exists an ex-plicit constant C , depending only on L , such that the following holds. Let J A, B K be an interval in Z ; write D = B − A and M = max {| A | , | B |} . Let F ∈ L ( Y ) bea rational fraction of degree at most d ≥ . Let S ⊂ J A, B K containing at least D/ elements and no poles of F . Let H ≥ max { , log(2 M ) } , and assume that(i) h ( F ( y )) ≤ H for every y ∈ S .(ii) D > max { d H log( dH ) , d d L } .Then we have h ( F ) ≤ H + C dH ) + 3 d log(2 M ) . In contrast, the height bound obtained from the interpolation algorithm isof the order of d H . Theorem 1.2 seems new even in the simpler case L = Q and is of independent interest.This paper is organized as follows. In Section 2, we recall the necessarybackground on PEL Shimura varieties. In Section 3, we define the modularequations associated with a choice of PEL setting. Section 4 is devoted to theproof of the degree bounds. Then we recall basic facts about heights, and prove2heorem 1.2, in Section 5. We finish the proof of height bounds for modularequations in Section 6. Notation.
Throughout the paper, the symbol C stands for an explicit con-stant depending only on the PEL setting. Its value may change from one lineto the next unless we label it explicitly as C , etc. Acknowledgements.
The author thanks Fabien Pazuki and his Ph.D. advi-sors, Damien Robert and Aurel Page, for helpful discussions and answering theauthor’s questions. The author also thanks Aurel Page for his careful proof-reading of the paper.
Our presentation is inspired by Milne’s [21], which serves as a general referencefor this section. These notes are themselves based on Deligne’s reformulationof Shimura’s works [10]. We use the following notation: if G is a connectedreductive algebraic group over Q , then • G der is the derived group of G , • Z is the center of G , • G ad = G/Z is the adjoint group of G , • T = G/G der is the largest abelian quotient of G , • ν : G → T is the natural quotient map, • G ad ( R ) + is the connected component of 1 in G ad ( R ) for the real topology, • G ( R ) + is the preimage of G ad ( R ) + in G ( R ) , and finally • G ( Q ) + = G ( Q ) ∩ G ( R ) + .We write A f for the ring of finite adeles of Q . Let ( B, ∗ ) be a simple Q -algebra with positive involution. The center F of B isa number field; let F be the subfield of invariants under ∗ . For simplicity, wemake the technical assumption that B is of type (A) or (C) [21, Prop. 8.3].Let ( V, ψ ) be a faithful symplectic ( B, ∗ ) -module. This means that V isa faithful B -module equipped with an nondegenerate alternating Q -bilinearform ψ such that for all b ∈ B and for all u, v ∈ V , ψ ( b ∗ u, v ) = ψ ( u, bv ) . Let GL B ( V ) denote the group of automorphisms of V respecting the actionof B , and let G be its algebraic subgroup defined by G ( Q ) = (cid:8) g ∈ GL B ( V ) | ψ ( gx, gy ) = ψ ( µ ( g ) x, y ) for some µ ( g ) ∈ F × (cid:9) .
3e warn the reader that our G is denoted G in [21, §8], and that consequentlythe definition of a PEL Shimura variety used here differs slightly from Milne’s.The group G is connected and reductive, and by [21, Prop. 8.7], its derivedgroup is G der = ker( µ ) ∩ ker(det) .Let x be a complex structure on V ( R ) , meaning an endomorphism of V ( R ) such that x ◦ x = − . We say that x is positive for ψ if it commutes withthe action of B and the form ( u, v ) ψ (cid:0) u, x ( v ) (cid:1) is symmetric and positivedefinite. Such a complex structure x exists [21, Prop. 8.14]. Define X + to bethe orbit of x under the action of G ( R ) + by conjugation; the space X + is ahermitian symmetric domain [21, Cor. 5.8]. We call the pair ( G, X + ) a simplePEL Shimura datum of type (A) or (C) , or simply a PEL datum .Let K be a compact open subgroup of G ( A f ) , and let K ∞ be the stabilizerof x in G ( R ) + . The PEL Shimura variety associated with ( G, X + ) of level K is the double coset Sh K ( G, X + )( C ) = G ( Q ) + \ ( X + × G ( A f )) /K = G ( Q ) + \ ( G ( R ) + × G ( A f )) /K ∞ × K. In the first description, G ( Q ) + acts on both X + and G ( A f ) by conjugation andleft multiplication respectively, and K acts on G ( A f ) by right multiplication.When the context is clear, we omit ( G, X + ) from the notation.The projection to the second factor induces a map with connected fibersfrom Sh K ( C ) to the double coset G ( Q ) + \ G ( A f ) /K , which is finite [21, Lem. 5.12].Let C be a set of representatives in G ( A f ) for this double coset. The connectedcomponent S c of Sh K ( C ) indexed by c ∈ C can be identified with Γ c \ X + , where Γ c = G ( Q ) + ∩ cKc − is an arithmetic subgroup of Aut( X + ) [21, Lem. 5.13].Thus, the Shimura variety has a natural structure of a complex analytic space,and is an algebraic variety by the theorem of Baily and Borel [21, Thm. 3.12].Since G der is simply connected, by [21, Thm. 5.17 and Lem. 5.20], the map ν induces an isomorphism G ( Q ) + \ G ( A f ) /K ≃ ν ( G ( Q ) + ) \ T ( A f ) /ν ( K ) . Therefore the set of connected components of Sh K ( C ) is a finite abelian group.Moreover, each connected component is itself a Shimura variety with underlyinggroup G der [21, Rem. 5.23].In fact, Sh K ( G, X + ) exists as an algebraic variety defined over the reflexfield , which is a number field depending only on G and X + [21, §12-14]. Ingeneral, the field of definition of an individual connected component of Sh K ( C ) may be larger. The Shimura variety Sh K ( G, X + )( C ) has a modular interpretation in terms of isogeny classes of abelian varieties with PEL structure [21, Thm. 8.17]. Afterchoosing lattices in V , we can rephrase it in terms of isomorphism classes of4uch varieties, in a flavor similar to [6, §2.6.2]. This second description is closerto the applications we have in mind.A lattice in a topological abelian group is a cocompact and discrete subgroup.Recall that the map Λ b Λ = Λ ⊗ b Z is a bijection between lattices in V andlattices in V ( A f ) ; its inverse is intersection with V ( Q ) .Let b Λ be a lattice in V ( A f ) that is stabilized by K , and let Λ = b Λ ∩ V ( Q ) .Let O be the largest order in B stabilizing Λ . We construct a standard polarizedlattice for each connected component as follows. For c ∈ C , write b Λ c = c ( b Λ ) , Λ c = b Λ c ∩ V ( Q ) . Since c respects the action of B on V ( A f ) , the order O is again the stabilizerof b Λ c , and thus of Λ c . Choose λ c ∈ Q × + such that ψ c = λ c ψ takes integer valueson Λ c × Λ c . Define Λ ⊥ c = { v ∈ V ( Q ) | ∀ w ∈ Λ c , ψ c ( v, w ) ∈ Z } . Then Λ ⊥ c is a lattice in V containing Λ c . We call the finite group T c = Λ ⊥ c / Λ c the polarization type of (Λ c , ψ c ) .We first formulate a modular interpretation using lattices. Let Z c be the setof isomorphism classes of tuples (Λ , x, ι, φ, ηK ) where • Λ is a free Z -module of rank dim V , • x ∈ End(Λ ⊗ R ) is a complex structure on Λ ⊗ R , • ι is an embedding O ֒ → End Z (Λ) , • φ : Λ × Λ → Z is a nondegenerate alternating Z -bilinear form on Λ , • ηK is a K -orbit of b Z -linear isomorphisms of O -modules b Λ → Λ ⊗ b Z ,satisfying the following condition: ( ⋆ ) there exists an isomorphism of O -modules a : Λ → Λ c , carrying ηK to cK and x to an element of X + , such that ∃ ζ ∈ µ (Γ c ) , ∀ u, v ∈ Λ , φ ( u, v ) = ψ c (cid:0) ζa ( u ) , a ( v ) (cid:1) . Isomorphisms between tuples are isomorphisms of O -modules f : Λ → Λ ′ thatsend x to x ′ , send ηK to η ′ K , and such that φ ( u, v ) = φ ′ (cid:0) ζf ( u ) , f ( v ) (cid:1) forsome ζ ∈ µ (Γ c ) .In particular, for every (Λ , x, ι, φ, ηK ) ∈ Z c , the complex structure x ispositive for φ , the Rosati involution defined by φ coincides with ∗ on B , theaction of B on Λ ⊗ Q leaves the complex structure x invariant, and the type ofthe polarization φ on Λ is T c . Theorem 2.1.
Let c ∈ C , let S c = Γ c \ X + be the associated connected compo-nent of Sh K ( G, X + )( C ) , and define Z c as above. Then the map Z c −→ S c (Λ , x, ι, φ, ηK ) [ axa − , c ] where a is as in ( ⋆ ) s well-defined and bijective. The inverse map is [ x, c ] (Λ c , x, ι, ψ c , cK ) . where ι is the natural action of O on Λ c .Proof. The proof is direct and omitted; the details are similar to [21, Prop. 6.3].We want to rephrase Theorem 2.1 using the language of abelian varieties.Giving an abelian variety A over C is the same as giving the lattice Λ = H ( A, Z ) and a complex structure on the universal covering Λ ⊗ R of A . Giving a polar-ization on A is the same as giving an alternating nondegenerate bilinear form φ taking integral values on Λ such that ( u, v ) φ ( u, iv ) is symmetric and positivedefinite. Then, endomorphisms of A correspond to endomorphisms of Λ thatrespect the complex structure, and Λ ⊗ b Z is canonically isomorphic to the globalTate module b T ( A ) = Y ℓ prime T ℓ ( A ) . When c ∈ C is fixed, we define a complex abelian variety with PEL structure to be a tuple ( A, φ, ι, ηK ) where • ( A, φ ) is a complex polarized abelian variety with polarization type T c , • ι is an embedding O ֒ → End( A ) such that the Rosati involution on B is ∗ , • ηK is a K -orbit of b Z -linear isomorphisms of O -modules b Λ → b T ( A ) ,satisfying the following condition: ( ⋆ ) there exists an isomorphism of O -modules a : H ( A, Z ) → Λ c , carrying φ to ψ c , carrying ηK to cK , and such that theinduced complex structure on V ( R ) belongs to X + .The difference with the setting of Theorem 2.1 is that isomorphisms of po-larized abelian varieties should respect the polarizations exactly , rather than upto an element of µ (Γ c ) . In general, µ (Γ c ) is not trivial, but there is the followingworkaround. If ε ∈ F × , then multiplication by ε defines an element in the centerof G ( Q ) , so it makes sense to define E = { ε ∈ F × | ε ∈ K } = { ε ∈ F × | ε ∈ Γ c } . Theorem 2.2.
Let c ∈ C , and assume that µ ( E ) = µ (Γ c ) . Then the map [ x, c ] (cid:0) V ( R ) / Λ c , ι, cK (cid:1) , where V ( R ) is seen as a complex vector space via x , and ι is the action of O induced by the action of B on V ( R ) , is a bijection between S c and the set ofisomorphism classes of complex abelian varieties with PEL structure.Proof. When defining Z c , we can impose ζ = 1 in condition ( ⋆ ) and strengthenthe notion of isomorphism between tuples to respect the polarizations exactly:indeed, multiplying a by some ε ∈ E leaves everything invariant except thealternating form, which gets multiplied by µ ( ε ) . The result follows then fromTheorem 2.1. 6 emark 2.3. In any case, µ ( E ) has finite index in µ (Γ c ) : indeed, if Z × F denotesthe unit group of F , we have µ ( E ) ⊂ µ (Γ c ) ⊂ Z × F and E contains a subgroup of finite index in Z × F . By [7, Th. 1], there existsa compact open subgroup M of µ ( K ) such that Z × F ∩ M = µ ( E ) . Define K ′ = K ∩ µ − ( M ) , and denote by Γ ′ c the associated arithmetic subgroups.Then E remains the same for K ′ , and for c ∈ G ( A f ) we have Γ ′ c = { γ ∈ Γ c | µ ( γ ) ∈ µ ( E ) } . Thus raising the level allows us to reach the situation of Theorem 2.2, at thecost of adding connected components. A modular form of weight w ∈ Z on Sh K ( G, X + )( C ) is a function f : G ( Q ) + \ ( G ( R ) + × G ( A f )) /K → C that satisfies suitable growth and holomorphy conditions [20, Prop. 3.2], andsuch that ∀ x ∈ G ( R ) + , ∀ g ∈ G ( A f ) , ∀ k ∞ ∈ K ∞ , f ([ xk ∞ , g ]) = ρ ( k ∞ ) w f ([ x, g ]) . Here ρ : K ∞ → C × is a certain canonical character of K ∞ . The weight of f isdenoted by wt( f ) . We also say that f is of level K .From a geometric point of view, there is a line bundle M on Sh K ( C ) such thatmodular forms of weight w are the holomorphic sections of M ⊗ w that are againholomorphic when extended to the Baily-Borel compactification of Sh K ( C ) . Infact, M is the inverse determinant of the tangent bundle on Sh K [1, Prop. 7.3].Let S be a connected component of Sh K ( C ) , and L its field of definition.A modular form of weight w on S is simply the restriction to S of a weight w modular form on Sh K ( C ) . Modular forms on S generate a graded ring. Thefollowing result is well known; since we did not find a precise reference in theliterature, we present a short proof. Theorem 2.4.
The graded ring of modular forms on S is generated by finitelymany elements defined over L , and modular forms of sufficiently high weightrealize a projective embedding of S . Every meromorphic function on S is aquotient of two modular forms of the same weight.Proof. As shown by Baily and Borel [1, Thm. 10.11], modular forms of suffi-ciently high weight give a projective embedding of S which is a priori definedover C . In other words, the line bundle M is ample on S . It can be definedover L , and remains ample as a line bundle on the variety S over L . Thereforethe graded L -algebra M w ≥ H ( S , M ⊗ w )
7s finitely generated, and consists of modular forms defined over L . They alsogive a projective embedding provided the weight is high enough. The secondpart of the statement is an easy consequence of the first.We can also consider modular forms that are symmetric under certain auto-morphisms of Sh K . Let Σ be a finite group of automorphisms of G that leaves G ( Q ) + , K , K ∞ , ν and the character ρ invariant. Then for every modular form f of weight w on S , and every σ ∈ Σ , the function f σ : [ x, g ] f ([ σ ( x ) , σ ( g )]) is a modular form of weight w on S . We say that f is symmetric under Σ if f σ = f for every σ ∈ Σ . Proposition 2.5.
Let Σ be a finite group of automorphism of G as above. Thensymmetric modular forms generate a graded ring that is also finitely generatedover L , and every symmetric modular function is the quotient of two symmetricmodular forms of sufficiently high weight.Proof. This is a consequence of Theorem 2.4 and Noether’s theorem [22] oninvariants under finite groups.
Let δ ∈ G ( A f ) , and let K ′ = K ∩ δKδ − . Consider the diagram Sh K ′ Sh δ − K ′ δ Sh K Sh Kp R ( δ ) p where the map R ( δ ) is [ x, g ] [ x, gδ ] . This diagram defines a correspon-dence H δ in Sh K × Sh K , called the Hecke correspondence of level δ , consistingof all pairs of the form (cid:0) p ( x ) , p ( R ( δ ) x ) (cid:1) for x ∈ Sh K ′ . Hecke correspondencesare algebraic, and are defined over the reflex field [21, Thm. 13.6].We define the degree of H δ to be the index d ( δ ) = [ K : K ′ ] = [ K : K ∩ δKδ − ] . It is finite as both K and K ′ are compact open in G ( A f ) , and is the degree ofthe map p : H δ → Sh K . We can also consider H δ as a map from Sh K to its d ( δ ) -th symmetric power, sending z ∈ Sh K to the set { z ′ ∈ Sh K | ( z, z ′ ) ∈ H δ } .It is easy to see how H δ behaves with respect to connected components: if z lies in the connected component indexed by t ∈ T ( A f ) , then its images lie inthe connected component indexed by t ν ( δ ) .In the modular interpretation, Hecke correspondences describe isogenousabelian varieties such that the isogeny is of a certain type. Let us describe the8onstruction. After multiplying δ by a suitable element in Q × , which does notchange H δ , we can assume that δ ( b Λ ) ⊂ b Λ . Write K = d ( δ ) G i =1 κ i K ′ . Let c ∈ C , and consider the lattice with PEL structure (Λ c , x, ι, ψ c , cK ) associ-ated with a point [ x, c ] ∈ S c by Theorem 2.1.Partition the orbit cK into K ′ -orbits cκ i K ′ . Each cκ i δ is then a b Z -linearembedding of O -modules b Λ ֒ → b Λ c ; it is well defined up to right multiplicationby δ − K ′ δ , hence by K . Let Λ i ⊂ Λ c be the lattice such that Λ i ⊗ b Z is theimage of this embedding. There is still a natural action of O on Λ i . Thedecomposition cκ i δK = q i c ′ K , with q i ∈ G ( Q ) + and c ′ ∈ C , is well defined, andthe element c ′ does not depend on i . Proposition 2.6.
Let δ ∈ G ( A f ) . Let c ∈ C and let S c be the associatedconnected component of Sh K ( C ) . Let z = [ x, c ] ∈ S c , and construct Λ i , q i , c ′ asabove. Then the image of z by the Hecke correspondence H δ is given by the d ( δ ) isomorphism classes of tuples with representatives (cid:16) Λ i , x, λ c ′ λ c ψ c (cid:0) µ ( q − i ) · , · (cid:1) , cκ i δK (cid:17) for ≤ i ≤ d ( δ ) . Proof.
The images of [ x, c ] via the Hecke correspondence are the points [ q − i x, c ′ ] of Sh K ( C ) . The relation cκ i δK = q i c ′ K shows that the map q − i sends thelattice Λ i to Λ c ′ . This map also respects the action of O , and sends the complexstructure x to q − i x . Finally, it sends the polarization ( u, v ) ψ c ( u, v ) on Λ i to ( u, v ) ψ c (cid:0) µ ( q i ) u, v (cid:1) on Λ c ′ .We define the isogeny degree of H δ as ℓ ( δ ) = (cid:0)b Λ /δ ( b Λ ) (cid:1) . Corollary 2.7.
Let δ ∈ G ( A f ) . Then, in the modular interpretation of The-orem 2.2, the Hecke correspondence H δ sends an abelian variety A with PELstructure to d ( δ ) abelian varieties A , . . . , A d ( δ ) such that for every i , there existsan isogeny A → A i of degree ℓ ( δ ) .Proof. By Proposition 2.6, the Hecke correspondence consists in taking d ( δ ) suitable sublattices Λ i ⊂ Λ c , and then descending the polarization to recoversomething isomorphic to the standard lattice Λ c ′ . Since the complex structureremains the same, and Λ c / Λ i ≃ b Λ /δ ( b Λ ) , the result follows.For later purposes, we state a relation between d ( δ ) and ℓ ( δ ) . Lemma 2.8.
We have d ( δ ) ≤ C ℓ ( δ ) C . roof. Since K is open, we can find an integer N ≥ such that (cid:8) g ∈ G ( A f ) ∩ GL( b Λ ) | g = 1 mod N b Λ (cid:9) ⊂ K. Then K ∩ δKδ − contains those elements g ∈ G ( A f ) ∩ GL( b Λ ) that are theidentity modulo b Λ = N b Λ ∩ N δ ( b Λ ) . This b Λ belongs to the set Y of sublatticesof index N dim V ℓ ( δ ) in b Λ . We have Y ≤
C ℓ ( δ ) C . The group K acts on Y ,and K ∩ δKδ − contains the stabilizer of b Λ ; the claim follows. Let ( G, X + ) be a PEL datum, let K be a compact open subgroup of G ( A f ) , andlet Σ be a finite group of automorphisms of G as in §2.3. Let n be the complexdimension of X + ; we assume that n = 0 . Let S , S ′ be connected componentsof Sh K ( G, X + )( C ) , and let L be a number field over which S and S ′ are defined.To complete the picture, we also need to choose invariants , which are coor-dinates on S , S ′ given by modular functions. Since the field L ( S ) of functionson S has transcendence degree n , the field L ( S ) Σ of functions on S that aresymmetric under Σ also has transcendence degree n . Choose a transcendencebasis ( j , . . . , j n ) of L ( S ) Σ , and another symmetric function j n +1 that gener-ates the remaining finite extension. On S , the function j n +1 satisfies a minimalrelation of the form j en +1 + e − X k =0 E k ( j , . . . , j n ) j kn +1 = 0 . ( E )We proceed similarly to define a basis of functions on S ′ : no confusion willarise if we also denote them by j , . . . , j n +1 . We refer to all the data definedup to now as the PEL setting . Throughout the paper, the symbol C refers to aconstant that depends on this data only. Let δ ∈ G ( A f ) defining a Hecke correspondence H δ that intersects S × S ′ non-trivially. We want to define explicit polynomials, called modular equations oflevel δ , describing H δ in the product S × S ′ .The ring of meromorphic functions on H δ , denoted by L ( H δ ) , is a finiteextension of degree ( d ( δ ) of L ( S ) Σ . We can identify it with the ring ofmodular functions of trivial weight on e S ⊂ Sh K ′ ( C ) , where K ′ = K ∩ δKδ − and e S is the preimage of S in Sh K ′ ( C ) . Let K ′′ = \ σ ∈ Σ σ ( K ′ ) = K ∩ \ σ ∈ Σ σ ( δ ) Kσ ( δ ) − . K ⋊ Σ on modular forms for K ′′ , given by ( k, σ ) · f : [ x, g ] f σ ([ x, gk ]) . For γ ∈ K ⋊ Σ , we write this action as f f γ . The modular forms invariantunder K ′ × { } (resp. K ⋊ Σ ) are exactly the elements of L ( H δ ) (resp. L ( S ) Σ ).The functions j k,δ : [ x, g ] j k ([ x, gδ ]) for ≤ k ≤ n + 1 belong to L ( H δ ) . We define the chain of subgroups K ⋊ Σ = K ⊃ K ⊃ · · · ⊃ K n +1 ⊃ K ′ where K m is the subgroup that leaves j ,δ , . . . , j m,δ invariant, and we write d i = [ K i − : K i ] . Definition 3.1.
The modular equations of level δ on S × S ′ are the tuple (Ψ δ, , Ψ δ, , . . . , Ψ δ,n +1 ) defined as follows: for each ≤ m ≤ n + 1 , set Ψ δ,m = X γ ∈ K /K m − m − Y i =1 Y γ i (cid:16) Y i − j γ i i,δ (cid:17)! Y γ m ∈ K m − /K m (cid:16) Y m − j γγ m m,δ (cid:17) where the middle product is over all γ i ∈ K /K i such that γ i = γ mod K i − ,but γ i = γ mod K i .Therefore, Ψ δ,m is a multivariate polynomial in the m variables Y , . . . , Y m .The expression for Ψ δ,m makes sense, because multiplying γ on the right by anelement in K m − only permutes the factors in the last product. Lemma 3.2.
The coefficients of Ψ δ,m lie in L ( j , . . . , j n +1 ) . The degree of Ψ δ,m in Y m is d m , and its degree in Y i for i < m is at most d i − .Proof. It is clear from the formula that the action of K leaves Ψ δ,m invariant.Hence the coefficients of Ψ δ,m are elements of L ( S ) Σ , and this field is generatedby j , . . . , j n +1 over L . The second statement is obvious from the formula.Using the equation ( E ) satisfied by j n +1 on S , there is a unique way towrite Ψ δ,m as an element of the ring L ( j , . . . , j n )[ j n +1 , Y . . . , Y m ] with degreeat most e − in j n +1 . We call this expression the canonical form of Ψ δ,m , andwe can consider its coefficients to be rational fractions in n variables J , . . . , J n . Proposition 3.3.
Let ≤ m ≤ n + 1 , and γ ∈ K /K m − . Then, up tomultiplication by an element in L ( j , . . . , j n +1 , j ,δ , . . . , j m − ,δ ) , we have Ψ δ,m ( j γ ,δ , . . . , j γm − ,δ , Y m ) = Y γ m ∈ K m − /K m ( Y m − j γγ m m ) . Proof.
This is straightforward from Definition 3.1.11roposition 3.3 has two consequences. First, modular equations vanishon H δ , as promised; and second, provided the multiplicative coefficient doesnot vanish, which is generically the case, Ψ δ,m provides all the possible valuesfor j m,δ once j , . . . , j n +1 and j ,δ , . . . , j m − ,δ are known. We can also defineother modular equations Φ δ,m for which there is true equality in Proposition 3.3,but they have a more complicated expression. In practice, using the Ψ δ,m is moreconvenient as they are typically smaller. Remark 3.4.
There is a geometric picture behind the definition of Ψ δ,m . Forsimplicity, assume that H δ is irreducible. Then L ( H δ ) is a field, and is generatedby the j i,δ for ≤ i ≤ n + 1 . We have a tower of function fields: L ( j , . . . , j n +1 , j ,δ , . . . , j n +1 ,δ ) = L ( H δ ) ... L ( j , . . . , j n +1 , j ,δ ) L ( S ) Σ . degree d n +1 degree d degree d Proposition 3.3 implies that up to scaling, Ψ δ,m ( j ,δ , . . . , j m − ,δ , Y m ) is theminimal polynomial of j m,δ over L ( j , . . . , j n +1 , j ,δ , . . . , j m − ,δ ) . Therefore itis a defining equation for the m -th floor of the tower above.If j ,δ is a generator of the whole field extension L ( H δ ) /L ( S ) Σ , then for every ≤ m ≤ n + 1 , we have d m = 1 and the polynomial Ψ δ,m is just the expressionof j m,δ in terms of j , . . . , j n +1 , and j ,δ .Using a nontrivial Σ increases the degree of modular equations. This alsohas a geometric interpretation: modular equations describe the Hecke corre-spondence H δ and its conjugates under Σ simultaneously. The Siegel modular varieties are prominent examples of PEL Shimura varieties.They classify abelian varieties of dimension g with a certain polarization andlevel structure. Another example is given by the Hilbert modular varieties,which do the same with an additional real multiplication embedding. In thissubsection and the next, we explain how these examples fit in the general settingof PEL Shimura varieties. In particular, we show that modular equations ofSiegel and Hilbert type in dimension 2 [18, 19] are special cases of modularequations as defined above.Let g ≥ . The Siegel modular variety of dimension g is obtained by taking B = Q , with trivial ∗ , and taking the symplectic module ( V, ψ ) to be V = Q g ∀ u, v ∈ V, ψ ( u, v ) = u t (cid:18) I g − I g (cid:19) v. Then G = GSp g ( Q ) . The Q -algebra B is simple of type (C). We can choose X + to be the set of all complex structures on V ( R ) that are positive for ψ , and wehave G ( R ) + = { g ∈ G ( R ) | µ ( g ) > } . The reflex field is Q [21, §14]. In fact, X + can be identified with the Siegelupper half-space H g endowed with the classical action of GSp g ( Q ) .Choose positive integers D | · · · | D g , and let Λ be the lattice generatedby the vectors e , . . . , e g , D e g +1 , . . . , D g e g . Then the polarization ψ has type ( D , . . . , D g ) on Λ . Let K be a compact open subgroup of G ( A f ) that stabilizes Λ ⊗ b Z , and let S denote the connected component of Sh K ( C ) above ∈ G ( A f ) .This component is identified with Γ \H g , where Γ = GSp g ( Q ) + ∩ K = Sp g ( Q ) ∩ K. By Theorem 2.2, S is a moduli space for polarized abelian varieties withpolarization type ( D , . . . , D g ) and level K structure such that H ( A, Z ) is iso-morphic to Λ with its additional data. This modular interpretation coincideswith the classical one [3, §8.1]. Also, modular forms on S are Siegel modularforms in the classical sense.Taking g = 1 and D = 1 , we find the classical modular curves, which arequotients of the upper half plane H by congruence subgroups of SL ( Z ) .We now focus on the special case given by g = 2 , D = D = 1 , Λ = Z g , K = GSp g ( b Z ) . Then Sh K ( C ) has only one connected component, and classifies principally po-larized abelian surfaces. Modular forms on Sh K are Siegel modular forms oflevel Γ(1) = Sp ( Z ) . As shown by Igusa [14], the graded ring of modular formsis generated by four elements of respective weights 4, 6, 10, 12. These can betaken to be I , I ′ , I , I in Streng’s notation [27, p. 42]. The function field of Sh K is generated by three algebraically independent Igusa–Streng invariants : j = I I ′ I , j = I I I , j = I I . Let ℓ be a prime, and consider the Hecke correspondence of level δ = (cid:18) ℓ (cid:19) as a 4 × × . Then the group K ∩ δKδ − ∩ G ( Q ) + is the usual group Γ ( ℓ ) , and the degreeof H δ is d ( δ ) = ℓ + ℓ + ℓ + 1 . ℓ -isogeous to a given one; the degree of these isogenies is ℓ ( δ ) = ℓ . In this case,the function j ,δ generates the function field on the Hecke correspondence, sothat d = d ( δ ) and d = d = 1 , in the notation of §3.2. The modular equationsfrom Definition 3.1 are the usual modular equations of Siegel type and level ℓ .They have been computed for ℓ = 2 and ℓ = 3 [18]. Let F be a totally real number field of degree g over Q , and let B = F withtrivial ∗ . The Q -algebra B is simple of type (C). Let V = F , which is a Q -vector space of dimension g , and define the symplectic form ψ by ∀ a, b, c, d ∈ F, ψ (cid:0) ( a, b ) , ( c, d ) (cid:1) = Tr F/ Q ( ad − bc ) . Then ( V, ψ ) is a faithful symplectic ( B, ∗ ) -module, where B acts on V by mul-tiplication. The associated algebraic group is G = GL ( F ) . The g real embed-dings of F induce an identification G ( R ) = g Y i =1 GL ( R ) . The subgroup G ( R ) + consists of matrices with totally positive determinant.There is a particular complex structure x ∈ G ( R ) on V ( R ) given by x = (cid:18)(cid:18) − (cid:19)(cid:19) ≤ i ≤ g . Let X + be the G ( R ) + -conjugacy class of x . Then ( G, X + ) is called a HilbertShimura datum . Its reflex field is Q : see [29, §X.4] when g = 2 , and [21, Ex. 12.4]in general. The domain X + can be identified with H g , where H is the complexupper half-plane, endowed with the action of GL ( R ) on each coordinate.Let Z F be the integer ring of F , and take Λ = Z F ⊕ Z ∨ F , where Z ∨ F is thedual of Z F with respect to the trace form. Then the stabilizer of Λ in B is Z F .Let K be a compact open subgroup of GL(Λ ⊗ A f ) . Remark 3.5.
In this setting, µ (Γ c ) is not equal to µ ( E ) in general. For instance,if K = GL(Λ ⊗ A f ) , and c = 1 is the trivial class, then Γ c = G ( R ) + ∩ K = { g ∈ GL(Λ ) | det( g ) is totally positive } , so µ (Γ c ) is the set of totally positive units in Z F . On the other hand, µ ( E ) isthe set of all squares of units. For instance, if g = 2 , then µ ( E ) = µ (Γ c ) if andonly if the fundamental unit in Z F has negative norm.We now assume that K has been chosen such that G ( Q ) + ∩ K = (cid:8) g ∈ GL ( Z F ⊕ Z ∨ F ) | µ ( g ) ∈ Z × F (cid:9) . Sh K ( G, X + )( C ) has several connected components: thenarrow class group of F appears in π (Sh K ( C )) [29, Cor. I.7.3]. Let S be theconnected component associated with c = 1 . Then there is a natural isomor-phism S = ( G ( Q ) + ∩ K ) \H g ≃ SL ( Z F ⊕ Z ∨ F ) \H g . By Theorem 2.2, the component S parametrizes polarized abelian varieties withreal multiplication by Z F and level K structure such that H ( A, Z ) is isomorphicto Λ with its additional data. Modular forms of weight w on S are classicalHilbert modular forms of weight ( w, w ) and level Γ(1) = SL ( Z F ⊕ Z ∨ F ) .Now consider the special case g = 2 , and let Σ = { , σ } , where σ is theinvolution of G coming from real conjugation in F . On G ( R ) + , the involution σ acts as permutation of the two factors. Modular forms that are symmetricunder Σ are symmetric Hilbert modular forms in dimension 2 in the usual sense.Let b be a prime ideal of Z F , and define δ = (cid:18) b (cid:19) in G ( A f ) . The Hecke correspondence of level δ has degree d ( δ ) = N F/ Q ( b ) + 1 , andparametrizes isogenies of degree ℓ ( δ ) = N F/ Q ( b ) . One can check that H δ inter-sects S × S nontrivially if and only if b is trivial in the narrow class groupof F , i.e. b = ( β ) is principal and generated by a totally positive element.Now assume that β ∈ Z F is totally positive and prime, and let δ = (cid:18) β (cid:19) . One possibility is to use, as invariants on S , the pullback of Igusa invariants bythe forgetful map to the Siegel threefold. They are symmetric with respect to Σ ,and the equation relating these three invariants is the equation of the associatedHumbert surface. In this case, the modular equations describe simultaneously β - and σ ( β ) -isogenies [19].In special cases, the field of Σ -invariant functions can be generated by twoelements called Gundlach invariants . This reduction of the number of vari-ables is interesting in practice. For instance, if F = Q ( √ , the graded ringof symmetric Hilbert modular forms is free over three generators F , F , F ofrespective weights , , and [12]; therefore, K ( S ) Σ = Q ( g , g ) with g = F F , g = F F F , and g , g are algebraically independent. The associated modular equationshave been computed up to N F/ Q ( β ) = 59 [17]. They also describe both β - and σ ( β ) -isogenies. 15 Degree estimates for modular equations
We fix a PEL setting as in §3.1. Let δ ∈ G ( A f ) such that δ ( b Λ ) ⊂ b Λ , and suchthat the Hecke correspondence H δ intersects S × S ′ nontrivially. In §3.2, wedefined the modular equations Ψ δ, , . . . , Ψ δ,n +1 ; they are multivariate polyno-mials in Y , . . . , Y n +1 describing H δ and its conjugates under Σ . When we writethese modular equations in canonical form, the coefficients of Ψ δ,m are uniquelydetermined rational fractions in L ( J , . . . , J n ) . The goal of this section is toprove the degree estimates about them given in Theorem 1.1, and their explicitvariants for abelian surfaces. Recall that the symbol C (or C , etc.) denotes aconstant that depends only on the setting. Ψ δ,m For each ≤ i ≤ n + 1 , fix a modular form χ i invariant under Σ such that χ i j i is holomorphic. This is possible by Proposition 2.5. Recall the notation K ′ = K ∩ δKδ − , K = K ⋊ Σ used in §3.2. For every i , the function χ i,δ : [ x, g ] χ i ([ x, gδ ]) is a modular form of level K ′ and weight wt( χ i ) on S . There is a natural actionof K on modular forms of level K ′ , and we define g δ,m for ≤ m ≤ n + 1 as g δ,m = m Y i =1 Y γ ∈ K /K ′ χ γi,δ . Lemma 4.1.
For every ≤ m ≤ n +1 , the function g δ,m is a nonzero symmetricmodular form on S , and wt( g δ,m ) = ( d ( δ ) m X i =1 wt( χ i ) . Proof.
Acting by an element of K ⋊ Σ only permutes the factors in the productdefining g δ,m , so g δ,m is invariant under the action of K and Σ . Moreover g δ,m is a modular form of level K ′ of weight P mi =1 K /K ′ ) wt( χ i ) , and is nonzerobecause each χ γi,δ is. Since K /K ′ ) = ( d ( δ ) , the result follows. Lemma 4.2.
For every ≤ m ≤ n + 1 , the multivariate polynomial g δ,m Ψ δ,m has holomorphic coefficients.Proof. This is immediate from the formula in Definition 3.1.In other words, the function g δ,m is a common denominator for the coeffi-cients of Ψ δ,m .When the invariants have similar denominators, it is possible to make abetter choice for g δ,m . The proof is easy and omitted.16 roposition 4.3. Assume that there is a modular form χ such that for every ≤ i ≤ n + 1 , we have χ i = χ α i for some integer α i ≥ . Let ≤ m ≤ n + 1 ,and define g δ,m = (cid:16) Y γ ∈ K χ γδ (cid:17) α , where α = max ≤ i ≤ m α i . Then g δ,m is a nonzero symmetric modular form on S , and wt( g δ,m ) = ( d ( δ ) α wt( χ ) . Moreover, the multivariate polynomial g δ,m Ψ δ,m has holomorphic coefficients. By §4.1, each coefficient of Ψ δ,m can be expressed as the quotient of two holo-morphic modular forms of the same bounded weight w . We show that whenwe rewrite such a quotient in terms of the chosen invariants, the degree of therational fractions we obtain is bounded linearly in w . Proposition 4.4.
Let f, g be symmetric modular forms on S of weight w , andassume that g is nonzero. Then there exists polynomials P, Q ∈ L [ J , . . . , J n +1 ] with total degree at most C w such that fg = P ( j , . . . , j n +1 ) Q ( j , . . . , j n +1 ) . Moreover, Q can be chosen independently of f .Proof. Let f k for ≤ k ≤ r be nonzero generators over L for the graded ring ofsymmetric modular forms, with respective weights w k . For each ≤ k ≤ r − ,let β k ≥ be the minimal integer such that β k w k ∈ Z w k +1 + · · · + Z w r . Then we can find nonzero modular forms λ k , ξ k ∈ L [ f k +1 , . . . , f r ] such that wt( λ k ) − wt( ξ k ) = β k w k . The modular function ξ k f β k k /λ k is symmetric under Σ ,so we can find P k , Q k ∈ L [ J , . . . , J n +1 ] such that ξ k f β k k λ k = P k ( j , . . . , j n +1 ) Q k ( j , . . . , j n +1 ) . We claim that the conclusion of the proposition holds with C r − X k =1 β k w k max (cid:8) deg( P k ) , deg( Q k ) (cid:9) k − Y l =1 (cid:18) ξ l ) β l w l (cid:19)! . Let f , g be as in the proposition. Then f and g can be expressed as a sumof monomial terms of the form cf α · · · f α r r with r X k =1 α k w k = w.
17n order to rewrite the fraction
P/Q = f /g (currently expressed as a rationalfraction of the modular forms f k ) as a fraction of invariants, we proceed asfollows. Set z = w and d = 0 . For k = 1 up to r − , do: ( s k ) Set s k ∈ N to be minimal such that z k − − s k w k ∈ h w k +1 . . . , w r i . ( a k ) Set a k = (cid:22) z k − β k w k (cid:23) . ( S k ) Divide P and Q by f s k k . ( R k ) Replace each occurence of f β k k by λ k P k ξ k Q k in P and Q . ( M k ) Multiply P and Q by ξ a k k Q a k k . ( z k ) Set z k = z k − − s k w k + a k wt( ξ k ) . ( d k ) Set d k = d k − + a k max { deg( P k ) , deg( Q k ) } .Finally, in step ( S r ) , simplify the remaining occurences of f r . We prove thefollowing statement ( ⋆ ) k by induction for every ≤ k ≤ r : Before step ( s k ) , P and Q are elements of the ring L [ J , . . . , J n +1 ][ f k , . . . , f r ] of weight z k − , with total degree at most d k − in J , . . . , J n +1 , such that fg = P ( j , . . . , j n +1 ) Q ( j , . . . , j n +1 ) . The statement ( ⋆ ) is true by definition of z and d ; assume that ( ⋆ ) k istrue. Then we see, in order, that • z k − must belong to h w k , . . . , w r i , so s k is well defined. • In each monomial of P and Q , the exponent of f k must be aβ k + s k forsome integer a ≤ a k . Therefore step ( S k ) is an exact division, and afterstep ( R k ) there are no more occurences of f k in P or Q . • After step ( M k ) , P and Q are elements of L [ J , . . . , J n +1 ][ f k +1 , . . . , f r ] ofweight z k − − s k w k + a k wt( ξ k ) = z k . None of those steps changed the fact that ( P/Q )( j , . . . , j n +1 ) equals f /g .It remains to show that the degree of P, Q in J , . . . , J n +1 is bounded by d k after step ( M k ) . This comes from the observation that in steps ( R k ) – ( M k ) , weonly multiply polynomials in J , . . . , J n +1 already present by P bk Q a k − bk for some ≤ b ≤ a k , and rearranging terms afterwards cannot increase the total degree.This proves ( ⋆ ) k for all ≤ k ≤ r .Similarly, after step ( S r ) , all the occurences of f r disappear. Therefore,at the end of this rewriting procedure, we obtain polynomials P, Q with totaldegree at most d r − such that fg = P ( j , . . . , j n +1 ) Q ( j , . . . , j n +1 ) .
18y induction, we obtain z k ≤ w k Y l =1 (cid:18) ξ l ) β l w l (cid:19) and d r − ≤ r − X k =1 wβ k w k max { deg( P k ) , deg( Q k ) } k − Y l =1 (cid:18) H l ) β l w l (cid:19)! = C w. The algorithm runs independently on the numerator and the denominator, sothe polynomial Q is independent of f . Proposition 4.5.
Let
P, Q ∈ L [ J , . . . , J n +1 ] with total degree at most d , andassume that Q ( j , . . . , j n +1 ) is not identically zero. Write the fraction P/Q incanonical form using equation ( E ) : P ( j , . . . , j n +1 ) Q ( j , . . . , j n +1 ) = e − X k =0 R k ( j , . . . , j n ) j kn +1 . Then deg R k ≤ C d for every ≤ k ≤ e − .Proof. We work in the ring L ( J , . . . , J n )[ J n +1 ] modulo the equation ( E ), whichwe write as E = 0 . In the proof, degrees and coefficients are taken with respectto the variable J n +1 unless otherwise specified. First, we invert the denomina-tor Q . Let R = Res J n +1 ( Q, E ) ∈ L [ J , . . . , J n ] . Using the generic expression of resultants, we can find Bézout coefficients
U, V ∈ L [ j , . . . , j n +1 ] such that R = U Q + V E.
The inverse of Q modulo E is U/R , so we have P ( j , . . . , j n +1 ) Q ( j , . . . , j n +1 ) = U ( j , . . . , j n +1 ) P ( j , . . . , j n +1 ) R ( j , . . . , j n ) . The resultant R has a polynomial expression of degree e = deg( E ) in thecoefficients of Q , and deg( Q ) in the coefficients of E . The same is true for everycoefficient of U , with e replaced by e − . Since the total degree of Q is at most d ,and E is part of the setting, the total degrees of U and R in J , . . . , J n +1 arebounded by some Cd ; the same is true for the numerator U P .Now, we reduce
U P modulo E in order to obtain a polynomial of degree atmost e − in J n +1 . We can decrease the degree by 1 using relation ( E ): j en +1 = − e − X k =0 E k ( j , . . . , j n ) j kn +1 . C . Hence, after theeuclidean division, total degrees remains bounded by Cd .We are now ready to prove part 1 of Theorem 1.1. Theorem 4.6.
Let F be a rational fraction occurring as a coefficient of one ofthe modular equations Ψ δ,m in canonical form. Then the total degree of F isbounded by C d ( δ ) . The same estimate holds if we require all the F ’s to sharea common denominator.Proof. By Lemmas 4.1 and 4.2, each coefficient appearing in one of the Ψ δ,m isof the form f /g δ,n +1 , where f is holomorphic. By Lemmas 3.2 and 4.1, both f and g δ,n +1 are symmetric modular forms of weight w = ( d ( δ ) n +1 X i =1 wt( χ i ) . By Proposition 4.4, we can write fg δ,n +1 = P ( j , . . . , j n +1 ) Q ( j , . . . , j n +1 ) with deg( P ) , deg( Q ) ≤ C w , and Q is independent of f . By Proposition 4.5,the degrees of coefficients of Ψ δ,m in canonical form are bounded by C C w ;note that the denominator is still independent of f . The theorem follows with C C C n +1 X i =1 wt( χ i ) . Even in the case of abelian surfaces, our methods provide new results aboutthe degrees occuring in modular equations. We detail this in the Siegel casewith Igusa invariants (§3.3), when ℓ is a prime, and in the Hilbert case with F = Q ( √ and Gundlach invariants (§3.4), when β ∈ F is a totally positiveprime. The modular equations are denoted Ψ ℓ, , Ψ ℓ, , Ψ ℓ, and Ψ β, , Ψ β, respectively. Lemma 4.7.
In the dimension Siegel case with Igusa–Streng invariants, wecan take C / .Proof. We follow the notation used in the proof of Proposition 4.4. Generatorsfor the ring of modular forms are, f = I ′ , f = I , f = I and f = I . Wehave β = β = 1 , β = 5 , and the rewriting relations are I ′ = I I j , I = I I j , I = I j . w , as follows. First, multiplyby I ⌊ w/ ⌋ above and below, and second, rewrite sequentially I I ′ → I j , I I → I j , I → I j This removes all the occurences of I ′ and I without introducing new denomi-nators. The remaining occurences of I and I must simplify.Since I is never used on the left of a rewriting law, we can as well ignoreit when computing the weight of a monomial. Then, the greatest ratio (degreein j , j , j )/(weight) is given by I I → I j , with a ratio of / . Thereforewe can take C (cid:18) (cid:19) ·
110 = 16 . Lemma 4.8.
In the dimension Hilbert case with F = Q ( √ and Gundlachinvariants, we can take C / .Proof. The proof is almost the same as in the Siegel case. The rewriting algo-rithm for a quotient of symmetric modular forms of weight w is as follows: firstmultiply numerator and denominator by F ⌊ w/ ⌋ , then rewrite sequentially F F → F g , F → F g . This introduces no new denominators, and the remaining occurrences of F and F must simplify. Ignoring F , the first relation has the highest de-gree/weight ratio, which is / . Therefore we can take C / as above. Remark 4.9.
The proof of Proposition 4.4 would give C / and C / respectively. Informally, the reason why such an improvement is possible is thatthe numerator λ i is always a power of f r , and Q i = 1 . It seems tedious to writea general formula for C that gives the right values here. Proposition 4.10.
The coefficients of Ψ ℓ, (resp. Ψ ℓ, , Ψ ℓ, ) have total degreesbounded by d ( ℓ ) / (resp. d ( ℓ ) / ), where d ( ℓ ) = ℓ + ℓ + ℓ + 1 .Proof. The quantity d ( ℓ ) is the degree of the Hecke correspondence. We are inthe situation of Proposition 4.3, so we can choose common denominators g ℓ, , g ℓ, , g ℓ, with wt( g ℓ, ) = 10 d ( ℓ ) , wt( g ℓ, ) = wt( g ℓ, ) = 20 d ( ℓ ) and in fact g ℓ, = g ℓ, = ( g ℓ, ) . There is no relation between invariants in thiscase. By Lemma 4.7, the degree of the rational fractions in Ψ ℓ,m is boundedby wt( g ℓ,m ) / for ≤ m ≤ , and the result follows. Proposition 4.11.
The coefficients of Ψ β, and Ψ β, have total degrees boundedby d ( β ) / , where d ( β ) = N F/ Q ( β ) + 1 . roof. The quantity d ( β ) is the degree of the Hecke correspondence, and thegroup Σ has order 2. We are also in the situation of Proposition 4.3: we canchoose common denominators g β = g β of weight d ( β ) . As above, there isno relation between invariants, so the result follows from Lemma 4.8.The degree bounds in Propositions 4.10 and 4.11 are both reached exper-imentally. In the Siegel case with ℓ = 2 , the maximum degree is 25; in theHilbert case with N F/ Q ( β ) = 41 , the maximum degree is 140 [17]. An important information when manipulating modular equations, besides thedegree, is the size of all the coefficients that occur. The precise notion to useis that of heights of elements, polynomials and rational fractions over a numberfield L . In this section, we recall the definition of heights and give key resultsrelating the height of a fraction with the height of its values. The symbol C (and C , etc.) stands for a constant that depends only on L . Let L be a number field of degree d L over Q . Write V (resp. V ∞ ) for the setof all nonarchimedean (resp. archimedean) places of L , and write V = V ⊔ V ∞ .Let P (resp. P L ) be the set of primes in Z (resp. prime ideals in Z L ). For eachplace v of L , write d v = [ L v : Q v ] , where subscripts denote completion. Denoteby |·| v the normalized absolute value associated with v : when v ∈ V , and p ∈ P is the prime below v , we have | p | v = 1 /p . We denote the norm of ideals by N .The (absolute logarithmic Weil) height of projective tuples, affine tuples,elements, polynomials and rational fractions over L is defined as follows. Definition 5.1.
1. For a projective tuple ( a : · · · : a n ) ∈ P nL , we write h ( a : . . . : a n ) = X v ∈V d v d L log (cid:0) max ≤ i ≤ n | a i | v (cid:1) .
2. For an affine tuple ( a , . . . , a n ) ∈ L n , we write h ( a , . . . , a n ) = h (1 : a : · · · : a n ) = X v ∈V d v d L log (cid:0) max { , max ≤ i ≤ n | a i | v } (cid:1) . In particular, for y ∈ L , we have h ( y ) = h (1 : y ) = X v ∈V d v d L log (cid:0) max { , | y | v } (cid:1) .
22. Let P be a multivariate polynomial over L , and write P = X k =( k ,...,k n ) c k Y k · · · Y k n n . Let v ∈ V . We write | P | v = max k | c k | v and h ( P ) = X v ∈V d v d L log (cid:0) max { , | P | v } (cid:1) . In other words, h ( P ) is the height of the affine tuple formed by all thecoefficients of P . When p ∈ P L , we also write v p ( P ) = min k v p ( c k ) .
4. Let
F ∈ L ( Y , . . . , Y n ) , and choose coprime polynomials P, Q over L suchthat F = P/Q . Then we define h ( F ) as the height of the projective tuple formed by all its coefficients: if ( c k ) denotes the collection of all thecoefficients of P and Q , then h ( F ) = X v ∈V d v d L log (cid:0) max k | c k | v (cid:1) . Heights are well defined and do not depend on the ambient field. In par-ticular, the height of a fraction does not depend on the particular numeratoror denominator chosen. When working with heights, we use without furthermention the fact that X v ∈V ∞ d v d L = 1 . Heights satisfy the
Northcott property : for every bound H ∈ R , the numberof projective tuples ( a : · · · : a n ) ∈ P nL such that h ( a : · · · : a n ) ≤ H is finite[13, §B.2]. Informally, the height of an element y ∈ L (or a polynomial, etc.) isa good measure of how much information is needed to represent y . For instance,if y = p/q ∈ Q is in irreducible form, then h ( y ) = log (cid:0) max {| p | , | q |} (cid:1) .Assume that L admits a fundamental unit ε . Then, by the Northcott prop-erty, h ( ε n ) tends to infinity as n grows. In general, multiplying integers by unitschanges the height. This causes problems in some of the proofs below, so weintroduce a modified height as follows. Definition 5.2.
Let y ∈ Z L be nonzero. Then we define e h ( y ) = 1 d L log (cid:0) | N L/ Q ( y ) | (cid:1) = X v ∈V ∞ d v d L log | y | v . More generally, if P ∈ Z L [ Y , . . . , Y n ] is nonzero, we define e h ( P ) = X v ∈V ∞ d v d L log | P | v . e h for integers is invariant under multiplication by units.It does not satisfy the Northcott property in general. Still, there is a closerelation between e h and the classical height h . Proposition 5.3.
Let P ∈ Z L [ Y , . . . , Y n ] be nonzero. Then we have ≤ e h ( P ) ≤ h ( P ) . Equality holds on the right if and only if | P | v ≥ for every v ∈ V ∞ .Proof. If c ∈ Z L is a nonzero coefficient of P , then e h ( P ) ≥ e h ( c ) = 1 d L log (cid:0) | N L/ Q ( c ) | (cid:1) ≥ because N L/ Q ( c ) ∈ Z \{ } , so | N L/ Q ( c ) | ≥ . The rest is obvious. Proposition 5.4.
Let P ∈ Z L [ Y , . . . , Y n ] be nonzero. Then there is a unit ε ∈ Z × L such that h ( εP ) ≤ max (cid:8) C , e h ( P ) (cid:9) . Proof.
Let m = V ∞ . In R m , we define the hyperplane H s for s ∈ R by H s = (cid:8) ( t , . . . , t m ) ∈ R m | t + · · · + t m = s (cid:9) , and the convex cone ∆ s by ∆ s = (cid:8) ( t , . . . , t m ) ∈ R m | ∀ i, t i ≥ s (cid:9) . The image of Z × L by the logarithmic embedding Log = (cid:16) d v d L log | · | v (cid:17) v ∈V ∞ is a full rank lattice Λ in H ; let V be a fundamental cell of Λ .Then for every s ≥ C , the convex set H s ∩ ∆ contains a translate of V ,hence H s = Λ + ( H s ∩ ∆ ) . Translating in the direction (1 , . . . , , we also have the following property: forevery s ≥ , H s = Λ + ( H s ∩ ∆ − C /m ) . Let P ∈ Z L [ Y , . . . , Y n ] \{ } , and consider the point Log( P ) = (cid:16) d v d L log | P | v (cid:17) v ∈V ∞ ∈ R m . The sum of its coordinates is s P = e h ( P ) . If s P ≥ C , then there is a unit ε ∈ Z × L such that Log( P ) + Log( ε ) belongs to ∆ . Then | εP | v ≥ for every v ∈ V ∞ , so h ( εP ) = e h ( εP ) = e h ( P )
24y Proposition 5.3. On the other hand, if ≤ s P < C , then we can still find aunit ε such that d v log | εP | v ≥ − C m for all v ∈ V ∞ . Then h ( εP ) = X v ∈V ∞ d v d L log max { , | εP | v } ≤ log e h ( εP ) + X v ∈V ∞ C m ≤ C . This proves the claim with C C . Corollary 5.5.
Every principal ideal a of Z L has a generator a ∈ Z L such that h ( a ) ≤ max (cid:8) C , d L log N ( a ) (cid:9) . Proof.
Apply Proposition 5.4 with n = 0 and P an arbitrary generator of a . When L = Q , every fraction F can be written as a quotient P/Q , where P and Q are integer polynomials that are coprime over Z and have height at most h ( F ) :choose P , Q such that the gcd of all their coefficients is 1. When Z L is not aPID, this is not possible in general, but there is the following substitute. Definition 5.6.
Let
F ∈ L ( Y , . . . , Y n ) . Then among all the possible ways towrite F = P/Q with
P, Q ∈ Z L [ Y , . . . , Y n ] where P , Q are coprime over L ,there is one such that h ( Q ) is minimal, by the Northcott property. We call it a minimal form of F .By Proposition 5.4, if F has a minimal form P/Q such that h ( Q ) ≥ C ,then e h ( Q ) = h ( Q ) . Lemma 5.7.
Let
F ∈ L ( Y , . . . , Y n ) with minimal form P/Q , and let λ ∈ L × such that λP, λQ ∈ Z L [ Y ] . If h ( Q ) ≥ C , then | N L/ Q ( λ ) | ≥ .Proof. By minimality, h ( λQ ) ≥ h ( Q ) ≥ C , so after multiplying λ by a unit,we can assume that e h ( λQ ) = h ( λQ ) . This does not change | N L/ Q ( λ ) | . Then e h ( λ ) + e h ( Q ) = e h ( λQ ) = h ( λQ ) ≥ h ( Q ) = e h ( Q ) . Therefore log | N L/ Q ( λ ) | = e h ( λ ) ≥ .If P/Q is a minimal form of F , then P , Q are almost coprime over Z L . Proposition 5.8.
There is an ideal r of Z L , depending only on L , such thatthe following holds. Let F ∈ L ( Y , . . . , Y n ) with minimal form P/Q , and let a be an ideal of Z L dividing all the coefficients of P and Q . Then a divides r . roof. Let C be a set of ideals in Z L that are representatives for the class groupof L , and define C = max c ∈ C N ( s ) . Take F ∈ L ( Y ) with minimal form P/Q ,and let a be an ideal of Z L dividing all the coefficients of P and Q .First, assume that e h ( Q ) ≤ C , and let q be a nonzero coefficient of Q . Then log | N L/ Q ( q ) | ≤ d L C . Since a divides q , we have N ( a ) ≤ e d L C .Second, assume that e h ( Q ) ≥ C . Let c ∈ C belonging to the class of a , andlet λ ∈ L × be a generator of a − c . Since λP and λQ have integer coefficients,by Lemma 5.7, we have | N L/ Q ( λ ) | ≥ . Therefore N ( a ) ≤ N ( c ) ≤ C .We can take r to be the least common multiple of all ideals of Z L of normat most max { e d L C , C } .If P ∈ Q [ Y , . . . , Y n ] , then we can find a minimal denominator a ∈ Z suchthat aP has coefficients in Z and max { h ( a ) , h ( aP ) } ≤ h ( P ) . The analogousstatement in number fields is as follows. Proposition 5.9.
For every P ∈ L [ Y , . . . , Y n ] , there is an element a ∈ Z L such that aP ∈ Z L [ Y , . . . , Y n ] and max { h ( a ) , h ( aP ) } ≤ h ( P ) + C .Proof. Let C be a set of ideals in Z L that are representatives for the class groupof L , and let P ∈ L [ Y , . . . , Y n ] , which we may assume nonzero. Let a = Y p ∈V p max { , − v p ( P ) } be the denominator ideal of P . Then log N ( a ) = X p ∈V log max { , N ( p ) − v p ( P ) } = X p ∈V d p log max { , | P | p } ≤ d L h ( P ) . Let c ∈ C such that ca is principal. By Corollary 5.5, we can find a generator a of ca such that h ( a ) ≤ max (cid:8) C , d L log N ( ca ) (cid:9) ≤ h ( P ) + C , with C (cid:8) C , d L max c ∈ C log N ( c ) (cid:9) . Then aP has integer coefficients, and we also have h ( aP ) = X v ∈V ∞ d v d L log max { , | aP | v }≤ X v ∈V ∞ d v d L (cid:0) log max { , | P | v } + log max { , | a | v } (cid:1) = h ( P ) + h ( a ) − X v ∈V d v d L log max { , | P | v } = h ( P ) + h ( a ) − d L log N ( a ) ≤ h ( P ) + C . .3 Evaluation and roots The following proposition is a slight generalization of [13, Prop. B.7.1].
Proposition 5.10.
Let P ∈ L [ Y , . . . , Y n ] with total degree d , let ≤ m ≤ n ,and let y , . . . , y m ∈ L . Write Q = P ( y , . . . , y m , Y m +1 , . . . , Y n ) . Then h ( Q ) ≤ h ( P ) + m log( d + 1) + dh ( y , . . . , y n ) . More generally, if I ⊔ · · · ⊔ I r is a partition of J , m K , and d k is the total degreeof P in the variables Y i for i ∈ I k , then h ( Q ) ≤ h ( P ) + r X k =1 ( I k ) log( d k + 1) + r X k =1 d k h (cid:0) ( y i ) i ∈I k (cid:1) . Proof.
It is enough to prove the second statement. If v ∈ V , we have (cid:12)(cid:12) P ( y , . . . , y m , Y m +1 , . . . , Y n ) (cid:12)(cid:12) v ≤ | P | v r Y k =1 (cid:16) max (cid:8) , max i ∈I k | y i | v (cid:9)(cid:17) d k . If v ∈ V ∞ , the same estimate holds after multiplying by the number of possiblemonomials in Y , . . . , Y m , which is r Y k =1 ( d k + 1) I k . Taking logarithms and summing gives the result.We can use this result to bound the height of a monic polynomial by theheight of its roots.
Proposition 5.11.
Let Q ∈ L [ Y ] be monic of degree d , and let α , . . . , α d beits roots in the algebraic closure of L . Then h ( Q ) ≤ d X i =1 h ( α k ) + d log 2 . Proof.
Apply Proposition 5.10 on the multivariate polynomial P = d Y k =1 ( Y d +1 − Y k ) with m = d , y k = α k , and I k = { k } . Since the coefficients of P all belong to {− , , } , we have h ( P ) = 0 .Conversely, the height of a polynomial controls the height of its roots. Proposition 5.12.
Let P ∈ L [ Y ] be monic, and let α be a root of P . Then h ( α ) ≤ h ( P ) + log(2) . Proof.
This is a consequence of [2, Prop. 3.5].27 .4 Heights of polynomials from their values
The idea is to control the height of a polynomial P in terms of heights ofevaluations of P at special points. We take these to be integers in an inter-val J A, B K ⊂ Z . Our basic tool is the Lagrange interpolation formula. In therest of this section, we use the notation D = B − A and M = max {| A | , | B |} . Lemma 5.13.
Let y , . . . , y d +1 ∈ J A, B K be distinct, and write Q k = i ( Y − y k ) Q k = i ( y i − y k ) = 1 D ! Q i ( Y ) for every ≤ i ≤ d + 1 . Then Q i ∈ Z [ Y ] , and | Q i | ≤ D ! (2 M ) d .Proof. Since the denominator Q k = i ( y i − y k ) divides D ! , we have Q i = N i Y k = i ( Y − y i ) for some N i ∈ Z dividing D ! . Hence Q i ∈ Z [ Y ] . Moreover, if c is the coefficientof Y d − k in Q i , then | c | ≤ | N i | (cid:18) dk (cid:19) M k ≤ D ! 2 d M d . Lemma 5.14.
Let P ∈ L [ Y ] of degree d ≥ , and y , . . . , y d +1 distinct valuesin J A, B K . Assume that h ( P ( y i )) ≤ H for every i . Then we have h ( P ) ≤ ( d + 1) H + D log( D ) + d log(2 M ) + log( d + 1) . Proof.
Write the Lagrange interpolation formula: P = 1 D ! d +1 X i =1 P ( y i ) Q i ( Y ) in the notation of Lemma 5.13. If v ∈ V , then max { , | P | v } ≤ (cid:12)(cid:12)(cid:12)(cid:12) D ! (cid:12)(cid:12)(cid:12)(cid:12) v max { , | P ( y ) | v , . . . , | P ( y d +1 ) | v }≤ (cid:12)(cid:12)(cid:12)(cid:12) D ! (cid:12)(cid:12)(cid:12)(cid:12) v n +1 Y i =1 max { , | P ( y i ) | v } . If v ∈ V ∞ , then by Lemma 5.13, max { , | P | v } ≤ d +1 X i =1 | P ( y i ) | v d M d ≤ ( d + 1)2 d M d d +1 Y i =1 max { , | P ( y i ) | v } . Since h (1 /D !) = h ( D !) ≤ D log( D ) , taking logarithms and summing gives theresult. 28he bound on h ( P ) in Lemma 5.14 is roughly dH , not H . This causes troublewhen one wants to apply this result in a context of successive interpolations.However, we can get a better bound on h ( P ) provided that a height bound isestablished on more than d + 1 values of P . Lemma 5.15.
Let P ∈ L [ Y ] of degree d , and let v ∈ V (resp. v ∈ V ∞ ) . Thenthe number of elements y ∈ J A, B K satisfying | P ( y ) | v < | D ! P | v (cid:16) resp. | P ( y ) | v < | P | v (2 M ) d ( d + 1) (cid:17) is at most d .Proof. We argue by contradiction. Let ( y i ) ≤ i ≤ d +1 be distinct elements of J A, B K satisfying the given inequality. If v ∈ V , then the Lagrange interpo-lation formula and Lemma 5.13 give | D ! P | v ≤ max i | P ( y i ) | v < | D ! P | v which is a contradiction. If v ∈ V ∞ , the contradiction is | P | v ≤ (2 M ) d d +1 X i =1 | P ( y i ) | v < | P | v . Proposition 5.16.
Let J A, B K ⊂ Z . Write D = B − A and M = max {| A | , | B |} .Let P ∈ L [ Y ] be a polynomial of degree d ≥ , and let y , . . . , y d be distinctelements of J A, B K . Assume that h ( P ( y i )) ≤ H for every i . Then we have h ( P ) ≤ H + D log D + d log(2 M ) + log( d + 1) . Proof. If v ∈ V , by Lemma 5.15, we have | P ( y i ) | v ≥ | D ! P | v for at least d values of i . Therefore d Y i =1 max { , | P ( y i ) | v } ≥ | D ! P | dv , so log max { , | P | v } ≤ log (cid:12)(cid:12)(cid:12)(cid:12) D ! (cid:12)(cid:12)(cid:12)(cid:12) v + 1 d d X i =1 log max { , | P ( y i ) | v } . If v ∈ V ∞ , then at least d of the P ( y i ) satisfy | P ( y i ) | v ≥ | P | v / (2 M ) d ( d + 1) , so d Y i =1 max { , | P ( y i ) | v } ≥ | P | dv (cid:0) (2 M ) d ( d + 1) (cid:1) d and log max { , | P | v } ≤ d log(2 M ) + log( d + 1) + 1 d d X i =1 log max { , | P ( y i ) | v } . Since h (1 /D !) ≤ D log D , summing all the contributions yields the result.29 .5 Height of fractions from their values: preliminaries We now turn to the problem of estimating the height of rational fractions interms of the height of their values at certain points. Our first goal is to give aheight bound using the minimal number of interpolation points; then, we makepreparations for Theorem 1.2.
Lemma 5.17.
Let
P, Q ∈ Z L [ Y ] of degrees d P and d Q respectively, and let ≤ k ≤ min { d P , d Q } − . Let R be the k -th subresultant of P and U , V theassociated Bézout coefficients. Write s = d P + d Q . Then we have h ( R ) ≤ ( d Q − k ) h ( P ) + ( d P − k ) h ( Q ) + s − k s − k ) ,h ( U ) ≤ ( d Q − k − h ( P ) + ( d P − k ) h ( Q ) + s − k −
12 log( s − k − ,h ( V ) ≤ ( d Q − k ) h ( P ) + ( d P − k − h ( Q ) + s − k −
12 log( s − k − . Recall that R , U , V are polynomials over Z L such that U P + V Q = R and deg R ≤ k, deg U ≤ d Q − k − , deg V ≤ d P − k − . Proof.
Let r be a coefficient of R . Then r has an expression as a determinantof size d P + d Q − k ; its entries in the first d Q − k columns are coefficients of P ,and its entries in the last d P − k columns are coefficients of Q . Let v ∈ V ∞ .By Hadamard’s lemma, we can bound | r | v by the product of L -norms of thecolumns. Hence | r | v ≤ (cid:0)p d P + d Q − k | P | v (cid:1) d Q − k (cid:0)p d P + d Q − k | Q | v (cid:1) d P − k . This gives the desired height bound on R .The proof is the same for U (resp. V ): the coefficients are determinants ofsize d P + d Q − k − , with one column less coming from P (resp. Q ). Proposition 5.18.
Let J A, B K ⊂ Z . Write D = B − A and M = max {| A | , | B |} .Let F ∈ L ( Y ) of degree d ≥ . Let d P , d Q be the degrees of its numerator anddenominator respectively, and let s = d P + d Q . Let y , . . . , y s +1 be distinctelements of J A, B K that are not poles of F , and assume that h ( F ( y i )) ≤ H forevery i . Then we can write F = P/Q with
P, Q ∈ Z L [ Y ] such that deg P = d P , deg Q = d Q , and max { h ( P ) , h ( Q ) } ≤ ( d + 1)(2 d + 1) H + ( d + 1) D log( D ) + (4 d + 3 d ) log(2 M )+ (2 d + 2) log(2 d + 1) + ( d + 1) C . Proof.
We follow the interpolation algorithm [4, §7.1]. Let T ∈ L [ Y ] be thepolynomial of degree at most s interpolating the points ( y i , F ( y i )) . Accordingto Lemma 5.14, we have h ( T ) ≤ ( s + 1) H + D log( D ) + s log(2 M ) + log( s + 1) . ( ⋆ )30y Proposition 5.9, we can find a denominator a ∈ Z L such that T ′ = aT is anelement of Z L [ Y ] and max { h ( a ) , h ( T ′ ) } ≤ h ( T ) + C . Define Z = s +1 Y i =1 ( Y − y i ) ∈ Z [ Y ] . The coefficients of Z are bounded in absolute value by (2 M ) s +1 , so we have h ( Z ) ≤ ( s + 1) log(2 M ) . Let P ′ be the d P -th subresultant of T ′ and Z , and let Q ′ T ′ + V Z = P ′ be the associated Bézout equality. By Lemma 5.17, we have h ( P ′ ) ≤ ( d + 1) h ( T ′ ) + d ( s + 1) log(2 M ) + 2 d + 12 log(2 d + 1) ,h ( Q ′ ) ≤ d h ( T ′ ) + ( s + 1) d log(2 M ) + d log(2 d + 1) . We can take P = P ′ and Q = aQ ′ : they have the right degrees and satisfy max { h ( P ) , h ( Q ) } ≤ ( d + 1)( h ( T ) + C
9) + ( s + 1) d log(2 M ) + ( d + 1) log(2 d + 1) . Using the previous bound ( ⋆ ) on h ( T ) and the bound s ≤ d ends the proof.The reader may wish to skip the following lemmas until their use in the proofof Theorem 1.2 becomes apparent. Lemma 5.19.
Let D = B − A , let S ⊂ J A, B K containing at least D/ ele-ments, and let ≤ k < D/ . Then there is a subinterval of J A, B K of amplitudeat most k containing at least k + 1 elements of S .Proof. Assume the contrary. We can partition J A, B K in at most ( D/ k ) + 1 intervals of amplitude at most k , so D ≤ S ≤ k (cid:18) D k + 1 (cid:19) = D k. This is absurd because k < D/ . Lemma 5.20.
Let R ≥ be an integer. Then X p ∈P , p | R log pp − ≤ max (cid:8) , C log log R (cid:9) . Proof.
Let m be the number of prime factors in R , and ( p i ) the sequence ofprime numbers in increasing order. It is enough to prove the claim for theinteger R ′ = Q mi =1 p i , which has both a greater left hand side, since log( p ) / ( p − is a decreasing function of p , and a smaller right hand side, since R ′ ≤ R . Wecan assume m ≥ . Then m X i =1 log( p i ) p i − ≤ m X i =1 log( p i ) p i + m X i =1 log( p i ) p i ( p i − ≤ log( p m ) + 2 + C
31y Mertens’s first theorem [16], and because the second series converges. By [25],we have p m < m log m + m log log m if m ≥ ; so the rough bound p m ≤ m holds. The result follows since m ≤ log( R ′ ) . Lemma 5.21.
Let R ∈ Z L be nonzero. Then X p ∈P L , p | R p | p ∈P log( N ( p )) p − ≤ d L max (cid:8) , C
10 log log | N L/ Q ( R ) | (cid:9) . Proof. X p | R log( N ( p )) p − ≤ X p | N L/ Q ( R ) P p | p log( N ( p )) p − d L X p | N L/ Q ( R ) log( p ) p − ≤ d L max (cid:8) , C
10 log log | N L/ Q ( R ) | (cid:9) by Lemma 5.20. Lemma 5.22.
Let p ∈ P L over p ∈ P , and let L p be the p -adic completion of L .Let Q ∈ L p [ Y ] of degree d with integer coefficients, and assume that v p ( Q ) = 0 .Let y , . . . , y n be distinct values in J A, B K , and write D = B − A ; assume that D ≥ . Let β ∈ N . Then n X i =1 min { β, v p ( Q ( y i )) } ≤ d (cid:18) β + d L log( D )log N ( p ) + Dp − (cid:19) . Proof.
Let λ be the leading coefficient of Q , and let α , . . . , α d be the roots of Q in an algebraic closure of L p , where we extend | · | p and v p . Up to reindexation,we may assume that | α j | p ≤ for ≤ j ≤ t , and | α j | p > for t + 1 ≤ j ≤ d .For every i , we have | Q ( y i ) | p = | λ | p d Y i =1 | y i − α j | p = (cid:18) | λ | p d Y j = t +1 | α j | p (cid:19) t Y j =1 | y i − α j | p . We must have (cid:18) | λ | p d Y j = t +1 | α j | p (cid:19) ≥ , for otherwise all the coefficients of Q would be divisible by p . Therefore v p ( Q ( y i )) ≤ t X j =1 v p ( y i − α j ) . Let k ∈ N such that p k ≤ D < p k +1 . Since the y i are all distinct mod-ulo p k +1 , there are at most t ≤ d values of i such that v p ( y i − α j ) > k for32ome j . For these i ’s, we bound min { β, v p ( Q ( y i )) } by β . This accounts for theterm dβ in the lemma.For all other values of i (say i ∈ I ), we have v p ( y i − α j ) ≤ k , and thus v p ( y i − α j ) = Z k u ≤ v p ( y i − α j ) du. Any two y i that fall in the same p -adic disk { u ≤ v p ( y − α j ) } must coincidemodulo p ⌈ u ⌉ . Therefore, for a given α j , and a given u ∈ ] l, l + 1] , there are atmost ⌈ D/p l +1 ⌉ values of i for which y i can belong to this disk. Therefore X i ∈ I v p ( Q ( y i )) ≤ X i ∈ I t X j =1 v p ( y i − α j )= X i ∈ I t X j =1 k − X l =0 Z l +1 l u ≤ v p ( y i − α j ) du = t X j =1 k − X l =0 Z l +1 l X i ∈ I u ≤ v p ( y i − α j ) ! du ≤ t k − X l =0 (cid:24) Dp l +1 (cid:25) ≤ t k − X l =0 (cid:18) Dp l +1 + 1 (cid:19) ≤ tk + tDp − . We have t ≤ d , and k ≤ log p ( D ) ≤ d L log( D )log N ( p ) . This accounts for the two remaining terms in the lemma.
We are now ready to prove Theorem 1.2.
Theorem 5.23.
Let J A, B K ⊂ Z . Write D = B − A and M = max {| A | , | B |} .Let F ∈ L ( Y ) of degree at most d ≥ . Let S ⊂ J A, B K containing at least D/ elements and no poles of F . Let H ≥ max { , log(2 M ) } , and assume that(i) h ( F ( y )) ≤ H for every y ∈ S .(ii) D > max { d H log( dH ) , d d L } .Then we have h ( F ) ≤ H + C dH ) + 3 d log(2 M ) . Proof.
Let
P/Q be a minimal form of F , and let R = Res( P, Q ) . Let y , . . . , y n be distinct elements of S . For ≤ i ≤ n , we define ideals s i , n i and d i of Z L asfollows: s i = gcd( P ( y i ) , Q ( y i )) , ( P ( y i )) = n i s i , ( Q ( y i )) = d i s i . s i | R , and ( F ( y i )) = n i d − i . Since n i and d i are coprime, we have log N ( d i ) ≤ d L h ( F ( y i )) , so log | N L/ Q ( Q ( y i )) | ≤ log N ( s i ) + d L H. The proof runs as follows.1. Obtain a bound on the resultant R .2. Show that the product of all s i ’s has bounded norm.3. Show that s i is reasonably small at least for some values of i .4. Take care of archimedean places, and obtain a bound on h ( Q ) .5. Deduce a bound on h ( P ) and conclude.We can assume that h ( Q ) ≥ C . Otherwise, we can go directly to step 5. Step 1 . By Lemma 5.19 with k = 2 d , we can find a subinterval J A ′ , B ′ K of J A, B K with amplitude at most d containing d + 1 elements y , . . . , y d +1 of S . We use these y i as interpolation points in Proposition 5.18: we can write F = P ′ /Q ′ where P ′ , Q ′ have integer coefficients and satisfy h ( P ′ ) , h ( Q ′ ) ≤ d H + 22 d log(6 d ) + 7 d log(2 M ) + 2 dC ≤ C d H. To simplify the right hand side, we use the inequalities ≤ d , d ≤ D − ≤ M , log(2 M ) ≤ H , and ≤ H . We can take C
12 = 2 C .Lemma 5.17 allows us to bound the resultant of P ′ and Q ′ : h (cid:0) Res( P ′ , Q ′ ) (cid:1) ≤ d log(2 d ) + dh ( P ′ ) + dh ( Q ′ ) ≤ C d H with C
13 = 2 C
12 + 1 . In order to relate this to R , we use Lemma 5.7: we have P ′ = λP and Q ′ = λQ for some λ ∈ L × such that N ( λ ) ≥ . Therefore log | N L/ Q ( R ) | ≤ log (cid:12)(cid:12) N L/ Q (cid:0) Res( P ′ , Q ′ ) (cid:1)(cid:12)(cid:12) ≤ d L h (cid:0) Res( P ′ , Q ′ ) (cid:1) ≤ C d H with C
14 = d L C . Step 2 . Let p ∈ P L be a prime factor of R with valuation β p , and let p ∈ Z be the prime below p . Let r be the ideal from Proposition 5.8. We claim: n X i =1 v p ( s i ) ≤ n v p ( r ) + d (cid:18) β p + d L log( D )log N ( p ) + Dp − (cid:19) . ( ⋆ )We can work in the p -adic completion L p . Let π be a uniformizer of L p ,and let r = min { v p ( P ) , v p ( Q ) } . Then π r must divide r , so r ≤ v p ( r ) . Write e P = P/π r , e Q = Q/π r . Then one of e P and e Q is not divisible by π ; by symmetry,assume that π does not divide e Q . Then v p ( s i ) ≤ min (cid:8) β p , r + v p ( e Q ( y i )) (cid:9) ≤ v p ( r ) + min (cid:8) β p , v p ( e Q ( y i )) (cid:9) so the claim follows from Lemma 5.22.34nequality ( ⋆ ) bounds the p -adic valuation of the product of all s i ’s. Takingthe product over all prime divisors of R , we obtain n Y i =1 N ( s i ) ≤ N ( r ) n | N L/ Q ( R ) | d exp X p ∈P L , p | R p | p ∈P (cid:18) d d L log( D ) + D log( N ( p )) p − (cid:19) ≤ N ( r ) n | N L/ Q ( R ) | d exp (cid:16) d d L log( D ) log (cid:0) | N L/ Q ( R ) | (cid:1) + Dd L max (cid:8) , C
10 log log | N L/ Q ( R ) | (cid:9)(cid:17) . Indeed, there are at most log | N L/ Q ( R ) | prime ideals dividing R , and Lemma 5.21applies. Since log | N L/ Q ( R ) | ≤ C d H , we obtain log n Y i =1 N ( s i ) ! ≤ n log N ( r ) + C d H (cid:0) d L log( D ) (cid:1) + C Dd L log( C d H ) ≤ C (cid:0) d H log( D ) + D log( dH ) (cid:1) with C
15 = max (cid:8) d L C , log( N ( r )) + C d L (cid:0) C (cid:1)(cid:9) . Step 3 . We can now put into play the assumptions our assumptions about D and S being sufficiently large. Since D ≥ d H log( dH ) > exp(1) , and t/ log( t ) is increasing for t > exp(1) , we have D log( D ) ≥ d H dH ) / log( dH ) ≥ d H . Moreover S − d d L ≥ D − D D . Therefore log n Y i =1 N ( s i ) ! ≤ C D + D log( dH )) ≤ C log( dH )( S − d d L ) with C
16 = 12 C . This shows that in every subset of S − d d L elements of S ,at least one must satisfy log N ( s i ) ≤ C
16 log( dH ) . Hence we may assume that log N ( s i ) ≤ C
16 log( dH ) for every ≤ i ≤ r = d d L + 1 . This implies log (cid:12)(cid:12) N L/ Q (cid:0) Q ( y i ) (cid:1)(cid:12)(cid:12) ≤ C
16 log( dH ) + d L H. Step 4 . Let v ∈ V ∞ . By Lemma 5.15, the inequality | Q ( y i ) | v < | Q | v (2 M ) d ( d + 1) d values of i . Since V ∞ ≤ d L , we can find y ∈ S among the ( y i ) ≤ i ≤ r such that ∀ v ∈ V ∞ , | Q ( y ) | v ≥ | Q | v (2 M ) d ( d + 1) . Then h ( Q ) ≤ e h ( Q ) by Proposition 5.4, since h ( Q ) ≥ C ≤ d L log (cid:12)(cid:12) N L/ Q (cid:0) Q ( y ) (cid:1)(cid:12)(cid:12) + d log(2 M ) + log( d + 1) ≤ H + C d L log( dH ) + d log(2 M ) + log( d + 1) , in other words h ( Q ) ≤ H + C log( dH ) + d log(2 M ) with C
17 = 1 + C /d L . Step 5 . Since S ≥ d d L + 1 , by Lemma 5.15, we can find y ∈ S such that ∀ v ∈ V ∞ , | P | v ≤ (2 M ) d ( d + 1) | P ( y ) | v . Then h ( P ) ≤ h ( P ( y )) + d log(2 M ) + log( d + 1) because P ∈ Z L [ Y ] ≤ h ( Q ( y )) + H + d log(2 M ) + log( d + 1) as P ( y ) = Q ( y ) F ( y ) ≤ h ( Q ) + H + 2 d log(2 M ) + 2 log( d + 1) by Proposition 5.10 . The result follows with C C
17 + 2 since h ( F ) ≤ h ( P ) + h ( Q ) . Remark 5.24.
The proof gives an explicit value for C : we have C
14 = d L (4 C ,C
15 = max (cid:8) d L C , log( N ( r )) + C d L (cid:0) C (cid:1)(cid:9) with r as in Prop. 5.8 ,C C /d L . It would be interesting to know whether we can obtain an efficient boundon h ( F ) using only O ( d ) evaluation points, as was the case for polynomials,instead of e O ( d H ) . In this final section, we prove part 2 of Theorem 1.1 about the height of co-efficients in modular equations. We fix a PEL setting as in §3, and keep thenotation used there. We write S = Γ \ X + .36 .1 Heights of abelian varieties Different types of heights can be defined for an abelian variety A over L . The Faltings height h F ( A ) is defined in [11, §3] in terms of Arakelov degrees ofmetrized line bundles on A . If A is given a principal polarization L , and r ≥ is an even integer, we can also define the Theta height of level r of ( A, L ) ,denoted h Θ ,r ( A, L ) , as the projective height of level r theta constants of ( A, L ) [23, Def. 2.6]. Finally, if A is an abelian variety with PEL structure over L givenby a point z ∈ S where j , . . . , j n +1 are well defined, we can define the j -height of A as h j ( A ) = h (cid:0) j ( A ) , . . . , j n +1 ( A ) (cid:1) . We also write h F ( A ) = max { , h F ( A ) } and define h , h Θ ,r , h j similarly.The Faltings height behaves well with respect to isogenies. Proposition 6.1.
Let A , A ′ be abelian varieties over Q , and assume that anisogeny φ : A → A ′ exists. Then (cid:12)(cid:12) h F ( A ) − h F ( A ′ ) (cid:12)(cid:12) ≤
12 log(deg φ ) . Proof.
This is a consequence of [11, Lem. 5].Our goal here is to establish a relation between h j ( A ) and h F ( A ) when A isan abelian variety with PEL structure. Theta heights are an intermediate stepbetween concrete values of invariants and the Faltings height. Theorem 6.2 ([23, Cor. 1.3]) . For every integer g ≥ , and even integer r ≥ , there is a constant C ( g, r ) such that the following holds. Let ( A, L ) be aprincipally polarized abelian variety of dimension g defined over Q . Then (cid:12)(cid:12)(cid:12) h Θ ,r ( A, L ) − h F ( A ) (cid:12)(cid:12)(cid:12) ≤ C ( g, r ) log (cid:0) min { h F ( A ) , h Θ ,r ( A, L ) } + 2 (cid:1) . Proposition 6.3.
There is a nonzero polynomial P ∈ L [ Y , . . . Y n +1 ] such thatthe following holds. If A is the abelian variety with PEL structure associatedwith a point z ∈ S where j , . . . , j n +1 are well defined and P ( j , . . . , j n +1 ) = 0 ,then C h F ( A ) ≤ h j ( A ) ≤ C h F ( A ) . Proof.
By [21, Thm. 5.17], there is a finite covering S ′ of S which is a connectedShimura variety for the derived group G der . Write S ′ = Γ ′ \ X + where Γ ′ isa congruence subgroup of G der . Since G der ⊂ ker(det) , it embeds into thereductive group GSp g ( Q ) , where g = dim Q V . Therefore, by [21, Thm. 5.16],we can find a congruence subgroup Γ ′′ of G der and an even integer r ≥ suchthat Γ ′′ \ X + embeds in the moduli space A Θ ,r of principally polarized abelianvarieties of dimension g with level r Theta structure. We have a diagram37 S = e Γ \ X + S ′ = Γ ′ \ X + S ′′ = Γ ′′ \ X + A Θ ,r S p ′ p ′′ p where e Γ = Γ ′ ∩ Γ ′′ . The maps p , p ′ , p ′′ are finite coverings, and all the maps inthis diagram are algebraic.The modular interpretation is as follows. Let (Λ , ψ ) be the standard po-larized lattice associated with the connected component S , as in Theorem 2.1.We can find a sublattice Λ ′′ ⊂ Λ , and λ ∈ Q × such that (Λ ′′ , λψ ) is principallypolarized. A point z ∈ S defines a complex structure x on Λ ⊗ R = V ( R ) , upto action of Γ . Lifting to e z ∈ e S corresponds to considering x up to action of e Γ only, and this group leaves Λ ′′ and its level r Theta structure stable. Then theimage of e z in A Θ ,r is given by (Λ ′′ , x, λψ ) .In particular, if e x ∈ e S , and if A , A ′′ are the abelian varieties correspondingto e x in S and A Θ ,r respectively, then A and A ′′ are linked by an isogeny ofdegree d = / Λ ′′ ) . Hence, by Proposition 6.1 and Theorem 6.2, (cid:12)(cid:12) h F ( A ) − h Θ ,r ( A ′′ ) (cid:12)(cid:12) ≤ log( d )2 + C log (cid:18) min { h F ( A ) , h Θ ,r ( A ′′ ) } + 2 + log( d )2 (cid:19) ≤ C min { h F ( A ) , h Θ ,r ( A ′′ ) } . ( ⋆ )Denote by θ , . . . , θ k the Theta constants of level r . They define a projectiveembedding of A Θ ,r . Then the pullbacks of θ /θ , . . . , θ k /θ generate the functionfield of S ′′ . By definition, j , . . . , j n +1 are coordinates on S ; let f , . . . , f n +1 begenerators for the function field of e S . Since p and p ◦ p are finite coverings, thefunctions j , . . . , j n +1 have fractional expressions in terms of f , . . . , f n +1 , andthe same is true for the θ i /θ on S ′′ . These polynomial systems are genericallyof dimension 0, so they can be inverted using Gröbner bases. By the primitiveelement theorem, up to a change of coordinates on e S , we can assume that theseGröbner bases are in echelon form. We define e F to be the Zariski closed subsetof codimension 1 in e S where at least one of the denominators vanishes among allthe fractional expression considered, or where one of the f i is not well defined.Then U = S\ p ◦ p ( e F ) is open dense in S . Let P ∈ L [ j , . . . , j n +1 ] such that { P = 0 } ⊂ U .Let z ∈ S where j , . . . , j n +1 are well defined, and P ( j , . . . , j n +1 ) = 0 . Welook at the diagram above, from left to right. Lift z to a point e z ∈ e S ; byconstruction, e z / ∈ e F . By repeated applications of Propositions 5.10 and 5.12,we have h (cid:0) f ( e z (cid:1) , . . . , f n +1 ( e z )) ≤ C h (cid:0) j ( z ) , . . . , j n +1 ( z ) (cid:1) . Writing z ′′ = p ( e z ) , we also have h (cid:16) θ θ ( z ′′ ) , . . . , θ k θ ( z ′′ ) (cid:17) ≤ C h (cid:0) f ( e z ) , . . . , f n +1 ( e z ) (cid:1) . h Θ ,r ( A ′′ ) ≤ C h j ( A ) , so by equation ( ⋆ ) h F ( A ) ≤ C h j ( A ) . Going through the diagram from right to left gives the reverse inequality.From now on, we define U to be the Zariski open set in S where j , . . . , j n +1 are well defined and P ( j , . . . , j n +1 ) = 0 . Corollary 6.4.
Let A , A ′ be the abelian varieties with PEL structure associatedwith z, z ′ ∈ U . Assume that A and A ′ are related by an isogeny of degree ℓ . Then h j ( A ′ ) ≤ C ( h j ( A ) + log ℓ ) . Proof.
Combine Propositions 6.1 and 6.3. We may take C
19 = C . Remark 6.5.
We can presumably do better than Corollary 6.4. For instance,when studying j -invariants of isogenous elliptic curves, one can prove that | h ( j ( E )) − h ( j ( E ′ )) | is bounded by logarithmic terms [24, Thm. 1.1]. Thisis also the kind of bound provided by Theorem 6.2. However, the estimate inCorollary 6.4 is sufficient for our purposes, so we do not pursue this questionfurther. This problem should be solved before attempting to give meaningfulexplicit values for the constant C in Theorem 1.1. We keep the notation used in §3.2. Fix a Hecke correspondence H δ of de-gree d ( δ ) , and consider the modular equations Ψ δ,m of level δ between connectedcomponents S , S ′ of Sh K ( G, X + )( C ) . Written in canonical form, the Ψ δ,m canbe seen as elements of the ring L ( J , . . . , J n )[ J n +1 , Y , . . . , Y m ] .Let U (resp. U ′ ) be the open subset of S (resp. S ′ ) constructed as in §6.1.Define U δ ⊂ S to be the Zariski open set of all points [ x, g ] ∈ S such that [ x, g ] ∈ U , and [ x, gδγ ] ∈ U ′ for every γ ∈ K /K n +1 . Finally, we define V δ ⊂ L n to be the Zariski open set of all points ( j , . . . , j n ) where the equation ( E )(cf §3.1) has e distinct roots and the following property holds: if j n +1 is a rootof ( E ), then ( j , . . . , j n +1 ) are the invariants of some z ∈ U δ . In particular, thedenominators of modular equations do not vanish on V δ . Lemma 6.6.
There is a nonzero polynomial P δ ∈ L [ J , . . . , J n ] of total degreeat most Cd ( δ ) such that { P δ = 0 } ⊂ V δ .Proof. Let R be the the resultant of ( E ) and its derivative with respect to j n +1 .It is a rational fraction in j , . . . , j n of degree C . If R is well defined and doesnot vanish at ( j , . . . , j n ) , then the equation ( E ) has e distinct roots.Similarly, there is a polynomial Q ∈ L [ J , . . . , J n +1 ] such that every tuple ( j , . . . , j n +1 ) satisfying ( E ) and such that Q ( j , . . . , j n +1 ) = 0 lies in the image39f S . Taking the resultant with ( E ) with respect to j n +1 , we obtain a rationalfraction R ′ ( j , . . . , j n ) of degree C . If R ′ is well defined and does not vanish at j , . . . , j n , then for every root j n +1 of ( E ), the tuple ( j , . . . , j n +1 ) lies in theimage of S .Since U and U ′ only depend on the setting, the conditions defining U δ areequivalent to asking that a certain modular form λ on S of weight Cd ( δ ) doesnot vanish: this modular form is constructed as in §4.1. After increasing theweight by a constant, we can find a modular form ξ such that wt( λ ) = wt( ξ ) and the divisors of λ and ξ have no common codimension 1 components. ByPropositions 4.4 and 4.5, we can write λξ = e − X k =0 R k ( j , . . . , j n ) j kn +1 where the R k are rational fractions such that deg R k ≤ Cd ( δ ) for every k .Taking the resultant with ( E ) with respect to j n +1 yields a rational frac-tion R ′′ ( j , . . . , j n ) of degree still bounded by Cd ( δ ) . If R ′ , R ′′ are well de-fined and do not vanish at ( j , . . . , j n ) , then for every root j n +1 of ( E ), thetuple ( j , . . . , j n +1 ) comes from a point z ∈ U δ .Therefore we can take P δ to be the product of all numerators and denomi-nators of R , R ′ and R ′′ . Proposition 6.7.
Let ( j , . . . , j n ) ∈ V δ , and let ≤ m ≤ n + 1 . Then h (cid:0) Ψ δ,m ( j , . . . , j n ) (cid:1) ≤ C d ( δ ) (cid:0) h ( j , . . . , j n ) + log ℓ ( δ ) (cid:1) . Proof.
Let J be the set of roots of ( E ) at ( j , . . . , j n ) , and let j n +1 ∈ J .Let [ x, g ] be the point of S describing an abelian variety A with PEL struc-ture whose invariants are ( j , . . . , j n +1 ) . Then for every γ ∈ K /K m , thepoint [ x, gγδ ] describes an abelian variety A δ which is related to A by an isogenyof degree ℓ ( δ ) , by Corollary 2.7. Therefore, by Corollary 6.4, we have h (cid:0) j γ ,δ ([ x, g ]) , . . . , j γn +1 ,δ ([ x, g ]) (cid:1) ≤ C ( h (cid:0) j , . . . , j n +1 ) + log ℓ ( δ ) (cid:1) . We now take a closer look at the formula in Definition 3.1. We see that Ψ δ,m is the evaluation of a multivariate polynomial in the variables j γi,δ ([ x, g ]) for ≤ i ≤ m and γ ∈ K /K i . The number of variables is d + d d + · · · + d · · · d m ≤ m d ( δ ) , and each variable appears with degree 1. Therefore, by Proposition 5.10, h (cid:0) Ψ δ,m ( j , . . . , j n +1 ) (cid:1) ≤ m d ( δ ) log(2) + m d ( δ ) C (cid:0) h ( j , . . . , j n +1 ) + log ℓ ( δ ) (cid:1) ≤ Cd ( δ ) (cid:0) h ( j , . . . , j n +1 ) + log ℓ ( δ ) (cid:1) . In order to obtain Ψ δ,m ( j , . . . , j n ) , we interpolate a polynomial of degree e − in j n +1 where J is the set of interpolation points. By Propositions 5.10 and 5.12,we have h ( j n +1 ) ≤ Ch ( j , . . . , j n ) for every j n +1 ∈ J . Therefore, bluntly using the Lagrange interpolation formula is sufficient.40 .3 Heights of coefficients of modular equations
In this final subsection, we prove height bounds on modular equations usingProposition 6.7 and the results on heights of fractions from §5.
Definition 6.8.
We call an ( n, N , N ) -evaluation tree a rooted tree with depth n ,arity N at depths , . . . , n − , and arity N at depth n − , such that everyvertex but the root is labeled by an element of Z and the sons of every vertexare distinct.Let T be an ( n, N , N ) -evaluation tree, and let ≤ k ≤ n . The k -th evaluation set I k ( T ) of T is the set of points ( y , . . . , y k ) ∈ Z k such that y is a son of the root, and y i +1 is a son of y i for every ≤ i ≤ k − . We saythat T is bounded by M if every vertex is bounded by M in absolute value. Wesay that T has amplitude ( D , D ) if for every vertex y of depth ≤ r ≤ n − (resp. depth n − ) in T , the sons of y lie in an integer interval of amplitude atmost D (resp. D ).Finally, let T be an ( n, N , N ) -evaluation tree, let a = ( a , . . . , a n ) ∈ Z n ,and let F be a coefficient of Ψ δ,m for some ≤ m ≤ n + 1 . Write F = P/Q in irreducible form over L ( Y , . . . , Y n ) , and let d = deg( F ) ; assume that d ≥ .We say that T and a are valid evaluation data for F if the following conditionsare satisfied:1. T and a are bounded by some M such that M > max { d H log( d H ) , d d L } , where H = max (cid:8) , log(2 M ) , C d ( δ ) (cid:0) log( M ( M + 1)) + log ℓ ( δ ) (cid:1)(cid:9) . N = 2 d and N ≥ M/ .3. T has amplitude (4 d, M ) .4. For every ( y , . . . , y n ) ∈ I n ( T ) , the point ( j , . . . , j n ) = ( y y n + a , . . . , y n − y n + a n − , y n + a n ) belongs to V δ .5. For every ( y , . . . , y n − ) ∈ I n − ( T ) , the polynomials P and Q evaluatedat ( y Y + a , . . . , y n − Y + a n − , Y + a n ) are coprime in L [ Y ] .6. Q ( a , . . . , a n ) = 0 . Lemma 6.9.
Let F be a coefficient of Ψ δ,m of degree d ≥ . Then there existvalid evaluation data ( T, a ) for F with M = C (cid:0) d ( δ ) + log ℓ ( δ ) (cid:1) .Proof. According to Theorem 4.6, we have d ≤ Cd ( δ ) . Provided that C islarge enough, condition 1 in the definition above will be satisfied.41ince Q is a nonzero polynomial, and has degree at most d in Y , we can find a ∈ Z such that | a | ≤ M and the polynomial Q ( a , Y , . . . , Y n ) is nonzero. Iter-ating, we find a = ( a , . . . , a n ) ∈ Z n bounded by M such that Q ( a , . . . , a n ) = 0 .We now build the evaluation tree T down from the root. Let P δ be anequation for the complement of V δ as in Lemma 6.6, and define R δ = P δ ( Y Y n + a , . . . , Y n − Y n + a n − , Y n + a n ) which is a nonzero polynomial of degree at most Cd ( δ ) . Finally, denote R = Res( P, Q )( Y Y n + a , . . . , Y n − Y n + a n − , Y n + a n ) which is nonzero and has degree at most d . Therefore we can find d valuesfor y , bounded by Cd ( δ ) , and lying in an interval with amplitude at most d ,such that neither R δ nor R vanishes when evaluated at Y = y . We iterate thisto construct T up to depth n − with the right arity, bound and amplitude, suchthat the evaluations of R δ and R are nonzero at every ( y , . . . , y n − ) ∈ I n − ( T ) .Let ( y , . . . , y n − ) ∈ I n − ( T ) . Then, as before, at most Cd ( δ ) values for y n are forbidden as they make either R δ or R vanish. Therefore it is easy to see thatwe can complete the tree, perhaps at the cost of increasing C . Nonvanishingof R δ and R guarantees conditions 4 and 5 respectively.We are now ready to prove part 1 of Theorem 1.1. Theorem 6.10.
Let F ( j , . . . , j n ) be a coefficient of some Ψ δ,m in canonicalform. Then h ( F ) ≤ Cd ( δ ) log ℓ ( δ ) .Proof. By Lemma 6.9, we can find valid evaluation data ( T, a ) for F boundedby M = C (cid:0) d ( δ ) + log ℓ ( δ ) (cid:1) . After scaling P and Q , we can assume that in fact Q ( a , . . . , a n ) = 1 . Let ( y , . . . , y n − ) ∈ I n − ( T ) , and write e F ( Y ) = F ( y Y + a , . . . y n − Y + a n − , Y + a n ) . For every son y n of y n − in T , we have h (cid:0) y y n + a , . . . , y n − y n + a n (cid:1) ≤ log (cid:0) ( M + 1) M (cid:1) . Therefore by Proposition 6.7 h ( e F ( y n )) ≤ C d ( δ ) (cid:0) log(( M + 1) M ) + log ℓ ( δ ) (cid:1) = H. By construction, the hypotheses of Theorem 1.2 are fulfilled, so h ( e F ) ≤ H + C log( dH ) + 3 d log(2 M ) ≤ Cd ( δ ) log ℓ ( δ ) by Lemma 2.8 . Moreover, the quotient P ( y Y + a , . . . , y n − Y + a n − , Y + a n ) Q ( y Y + a , . . . , y n − Y + a n − , Y + a n )
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