Period relations for Riemann surfaces with many automorphisms
Luca Candelori, Jack Fogliasso, Christopher Marks, Skip Moses
aa r X i v : . [ m a t h . AG ] D ec PERIOD RELATIONS FOR RIEMANN SURFACES WITH MANYAUTOMORPHISMS
LUCA CANDELORI, JACK FOGLIASSO, CHRISTOPHER MARKS, AND SKIP MOSES
Abstract.
By employing the theory of vector-valued automorphic forms for non-unitarizable representations, we provide a new bound for the number of linear rela-tions with algebraic coefficients between the periods of an algebraic Riemann surfacewith many automorphisms. The previous best-known general bound for this num-ber was the genus of the Riemann surface, a result due to Wolfart. Our new boundsignificantly improves on this estimate, and it can be computed explicitly from thecanonical representation of the Riemann surface. As observed by Shiga and Wolfart,this bound may then be used to estimate the dimension of the endomorphism algebraof the Jacobian of the Riemann surface. We demonstrate with a few examples howthis improved bound allows one, in some instances, to actually compute the dimensionof this endomorphism algebra, and to determine whether the Jacobian has complexmultiplication. Introduction
The theory of vector-valued modular forms, though nascent in the work of variousnineteenth century authors, is a relatively recent development in mathematics. Perhapsthe leading motivation for working out a general theory in this area comes from two-dimensional conformal field theory, or more precisely from the theory of vertex operatoralgebras (VOAs). Indeed, in some sense the article [Zhu96] – in which Zhu proved thatthe graded dimensions of the simple modules for a rational VOA constitute a weaklyholomorphic vector-valued modular function – created a demand for understanding howsuch objects work in general, and what may be learned about them by studying therepresentations according to which they transform. Knopp and Mason [KM03, KM04,Mas07] were the first to give a systematic treatment of vector-valued modular forms,with significant contemporary contributions by Bantay and Gannon [BG07, Ban09] aswell. This initial motivation from theoretical physics leads one naturally to arithmeticconsiderations, and more recently a number of authors have utilized representationsof the modular group to study the bounded denominator conjecture for noncongruencemodular forms [Mas12, FM14, Mar15, FM16, FGM18], the vector-valued version ofwhich originated with Mason as well.In this article we utilize the vector-valued point of view for another arithmetic ap-plication, this time to the classical theory of Riemann surfaces and in particular to thestudy of the periods of an algebraic Riemann surface. More precisely, suppose X is acompact Riemann surface of genus g defined over Q . For any holomorphic differential ω defined over Q and any homology class [ γ ] ∈ H ( X, Z ), represented by a closed loop γ on X , the period associated to ω and γ is the complex number R γ ω . Following [Wol02], let V X := span Q (cid:26)Z γ ω : γ ∈ H ( X, Z ) , ω defined over Q (cid:27) , viewed as a Q -subspace of C . Since the dimension of the space of Q -differentials is g andthe rank of H ( X, Z ) is 2 g , we deduce that dim Q V X ≤ g . This bound is saturated,and in fact it is realized by generic compact Riemann surfaces of genus g .There are, however, special classes of algebraic curves where the above bound is asubstantial overestimate. For example, in [Wol02] Wolfart studies Riemann surfaceswith many automorphisms . These curves are characterized by the fact that if onefixes any desired genus and automorphism group, the resulting family of curves is zero-dimensional. For such a curve X , it is proven in [Wol02] that(1) dim Q V X ≤ g, reflecting the fact that the larger automorphism group imposes additional linear rela-tions among the periods.In this article we improve the bound (1), by inserting terms depending on the canon-ical representation of X . More precisely, let X be a curve with many automorphisms,let G be the group of automorphisms of X , and denote by ρ X : G op −→ GL(Ω ( X ))be the g -dimensional complex representation given by the action of g ∈ G on differentialsgiven by pull-back (note that pull-back is contravariant, thus the ‘op’ is needed to obtaina group representation). This is called the canonical representation of X , since Ω ( X ) isalso known as the canonical bundle of X . Its decomposition into irreducible characterscan readily be computed using the Chevalley-Weil formula [CW34] (see [Can18] for amodern account). Now any Riemann surface X with many automorphisms can actuallybe uniformized as a quotient [Wol02] X ( N ) := H /N, N ⊳ ∆where H denotes the complex upper half-plane and N ⊳ ∆ is a finite-index normalsubgroup of a co-compact Fuchsian triangle group ∆ = ∆( p, q, r ) = { δ , δ , δ ∞ : δ p = δ q = δ r ∞ = δ δ δ ∞ } acting as linear fractional transformations on H , so that G appears as the finite quotient∆ /N . In this presentation there are now three generators δ , δ , δ ∞ of G , which, byabuse of notation, represent the images of the corresponding generators of ∆. In termsof these generators, we prove: Theorem 1.1.
Let X be a curve with many automorphisms, uniformized as X = X ( N ) and with canonical representation ρ N := ρ X ( N ) . Then dim Q V X ≤ g − d − d − d ∞ where d x = dim C (ker( ρ N ( δ x ) − I g )) is the geometric multiplicity of the eigenvalue 1 in ρ N ( δ x ) , x ∈ { , , ∞} . ACOBIANS 3
In practice, the numbers d , d , d ∞ are easy to compute given the character of ρ N ,and they tend to be linear in g , providing a significant improvement over the bound(1). We demonstrate sample computations of the bound of Theorem 1.1 in Section 8below.The proof of Theorem 1.1 builds on the ideas of [CHMY18], and it relies cruciallyon the theory of vector-valued modular forms for non-unitarizable representations, aspioneered by Knopp and Mason [KM03], [KM04]. The proof of the Theorem, togetherwith the connection with modular forms, can be found in Sections 5 and 7 below.As demonstrated already by Shiga and Wolfart in [SW95], the knowledge of theinteger dim Q V X is essentially equivalent to the knowledge of the dimension of the Q -algebra of endomorphisms of Jac( X ) (see Theorem 3.2 below for a precise statement).There are many open questions about the endomorphism algebras of Jacobians, suchas Coleman’s conjecture regarding Riemann surfaces with complex multiplication andthe Ekedahl-Serre question regarding Riemann surfaces whose Jacobian is isogenous toa product of elliptic curves. These questions may be viewed as the main motivationfor studying linear relations between the periods of X . We demonstrate in Section8 how Theorem 1.1, in conjuction with the theorems of Shiga-Wolfart, allow one tocompute the decomposition of the Jacobians of some Riemann surfaces with manyautomorphisms. Acknowledgments.
The first author would like to thank Bill Hoffman and LingLong at Louisiana State University for helpful observations during the early stages ofthis project, as well as Tony Shaska at Oakland University of helpful conversations. Thethird author is extremely happy to acknowledge the ongoing contribution of GeoffreyMason to his research career, and this article is dedicated to him on the occasion of his70th birthday. The second, third, and fourth authors all benefited from internal grantsfunded by the Research Foundation and the College of Natural Sciences at CaliforniaState University, Chico.2.
Riemann surfaces with many automorphisms
Suppose X is a compact, connected Riemann surface. By the uniformization theoremof Riemann surfaces, X is isomorphic to a quotient Γ \U , where U is either the complexupper half-plane H , the Riemann sphere P , or the complex plane C and Γ ⊆ Aut( U ) isa discrete subgroup. By Riemann’s existence theorem, X also possesses the structureof an algebraic curve over C : that is, it can be defined as the zero locus of a collectionof polynomials with complex coefficients inside a suitable projective space. Supposethat these polynomials can be chosen to all have coefficients in the field of algebraicnumbers ¯ Q . Then we say that X is defined over ¯ Q . In this case, we can be more specificabout the group Γ uniformizing X : Theorem 2.1 (Belyi [Bel79], Wolfart [Wol02]) . Let X be a compact, connected Rie-mann surface defined over ¯ Q . Then X is isomorphic to Γ \ H , for some finite-indexsubgroup Γ of a co-compact Fuchsian triangle group ∆ ⊆ PSL ( R ) ≃ Aut( H ) . For any choice of positive integers p, q, r >
0, the triangle group ∆ = ∆( p, q, r )is defined abstractly as the infinite group generated by three elements δ , δ , δ ∞ with L. CANDELORI, J. FOGLIASSO, C. MARKS, AND S. MOSES presentation ∆ = ∆( p, q, r ) = { δ , δ , δ ∞ : δ p = δ q = δ r ∞ = δ δ δ ∞ } . The triangle group is
Fuchsian if it can be embedded in PSL ( R ), and it is co-compact ifthe corresponding quotient ∆ \ H is compact. In this article ∆ will always be Fuchsian,but not necessarily co-compact. For example, it is customary to allow any of p, q, r tobe ∞ whenever the corresponding generator is of infinite order, so that, for example,∆( ∞ , ∞ , ∞ ) = F = Γ(2) , ∆(2 , , ∞ ) = PSL ( Z ) . When this happens the triangle groups are no longer co-compact, in which case weapply a suitable compactification to the quotient ∆ \ H by adding cusps (as is the casefor modular curves , for example). In either case, we let X (∆) denote the correspond-ing Riemann surface, and similarly we denote by X (Γ) the compact Riemann surfacecorresponding to finite-index subgroups Γ ≤ ∆, each of which yields a finite ramifiedcover X (Γ) → X (∆).Theorem 2.1 is remarkable in that it allows one to study algebraic Riemann surfacesfrom the point of view of group theory and the representation theory of the trianglegroups ∆( p, q, r ), which are very much amenable to computations. For example, givena fixed triangle group ∆, we get a 1-1 correspondence { finite-index normal subgroups N ⊳ ∆ } ←→ { ramified Galois covers X ( N ) → X (∆) } . As is the case in algebraic number theory, where questions about general finite exten-sions
L/K of fields can be tackled by first studying
Galois extensions, here too thestudy of an algebraic Riemann surface X (Γ) → X (∆) can often be reduced to studying Galois covers, by taking a suitable ‘Galois closure’. By the above correspondence, thatis the same as restricting to normal subgroups N ⊆ ∆. For simplicity we will do so inthis article, keeping in mind that all the results presented can be extended to the caseof a general Riemann surface, not necessarily uniformized by a normal subgroup.Now, when N ⊳ ∆ is normal, the finite quotient G = ∆ /N acts as a group of auto-morphisms on X ( N ), which partially justifies the following: Definition 2.2 (Wolfart [Wol02]) . A Riemann surface X defined over ¯ Q has manyautomorphisms if there is an isomorphism X ≃ X ( N ), where N ⊳ ∆ is a normalsubgroup of a triangle group ∆.In general, let A ( G ) be the locus of genus g Riemann surfaces with automorphismgroup equal to G , viewed as a subvariety of the moduli space of all Riemann surfacesof genus g (see, e.g., [MSSV02]). Then X has many automorphisms if and only ifdim A ( G ) = 0.3. Endomorphism algebras of abelian varieties and transcendence
Suppose now that A is an abelian variety of dimension g > Q . Equiv-alently, A is a complex torus C g / Λ together with an embedding into projective spacedefined by polynomial equations with coefficients in ¯ Q . In this case, the g -dimensional C -vector space of holomorphic 1-forms for A contains the ¯ Q -vector space of 1-forms that ACOBIANS 5 are defined over ¯ Q , that is, those differentials that come by base-change from the reg-ular, K¨ahler differentials on the underlying algebraic variety. These ¯ Q -differentials canbe integrated over homology classes of cycles γ ∈ H ( A, Z ), and the resulting complexnumber Z γ ω ∈ C is a period of ω . This integration process is only possible by viewing A as a complexmanifold, a highly transcendental operation, and therefore periods tend to be transcen-dental numbers even if ω is defined over ¯ Q . To keep track of ‘how many’ transcendentalperiods we get, we make the following important definition: Definition 3.1 ([Wol02]) . Let A be an abelian variety defined over ¯ Q . The period ¯ Q -span of A is the ¯ Q -vector space V A := span ¯ Q (cid:26)Z γ ω : γ ∈ H ( A, Z ) , ω defined over ¯ Q (cid:27) ⊆ C If we regard C as an infinite-dimensional vector space over ¯ Q , then V A is seen to bea ¯ Q -subspace of C . It is clearly finite-dimensional: indeed, the ¯ Q -space of algebraicdifferential 1-forms for A is g -dimensional, by base-change, and the rank of H ( A, Z )as a free Z -module is 2 g . Therefore dim ¯ Q V A ≤ g for any abelian variety A defined over ¯ Q . We call this the trivial bound on dim ¯ Q V A .How can we get ‘extra’ linear ¯ Q -relations on V A ? If φ is a non-scalar endomorphismof A (defined over ¯ Q ) then it is easy to show that φ induces a ¯ Q -relation via its actionby pull-back on ω and its natural action on H ( A, Z ). The amazing fact is that all such linear relations are given in this way [SW95], as follows from W¨ustholz’ analyticsubgroup theorem. To state the precise result, letEnd ( A ) := End( A ) ⊗ Q be the endomorphism algebra of A . Theorem 3.2 (Shiga-Wolfart, [SW95]) . Suppose A is a simple abelian variety of di-mension g . Then dim ¯ Q V A = 2 g dim Q End ( A ) . More generally, if A is isogenous to a product A k × · · · × A k m m of simple abelian varities,each of dimension g i , then dim ¯ Q V A = m X i =1 g i dim Q End ( A i ) . Therefore, the knowledge of dim ¯ Q V A gives information about the dimension of theendomorphism algebra of A , and it is often enough to actually determine dim Q End ( A ). L. CANDELORI, J. FOGLIASSO, C. MARKS, AND S. MOSES
Example 3.3.
Suppose g = 1, so that A = E is an elliptic curve over ¯ Q . In this casethe space of ¯ Q -differentials is one-dimensional, say, generated by ω , and the integralhomology H ( E, Z ) is a rank 2 free Z -module generated by, say, γ and γ . The periodsof ω are the nonzero complex numbersΩ := Z γ ω, Ω := Z γ ω. Consider the ratio τ := Ω / Ω . Generically, this ratio will be transcendental, so thatdim ¯ Q V E = 2 and the elliptic curve attains the trivial bound of 2 g . By Theorem 3.2,this forces End ( E ) = Q , that is, the only endomorphisms of E act as scalars. On theother hand, when τ ∈ ¯ Q , then Theorem 3.2 implies that dim Q End ( E ) = 2. As is well-known [Sil86], in this case End( E ) is an order inside the ring of integers of an imaginaryquadratic extension of Q , and the elliptic curve is said to have complex multiplication .We thus recover the famous theorem of Schneider from 1936: τ ∈ ¯ Q ⇐⇒ E has complex multiplication . Example 3.4.
In general, when dim ¯ Q V A = 2 g then Theorem 3.2 implies that End( A ) = Z . 4. Jacobians with many automorphisms
Suppose now that X is a Riemann surface of genus g > J ( X ) be its Jacobian variety , the abelian variety of dimension g canonically associated to X bytaking the torus C g / Λ, where Λ is the rank 2 g lattice spanned by the periods of X . If X = X ( N ) → X (∆) has many automorphisms (Def. 2.2), then X is defined over ¯ Q , andits Jacobian variety J ( X ) is an abelian variety also defined over ¯ Q [Mil86]. In this case,by using the fact that the group of automorphisms of the Galois cover X ( N ) → X (∆)acts transitively on H ( X ( N ) , Z ), it is possible to give a sharper general bound on thedimension of the period ¯ Q -span of Jac( X ): Theorem 4.1 (Wolfart [Wol02]) . Suppose X is a Riemann surface of genus g > withmany automorphisms, let J ( X ) be its Jacobian variety and let V X := V Jac( X ) be theperiod ¯ Q -span of Jac( X ) . Then dim ¯ Q V X ≤ g. This bound on dim ¯ Q V X is so far the only known for a general Riemann surface withmany automorphisms. Though weak, it can already be used to deduce special structureswithin the endomorphism algebras of Jacobians with many automorphisms, as in thetwo examples below. Example 4.2.
Suppose X has genus g = 1, and that X has many automorphisms.Then dim ¯ Q V X = 1 so that Jac( X ) ≃ X has complex multiplication (compare to Ex-ample 3.3 in Section 3). Example 4.3.
In general, if X has many automorphisms then End(Jac( X )) = Z always(compare to Example 3.4 in Section 3). ACOBIANS 7 Vector-valued modular forms and periods
An excellent reference for the Riemann surface and automorphic function theorydescribed in this section is Shimura’s classic text [Shi94], and the vector-valued pointof view we utilize here emerges naturally out of work of Knopp and Mason [KM03,KM04, Mas07, Mas08] . We also note that the construction of Theorem 5.1 below is ageneralization of an idea used in Theorem 1.4 of the recent article [CHMY], where sucha result was proven in a different manner for subgroups of the modular group PSL ( Z ).Suppose N is a normal, finite index subgroup of a Fuchsian group of the first kind Γ ≤ PSL ( R ), and let H ∗ denote the union of H with the cusps of N in R ∪ {∞} (note thatif Γ is cocompact in PSL ( R ) then so is N , and this set of cusps is empty). Let X ( N ) = N \ H ∗ denote the compact Riemann surface X ( N ) associated to N , and assume that thegenus of X ( N ) is g ≥
1. The finite group G = Γ /N acts as automorphisms of the cover X ( N ) → X (Γ) and, via pullback (which is contravariant), acts on the vector space ofdifferential forms for X ( N ). This defines a g -dimensional linear representation G op → GL g ( C ), the canonical representation of the cover. We may lift this representation to agroup representation(2) ρ N : Γ op → GL g ( C )via the quotient map Γ → Γ /N . Let { ω , . . . , ω g } be a basis for the space Ω ( X ( N )) ofholomorphic 1-forms for X ( N ). Under the uniformization isomorphism X ( N ) ≃ N \ H ∗ ,we can lift ω , . . . , ω g to a basis { f , . . . , f g } of weight two cusp forms on N . The vector F := ( f , · · · , f g ) t is then a weight two vector-valued cusp form for the opposite representation ρ := ρ opp N .In other words, F ( τ ) is holomorphic in H , vanishes at any cusps that N may have, andfor each γ = (cid:18) a bc d (cid:19) ∈ Γ the functional equation(3) F ( γτ )( cτ + d ) − = ρ ( γ ) F ( τ )is satisfied.Fix a base-point τ ∈ X ( N ). For each 1 ≤ k ≤ g, let u k ( τ ) = Z ττ f k ( z ) dz. These functions have an important transformation property:
Theorem 5.1. U = ( u , . . . , u g , t is a holomorphic vector-valued automorphic func-tion for a representation π N : Γ → GL g +1 ( C ) arising from a non-trivial extension ofthe form → ρ → π N → → . Proof.
Writing U ( τ ) = (cid:16)R ττ F ( z ) dz, (cid:17) t , we first note that the functional equation (3)for F may be written as F ( γτ ) d ( γτ ) = ρ ( γ ) F ( τ ) dτ. L. CANDELORI, J. FOGLIASSO, C. MARKS, AND S. MOSES
For each γ ∈ PSL ( Z ), set(4) Ω( γ ) = Z γτ τ F ( z ) dz. Then for any γ ∈ Γ we have U ( γτ ) = (cid:18)R γττ F ( z ) dz (cid:19) = (cid:18)R γτ τ F ( z ) dz + R γτγτ F ( z ) dz (cid:19) = (cid:18) Ω( γ ) + R ττ F ( γz ) d ( γz )1 (cid:19) = (cid:18) Ω( γ ) + ρ ( γ ) R ττ F ( z ) dz (cid:19) = π N ( γ ) U ( τ )where we define π N ( γ ) = (cid:18) ρ ( γ ) Ω( γ )0 1 (cid:19) ∈ GL g +1 ( C ) . One sees now that π N is a representation of PSL ( Z ) if and only if the relationΩ( γ γ ) = ρ ( γ )Ω( γ ) + Ω( γ )obtains for any γ , γ ∈ PSL ( Z ). Similar to the above computation, we haveΩ( γ γ ) = Z ( γ γ ) τ τ F ( z ) dz = Z γ τ τ F ( z ) dz + Z γ ( γ τ ) γ τ F ( z ) dz = Ω( γ ) + Z γ τ τ F ( γ z ) d ( γ z )= Ω( γ ) + ρ ( γ ) Z γ τ τ F ( z ) dz = Ω( γ ) + ρ ( γ )Ω( γ )so π N is indeed a representation. To see that this extension of ρ by the trivial characterof Γ does not split, we simply need to observe that if it did, then ker( π N ) would contain N . This would imply that the Riemann surface X ( N ) admits nonconstant holomorphicfunctions, namely the u k , but this is impossible since X ( N ) is compact. This completesthe proof of the theorem. (cid:3) For each γ ∈ N , the map τ γτ defines a closed loop on X ( N ). One observes thatthe proof of Proposition 1.4 in [Man72] remains valid for any Fuchsian group of the firstkind in PSL ( R ), so there is a surjective homomorphism of groups N → H ( X ( N ) , Z )given by γ → [ τ γτ ] whose kernel is generated by the commutator subgroup N ′ together with the elliptic and parabolic elements of N . Thus there is some set γ j , ACOBIANS 9 ≤ j ≤ g , of hyperbolic elements of N that surjects onto a basis of H ( X ( N ) , Z ),and using the above notation we find that the vectors Ω( γ j ) ∈ C g generate the periodlattice Λ that defines the Jacobian of X ( N ). On the other hand, from the proof ofTheorem 5.1 we see for each j we have π N ( γ ) = (cid:18) ρ ( γ ) Ω( γ )0 1 (cid:19) , so in fact these vectors appear in the matrices defining the extension representation π N . Thus, when these periods are algebraic Theorem 5.1 allows one to compute themexplicitly, as was already demonstrated for modular curves in [CHMY18]. Even whenthey cannot be computed explicitly (since in general at least some of them are tran-scendental), the existence and analysis of π N yields important information about theperiods, as we demonstrate below in Section 7.6. Group cohomology
Let N be a normal subgroup of finite index in a Fuchsian group Γ of the first kind,and let ρ N be the canonical representation associated to the cover X ( N ) → X (Γ).Theorem 5.1 shows that the periods of the curve X ( N ) are the matrix entries of anon-trivial extension π N of the trivial representation by ρ N . In this section we brieflyrecall how such extensions are classified by group cohomology .Let ρ : Γ → GL( V ) be a finite-dimensional complex representation of Γ. For anyinteger n ≥
1, denote by C n (Γ , ρ ) the group of functions Γ n → V (the n-cochains ) andlet d n +1 : C n → C n +1 be the coboundary homomorphism given by d n +1 κ ( g , . . . , g n +1 ) = ρ ( g ) κ ( g , . . . , g n ) + n X i =1 ( − i κ ( g , . . . , g i − , g i g i +1 , . . . , g n +1 )++ ( − n +1 κ ( g , . . . , g n )For n ≥ Z n (Γ , ρ ) := ker d n +1 be the group of n-cocyles and for n ≥ B n (Γ , ρ ) :=im d n be the group of n-coboundaries . Since d n +1 ◦ d n = 0, we have B n ⊂ Z n for all n ≥ C • , d • ) forms a cochain complex ) and we can define the n-th group cohomology by H (Γ , ρ ) := V ρ = { v ∈ V : ρ ( v ) = v } H n (Γ , ρ ) := Z n (Γ , ρ ) B n (Γ , ρ ) , n ≥ . In particular, for n = 1 we have: Z (Γ , ρ ) = { κ : Γ → V | κ ( g g ) = ρ ( g ) κ ( g ) + κ ( g ) } B (Γ , ρ ) = { κ : Γ → V | κ ( g ) = ρ ( g ) v − v for some v ∈ V } . Let now Ext ( ρ,
1) be the set of isomorphism classes of extensions of Γ-representations ρ → π →
1, where an isomorphism π ≃ π ′ is an isomorphism of Γ-representations whichrestricts to an isomorphism ρ ≃ ρ ′ . Given any π ∈ Ext ( ρ, π such that π ( g ) = (cid:18) ρ ( g ) κ π ( g )0 1 (cid:19) , as in the proof of Theorem 5.1. An easy computation shows that κ π ∈ Z (Γ , ρ ) andthat its class in H completely classifies the extension π : Proposition 6.1.
The map π κ π gives a bijection Ext ( ρ, ←→ H (Γ , ρ ) . (cid:3) Note that the group structure on H (Γ , ρ ) goes over naturally to give the Baer sum ofextensions on Ext ( ρ, C -vector spaces. The zerovector in Ext ( ρ, B (Γ , ρ ), is given by thoserepresentations π which are completely reducible, π ≃ ρ ⊕
1. In particular, note thatif Γ is a finite group then H (Γ , ρ ) = 0 always, since in this case the category of Γ-representations is semi-simple. Going back to Theorem 5.1, for example, we see thatthe canonical representation ρ N cannot have any non-trivial extensions π N of it by 1 ifviewed as a representation of G := Γ /N . However, when lifted to a representation ofΓ, Theorem 5.1 says that a canonical non-trivial extension class π N ∈ H (Γ , ρ N )becomes available, and is determined by the periods of X ( N ). In the next section westudy the properties of this class and connect it to the endomorphism algebra of theJacobian of X ( N ).7. A sharper bound on the dimension of the period ¯ Q -span of Riemannsurfaces with many automorphisms We now go back to the case of triangle groups ∆ = ∆( p, q, r ). Let
N ⊳ ∆ and let ρ N be the canonical representation of X ( N ) → X (∆), lifted to a representation of ∆ as in(2). As explained in Section 6, Theorem 5.1 gives a canonical class π N = 0 ∈ H (∆ , ρ N )encoding the periods of X ( N ). Using this new invariant of the curve X ( N ), we obtainthe following bound: Theorem 7.1.
Suppose X ( N ) → X (∆) is a Riemann surface of genus g > withmany automorphisms, and let ρ N : ∆ op → GL g ( C ) be its canonical representation. Let V X ( N ) := V Jac( X ( N )) be the span of ¯ Q -periods of Jac( X ( N )) . Then dim ¯ Q V X ( N ) ≤ g − d − d − d ∞ where d x = dim C (ker( ρ N ( δ x ) − I g )) is the geometric multiplicity of the eigenvalue 1 in ρ N ( δ x ) , x ∈ { , , ∞} .Proof. First, let˜∆ := ∆( p, q, ∞ ) = { δ , δ : δ p = 1 , δ q = 1 } ≃ Z /p Z ∗ Z /q Z . Let δ ∞ := δ − δ − ∈ ˜∆ and let K denote the smallest normal subgroup of ˜∆ containing δ r ∞ . Taking the quotient by K determines a surjective homomorphism ˜∆ → ∆, andwe can lift ρ N to a representation of ˜∆ (we will abuse notation and denote both ρ N and its lift to ˜∆ by the same symbol). We need the following lemma about the groupcohomology of ˜∆. ACOBIANS 11
Lemma 7.2.
Let ρ : ˜∆ → GL( V ) be any g -dimensional complex representation with V ρ = { } . Let A = ρ ( δ ) , B = ρ ( δ ) and let V A , V B be the eigenspaces of eigenvalue 1of A, B , respectively. Then there is canonical isomorphism of complex vector spaces VV A + V B δ −→ H ( ˜∆ , ρ ) given as follows: write v = v − v ∈ VV A + V B and let δ ( v ) = κ v be the 1-cocycle in Z ( ˜∆ , ρ ) determined by κ v ( A ) = ρ ( A ) v − v , κ v ( B ) = ρ ( B ) v − v . In particular, dim H ( ˜∆ , ρ ) = g − d − d .Proof. The group cohomology of a free product with two factors can be computed viathe Mayer-Vietoris sequence, which in the case of ˜∆ ≃ Z /p Z ∗ Z /q Z gives the exactsequence of vector spaces(5) V ρ → V A ⊕ V B ( v A ,v B ) v A − v B −−−−−−−−−−→ V δ −→ H ( ˜∆ , ρ ) → H ( Z /p Z , ρ ) ⊕ H ( Z /q Z , ρ ) , where δ is the map given in the statement of the Lemma. We want to show that δ is anisomorphism. By assumption V ρ = { } . We also have that H ( Z /N Z , ρ ) = 0 for any N ≥
1. Indeed, if the group Z /N Z is generated by γ , then any extension ρ → π → Z /N Z -representations is determined by the matrix π ( γ ) = (cid:18) ρ ( γ ) κ ( γ )0 1 (cid:19) . But π ( γ ) has finite order (dividing N ) which means it is diagonalizable, and κ ( γ )differs from 0 by a coboundary. Therefore H ( Z /p Z , ρ ) = H ( Z /q Z , ρ ) = 0 and δ is anisomorphism. (cid:3) Since the fixed vectors of ρ N are the differential forms on X (∆), and since X (∆) hasgenus zero, ρ N satisfies the hypothesis of Lemma 7.2. Using this Lemma, we may definea canonical basis for H ( ˜∆ , ρ N ) as follows: let a = dim V A = d , b = dim V B = d andchoose bases v A = { v A , . . . , v Aa } , v B = { v B , . . . , v Bb } for V A , V B , respectively. Since V A ∩ V B = V ρ N = { } , we may extend v A ∪ v B to a basis { v , . . . , v d } ∪ v A ∪ v B of V ,where d := g − a − b . With respect to this choice of basis, the ‘fundamental cocycles’ κ , . . . , κ d , determined by κ ( δ ) = = e , κ ( δ ) = , . . . , κ d ( δ ) = = e d , κ d ( δ ) = form a canonical basis for H ( e ∆ , ρ N ). We next need a lemma which determines the‘field of definition’ of the κ i ’s. Lemma 7.3.
Let ρ : e ∆ → GL( V ) be any finite-dimensional complex representation offinite image and with V ρ = { } . Suppose dim H ( e ∆ , ρ ) = d. Then there is a choice ofbasis for V such that the fundamental coycles κ i have entries in Q ( ζ e ) , where ζ e is aprimitive e -th root of unity and e is the exponent of the finite group e ∆ / ker ρ .Proof. Since ρ is of finite image, a basis β can be chosen so that the matrix entriesof ρ belong to Q ( ζ e ), where ζ e is a primitive e -th root of unity and e is the exponentof the finite group e ∆ / ker ρ [Bra45]. From this basis β we may compute a basis for V A + V B and extend it to a basis of V to get fundamental cocyles κ i ( A ) , κ i ( B ), withoutchanging the field of definition Q ( ζ e ) of the entries of ρ , by basic linear algebra. Since κ i ( A ) , κ i ( B ) ∈ Z g , the entries of a general vector κ i ( γ ), γ ∈ e ∆, are polynomial in thematrix entries of ρ , thus they are also contained in Q ( ζ e ). (cid:3) Let now π N be the representation of Theorem 5.1, viewed as a representation of e ∆ → ∆, and let κ π N be Ω, the 1-cocycle (4) determined by π N as shown the proof ofthat theorem. Since ρ N has no fixed vectors, we can write Ω in terms of the fundamentalcocycles κ i , Ω = d X i =1 λ i κ i , λ i ∈ C where d = dim H ( e ∆ , ρ N ). As discussed after the proof of Theorem 5.1, the periods ofthe curve X ( N ) are precisely the entries of Ω( γ ), where γ runs through the hyperbolicelements of N . The periods of X ( N ) are thus the entries of the vectorsΩ( γ ) = d X i =1 λ i κ i ( γ ) . Since ρ N has finite image, a basis can be chosen so that the entries of κ i ( γ i ) are con-tained in Q ( ζ e ), where e is the exponent of the finite group G = ∆ /N , by Lemma 7.3.Therefore, the periods can be expressed as Q -linear combinations of complex numbers λ , . . . , λ d ∈ C (which are potentially linearly dependent over Q ). By the result ofLemma 7.2, we deduce that(6) dim Q V X ( N ) ≤ d = dim H ( e ∆ , ρ N ) = g − d − d which proves Theorem 7.1 for the larger group e ∆. In order to deduce the sharperresult for ∆, we use the fact that π N is actually a representation of ∆, and therefore ACOBIANS 13 π N ( δ − r ∞ ) = I g +1 . In particular,0 = Ω( δ − r ∞ )= r − X j =0 ρ N ( δ j ∞ )Ω( δ − ∞ )= r − X j =0 ρ N ( δ j ∞ )Ω( δ δ )= r − X j =0 ρ N ( δ j ∞ )Ω( δ ) (since Ω( δ ) = d X i =1 λ i κ i ( δ ) = 0)= d X i =1 λ i T κ i ( δ )where for ease of notation we let T := P r − j =0 ρ N ( δ j ∞ ). Since ρ N is a representationof ∆, ρ N ( δ ∞ ) is of order dividing r . It is therefore diagonalizable as ρ N ( δ ∞ ) ∼ diag( α , . . . , α g ), where each α i is an r -th root of unity. By a standard computationwith roots of unity it follows that T has the block decomposition T ∼ (cid:18) r I d ∞
00 0 (cid:19) with respect to a basis of eigenvectors for ρ N ( δ ∞ ), where the basis for the eigenspace ofeigenvalue α = 1 is placed first. Let M be the change-of-basis matrix corresponding tothis choice of basis. Then the entries of M can be chosen to be in Q , since the entriesof ρ N ( δ ∞ ) are. We thus have the vector equation0 = d X i =1 λ i M − (cid:18) r I d ∞
00 0 (cid:19)
M κ i ( δ )which shows that we have an additional d ∞ -many independent Q -linear relations amongthe λ i ’s. Combined with (6), we get the sharper bounddim Q V X ( N ) ≤ g − d − d − d ∞ , which proves Theorem 7.1. (cid:3) Remark 7.4.
Consider the inflation-restriction exact sequence of vector spaces0 → H (∆ , ρ N ) → H ( e ∆ , ρ N ) Res → H ( K, ρ N ) ∆ → K ֒ → e ∆ → ∆, where Res( κ ) = [ κ ( δ ∞ ) r ]. Thenby the proof of Theorem 7.1 it is clear that dim H (∆ , ρ N ) = g − d − d − d ∞ so thebound of Theorem 7.1 can be stated more concisely asdim Q V X ( N ) ≤ dim H (∆ , ρ N ) . Remark 7.5.
The bound of Theorem 7.1 tends to be much smaller than that of The-orem 4.1. Indeed, for ∆ = ∆( p, q, r ) ‘on average’ we expect thatdim C H (∆ , ρ N ) ∼ g − g/p − g/q − g/r ≪ g. The numbers d , d , d ∞ are easy to compute. Indeed, the character χ N of ρ N can becomputed explicitly using the Chevalley-Weil formula [CW34] (see [Can18] for a modernaccount), and the numbers d , d , d ∞ can then be computed from the knowledge of χ N and of the character table of the finite group G = ∆ /N . These computations can easilybe implemented in GAP or MAGMA, as we show in the next section.8. Examples
Example 8.1. (Bolza surface) In genus two, the Riemann surface with largest au-tomorphism group is known as the
Bolza surface . This Riemann surface has manyautomorphisms, and it can be uniformized as X ( N ), by an index 48 normal subgroup N ⊳ ∆(2 , , G ≃ GL(2 , ρ N in this case is irreducible, which implies that J ( X ) ∼ A k for a simple abelian variety A [Wol02]. The only possibilities are either dim A = 1, k = 2 or dim A = 2 , k = 1. We havedim H (∆(2 , , , ρ N ) = 1and so by Theorems 7.1 and 3.2dim ¯ Q V X = 2 dim A dim Q End ( A ) = 1 . If dim A = 2, then this can only happen if dim Q End ( A ) = 8, which is impossible sincedim Q End ( A ) ≤ A . Therefore Jac( X ) ∼ E for some ellipticcurve E . Moreover, by Example 3.3 we deduce that E has complex multiplication, sothat X has complex multiplication as well. The period matrix of X can be computedexplicitly using our methods, as in [CHMY18]. Example 8.2 (Klein quartic) . There is a unique normal subgroup
N ⊳ ∆(2 , ,
7) = ∆ ofindex 168. The corresponding curve X = X ( N ) is the famous ‘Klein quartic’, of genus g = 3. By Chevalley-Weil, the canonical representation ρ N of this curve is irreducible,and dim H (∆ , ρ N ) = 1 . Since ρ N is irreducible, this means that Jac( X ) ∼ A k , where A is a simple abelianvariety [Wol02]. Since dim Jac( X ) = g = 3, either dim A = 1 and k = 3 or dim A = 3and k = 1. We rule out the latter as follows. Using Thm. 3.2 and Thm. 7.1 we deducethat dim ¯ Q V X = 2 dim A dim Q End ( A ) = 1 . If dim A = 3, we get dim Q End ( A ) = 18. This is impossible, since for any abelianthreefold, dim Q End ( A ) ≤
6. Therefore dim A = 1 , k = 3 and Jac( X ) ∼ A is totally ACOBIANS 15 decomposable and dim Q End ( A ) = 2, i.e., A is a CM elliptic curve. Thus J ( X ) is alsoCM, and again it has totally decomposable Jacobian. Example 8.3 (Macbeath curve) . There is a unique normal subgroup
N ⊳ ∆(2 , ,
7) = ∆of index 504. The corresponding curve X = X ( N ) is the famous ‘Macbeath curve’,of genus g = 7. Again by Chevalley-Weil, this curve has an irreducible canonicalrepresentation ρ N with dim H (∆ , ρ N ) = 2 . Since ρ N is irreducible, this means again that Jac( X ) ∼ A k , with A simple. Sincedim Jac( X ) = g = 7, either dim A = 1 and k = 7 or dim A = 7 and k = 1. UsingTheorem 3.2 and Theorem 7.1 we deduce thatdim A dim Q End ( A ) ≤ . If dim A = 7, we get dim Q End ( A ) ≥
49, which is impossible. Therefore Jac( X ) ∼ A with A = E an elliptic curve, so that Jac( X ) is totally decomposable. The questionof whether E has complex multiplication or not was posed by Berry and Tretkoff in[BT92]. Later Wolfart [Wol02] proved that E does not have CM by finding an explicitWeierstrass equation for E . Therefore(7) 2dim A dim Q End ( A ) = 2 = dim H (∆ , ρ N ) . Example 8.4.
In genus 14, there are three Hurwitz curves with automorphism groupisomorphic to PSL(2 , , , H ( X ( N ) , ρ N ) = 2 . Writing Jac( X ) ∼ A k , we must again have(8) dim A dim Q End ( A ) ≤ , as for the Macbeath curve. This equality rules out dim A = 7 ,
14 but both dim A = 1 , X ) ∼ E for A = E an elliptic curve,using the group algebra decomposition of Jac( X ) [BL04, 13.6], [Pau11]. By (8) wecannot conclude whether E has complex multiplication or not. It would be interestingto determine whether or not each of the three genus 14 Hurwitz curves has complexmultiplication or, equivalently, whether or not the equality (7) is attained for theseHurwitz curves. Remark 8.5.
As shown in the above examples, it is possible to use our Theorem 7.1to give a sufficient criterion for the Riemann surface X to have complex multiplication .This criterion is similar in spirit to that of Streit [Str01]. It would be interesting towork out the exact relation between the two methods to detect complex multiplication. Remark 8.6.
Using the group algebra decomposition of Jac( X ) [BL04, 13.6] it ispossible to give bounds for dim V X via the Shiga-Wolfart Theorem 3.2. It would beinteresting to determine precisely which part of the group algebra decomposition isdetermined by the bounds given in Theorem 7.1. References [Ban09] P. Bantay. Vector-valued modular forms. In
Vertex operator algebras and related areas ,volume 497 of
Contemp. Math. , pages 19–31. Amer. Math. Soc., Providence, RI, 2009.[Bel79] G. V. Belyi. Galois extensions of a maximal cyclotomic field.
Izv. Akad. Nauk SSSR Ser.Mat. , 43(2):267–276, 479, 1979.[BG07] Peter Bantay and Terry Gannon. Vector-valued modular functions for the modular groupand the hypergeometric equation.
Commun. Number Theory Phys. , 1(4):651–680, 2007.[BL04] Christina Birkenhake and Herbert Lange.
Complex abelian varieties , volume 302 of
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemat-ical Sciences] . Springer-Verlag, Berlin, second edition, 2004.[Bra45] Richard Brauer. On the representation of a group of order g in the field of the g -th rootsof unity. Amer. J. Math. , 67:461–471, 1945.[BT92] Kevin Berry and Marvin Tretkoff. The period matrix of Macbeath’s curve of genus seven.In
Curves, Jacobians, and abelian varieties (Amherst, MA, 1990) , volume 136 of
Contemp.Math. , pages 31–40. Amer. Math. Soc., Providence, RI, 1992.[Can18] Luca Candelori. The Chevalley-Weil formula for orbifold curves.
SIGMA Symmetry Inte-grability Geom. Methods Appl. , 14:Paper No. 071, 17, 2018.[CHMY18] Luca Candelori, Tucker Hartland, Christopher Marks, and Diego Y´epez. Indecomposablevector-valued modular forms and periods of modular curves.
Res. Number Theory , 4(2):Art.17, 24, 2018.[CW34] C. Chevalley and A. Weil. ¨Uber das verhalten der integrale 1. gattung bei automorphismendes funktionenk¨orpers.
Abh. Math. Sem. Univ. Hamburg , 10(1):358–361, 1934.[FGM18] Cameron Franc, Terry Gannon, and Geoffrey Mason. On unbounded denominators andhypergeometric series.
J. Number Theory , 192:197–220, 2018.[FM14] Cameron Franc and Geoffrey Mason. Fourier coefficients of vector-valued modular formsof dimension 2.
Canad. Math. Bull. , 57(3):485–494, 2014.[FM16] Cameron Franc and Geoffrey Mason. Three-dimensional imprimitive representations of themodular group and their associated modular forms.
J. Number Theory , 160:186–214, 2016.[KM03] Marvin Knopp and Geoffrey Mason. On vector-valued modular forms and their Fouriercoefficients.
Acta Arith. , 110(2):117–124, 2003.[KM04] Marvin Knopp and Geoffrey Mason. Vector-valued modular forms and Poincar´e series.
Illinois J. Math. , 48(4):1345–1366, 2004.[Man72] Ju. I. Manin. Parabolic points and zeta functions of modular curves.
Izv. Akad. Nauk SSSRSer. Mat. , 36:19–66, 1972.[Mar15] Christopher Marks. Fourier coefficients of three-dimensional vector-valued modular forms.
Commun. Number Theory Phys. , 9(2):387–412, 2015.[Mas07] Geoffrey Mason. Vector-valued modular forms and linear differential operators.
Int. J.Number Theory , 3(3):377–390, 2007.[Mas08] Geoffrey Mason. 2-dimensional vector-valued modular forms.
Ramanujan J. , 17(3):405–427, 2008.[Mas12] Geoffrey Mason. On the Fourier coefficients of 2-dimensional vector-valued modular forms.
Proc. Amer. Math. Soc. , 140(6):1921–1930, 2012.[Mil86] J. S. Milne. Jacobian varieties. In
Arithmetic geometry (Storrs, Conn., 1984) , pages 167–212. Springer, New York, 1986.[MSSV02] K. Magaard, T. Shaska, S. Shpectorov, and H. V¨olklein. The locus of curves with pre-scribed automorphism group.
RIMS Kyoto Series , (1267):112–141, 2002. Communicationsin arithmetic fundamental groups (Kyoto, 1999/2001).[Pau11] Jennifer Paulhus. Jacobian decomposition of the hurwitz curve of genus 14. , 2011.
ACOBIANS 17 [Shi94] Goro Shimura.
Introduction to the arithmetic theory of automorphic functions , volume 11 of
Publications of the Mathematical Society of Japan . Princeton University Press, Princeton,NJ, 1994. Reprint of the 1971 original, Kanˆo Memorial Lectures, 1.[Sil86] Joseph H. Silverman.
The arithmetic of elliptic curves , volume 106 of
Graduate Texts inMathematics . Springer-Verlag, New York, 1986.[Str01] M. Streit. Period matrices and representation theory.
Abh. Math. Sem. Univ. Hamburg ,71:279–290, 2001.[SW95] Hironori Shiga and J¨urgen Wolfart. Criteria for complex multiplication and transcendenceproperties of automorphic functions.
J. Reine Angew. Math. , 463:1–25, 1995.[Wol02] J¨urgen Wolfart. Regular dessins, endomorphisms of Jacobians, and transcendence. In
Apanorama of number theory or the view from Baker’s garden (Z¨urich, 1999) , pages 107–120.Cambridge Univ. Press, Cambridge, 2002.[Zhu96] Yongchang Zhu. Modular invariance of characters of vertex operator algebras.
J. Amer.Math. Soc. , 9(1):237–302, 1996.
Department of Mathematics, Wayne State University, 656 W Kirby, Detroit, MI48202, USA
E-mail address : [email protected] Department of Mathematics and Statistics, California State University, Chico, 400West First Street, Chico, CA 95929, USA
E-mail address : [email protected]; Department of Mathematics and Statistics, California State University, Chico, 400West First Street, Chico, CA 95929, USA
E-mail address : [email protected] Department of Mathematics and Statistics, California State University, Chico, 400West First Street, Chico, CA 95929, USA
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