Shimura Degrees, New Modular Degrees, and Congruence Primes Alyson Deines CCR La Jolla October 2, 2015 Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 1 / 34
Elliptic Curve Parameterization � We can parameterize modular elliptic curves by modular curves and Shimura curves. � It’s often difficult to write down the map, but the degree is accessible. � We can usually find the optimal quotient. � This information gives us another way to study all of these objects, and even the related modular forms by way of congruence numbers. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 2 / 34
Modular Elliptic Curves �� � � a b � Γ 0 ( N ) = ∈ GL 2 ( Z ) : c ≡ 0 (mod N ) c d � X 0 ( N ) - the modular curve Γ 0 ( N ) \ H ∪ cusps � J 0 ( N ) - Jacobian of X 0 ( N ) � E - a modular elliptic curve over Q of conductor N , with E = C / Λ � f E - the modular form in S 2 ( N ) associated to E with Fourier coefficients a n . Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 3 / 34
Modular Elliptic Curves E is modular, so we have the following surjective map: π : X 0 ( N ) → E given by τ ∈ X 0 ( N )( C ) � i ∞ ∞ a n f ( τ ′ ) d τ ′ = n e 2 π in τ ∈ C / Λ . � π ( τ ) = − 2 π i τ n =1 Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 4 / 34
Modular Degree Let π : X 0 ( N ) → E be the modular parameterization. We have such map for any curve isogenous to E . Definition The modular degree of E is the minimal such degree. Definition The optimal quotient is the curve E in the isogeny class which gives the minimal degree. Alternatively, the optimal quotient is the curve E in the isogeny class such that the map J 0 ( N ) → E has connected kernel. Definition If E is an optimal quotient of J 0 ( N ), π : J 0 ( N ) → E , π ∨ : E → J 0 ( N ) π ◦ π ∨ ∈ End( E ) is multiplication by an integer m E . This integer m E is called the modular degree of E . Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 5 / 34
Shimura Curves Let F be a totally real number field. Fix B an indefinite quaternion algebra over F of discriminant D and O ⊂ B an Eichler order of level M . � Define Γ D 0 ( M ) to be the group of norm-1 units in O . � Our Shimura curve is X D 0 ( M ) = Γ D 0 ( M ) \ H . � We denote its Jacobian by J D 0 ( M ). Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 6 / 34
Quaternionic Modular Forms Definition A quaternionic modular form of weight k on Γ D 0 ( M ) is a holomorphic function f on H such that � � a b f ( γτ ) = ( c γ + d ) k f ( τ ) for all γ = ∈ Γ D 0 ( M ) . c d The space of such forms is denoted by M D k ( M ), and cusp forms by S D k ( M ). Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 7 / 34
New and Old spaces Let D , M , and N be positive integers such that N = DM (or ideals in a totally real number field F ). Then for f ( τ ) ∈ S ¯ 2 ( M ) and r | D , f ( r τ ) ∈ S ¯ 2 ( N ). Thus we have maps S ¯ 2 ( M ) → S ¯ 2 ( N ) for each r | D . Combining these maps gives φ M : ⊕ r | D S ¯ 2 ( M ) → S ¯ 2 ( N ) . Definition 2 ( N ) D -old . The The image of φ M is called the D-old subspace S ¯ 2 ( N ) D -old in S ¯ orthogonal complement of S ¯ 2 ( N ) with respect to the 2 ( N ) D -new . Petersson inner product is called the D-new subspace S ¯ Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 8 / 34
Jacquet-Langlands correspondance Theorem (Eichler-Shimura-Jacquet-Langlands) There is an injective map of Hecke modules S D 2 ( M ) ֒ → S ¯ 2 ( N ) where N = DM, whose image consists of those cusp forms which are new at all primes p | D. In general there is a non-canonical isomorphism S D 2 ( N ) D − new . 2 ( M ) ≈ S ¯ Working over Q , let J D -new ( N ) be the D -new part of J 0 ( N ). 0 Corollary The Jacobians J D-new ( N ) and J D 0 ( M ) are isogenous. 0 Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 9 / 34
Degree of Parameterization We have a paramterization for both J-new and Shimura Jacobians. � E - a modular elliptic curve defined over F of conductor N . � J - either J D 0 ( M ) (or J D -new ( N ) . ) 0 � π : J → E where E is the optimal quotient. � The Shimura degree (or D-new degree) is the degree of π . Definition The endomorphism π ◦ π ∨ ∈ End( E ) is multiplication by an integer. This integer is called the Shimura degree (or D-new degree) , δ D ( M ) (or δ D -new ( N )), of the elliptic curve E . Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 10 / 34
Idea for studying Shimura Degrees � Examine character groups of E and J locally, i.e., at primes dividing N = DM . � Use a short exact sequence of Grothendieck to rewrite the degree of parameterization in terms of computable invariants. � Use dual graphs to view character groups as Hecke modules. � Use Ribet’s level-lowering sequence to compute Shimura degrees and make comparisons. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 11 / 34
Local objects Let A be a principally polarized abelian variety over F (either J , Shimura jacobian or new-modular jacobian, or E , elliptic curve) and p | N = DM : � A p - Néron model � Φ p ( A ) = A p / A 0 p - Component Group � T p ( A ) - Toric part of A p � X p ( A ) = Hom( T p ( A ) , G m ) - Character Group Theorem (Grothendieck) There is a natural exact sequence 0 → X p ( A ) α − → Hom ( X p ( A ) , Z ) → Φ p ( A ) → 0 in which α is obtained from the monodromy pairing u A , p by ( α ( x ))( y ) = u A , p ( x , y ) . Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 12 / 34
Alternate Description of Shimura Degree A �→ X p ( A ) is functorial, so induces maps: π ∗ : X p ( E ) → X p ( J ) π ∗ : X p ( J ) → X p ( E ) then π ∗ ◦ π ∗ : X p ( E ) → X p ( E ) is multiplication by δ D ( M ) on X p ( E ). Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 13 / 34
Diagram Chasing In particular we have 0 → X p ( J ) → Hom( X p ( J ) , Z ) → φ p ( J ) → 0 ↓↑ ↓ ↓ 0 → X p ( E ) → Hom( X p ( E ) , Z ) → φ p ( E ) → 0 As X p ( E ) injects into X p ( J ), let L p ( E ) denote the saturation of π ∗ X p ( E ). Alternatively, L p ( E ) = { x ∈ X p ( J ) : T n x = a n ( f E ) x for all n coprime to N } Note: L p ( E ) depends only on the isogeny class of E and not on E itself. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 14 / 34
Formula for Shimura Degree Let g p be a generator of L p ( E ) and π ∗ : Φ p ( J ) → Φ p ( E ). Define the following notation: h p = u J , p ( g p , g p ) , ¯ c p = #Φ p ( E ) , i p = #image( π ∗ ) , j p = #coker( π ∗ ) . Theorem ( F = Q due to Takahashi) The # image ( π ∗ ) divides u J ( g p , g p ) and δ D ( M ) = u J , p ( g p , g p ) = h p ¯ # image ( π ∗ ) · # coker ( π ∗ ) = h p j p c p . i 2 i p p Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 15 / 34
Hecke Modules - something we can compute Let H be the definite quaternion algebra of discriminant D with Eichler order O ( M ) of level M . � The Brandt module Br( D , M ) = Z [Cl R ( O ( M ))]. � The Hecke module X ( D , M ) = Br( D , M ) 0 . � Computable due to an algorithm of Kirschmir and Voight. � Inner product: � [ I ] , [ J ] � = δ [ I ] , [ J ] ω I / 2 where δ [ I ] , [ J ] = 1 if [ I ] = [ J ] and 0 otherwise and ω I = # O L ( I ) × / Z × F . � Hecke operators are matrices with entries: � � x ∈ I i I − 1 : nrm( xI i I − 1 T ( p ) i , j = # ) = ( p ) . j j Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 16 / 34
Level Lowering Sequence Theorem (Buzzard over Q ) When N = DMp, X p ( J D 0 ( pM )) = X ( Dp , M ) . Theorem (Ribet, Buzzard over Q ) We have the following short exact sequence of Hecke modules 0 → X p ( J Dpq ( M )) → X q ( J D 0 ( Mpq )) → X q ( J D 0 ( Mq )) ×X q ( J D 0 ( Mq )) → 0 . 0 Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 17 / 34
Computing Character Groups of Jacobians Shimura Curves There are two cases, p divides the level p | pM and p divides the discriminant p | pD with p || N = DMp . � If p | Mp : Let H be the definite quaternion algebra of discriminant pD with Eichler order O ( M ) of level M . Then 0 ( M )) ∼ X p ( J D = X ( Dp , M ). � If p | Dp : Let H be the quaternion algebra ramified at all infinite places of discriminant D with Eichler orders O ( M ) of level M and O ( Mp ) of level Mp . Then 0 → X p ( J Dpq ( M )) → X ( Dq , Mp ) → X ( Dq , M ) × X ( Dq , M ) → 0 . 0 Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 18 / 34
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