Shimura Degrees, New Modular Degrees, and Congruence Primes

Transcription

1 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

2 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

3 Modular Elliptic Curves {( ) } a b Γ 0 (N) = GL c d 2 (Z) : c 0 (mod N) 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

4 Modular Elliptic Curves E is modular, so we have the following surjective map: given by τ X 0 (N)(C) π(τ) = 2πi i τ π : X 0 (N) E f (τ )dτ = n=1 a n n e2πinτ C/Λ. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 4 / 34

5 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

6 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 X0 D(M) = ΓD 0 (M) \ H. We denote its Jacobian by J0 D(M). Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 6 / 34

7 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 ( ) f (γτ) = (cγ + d) k a b f (τ) for all γ = Γ D 0 (M). c d The space of such forms is denoted by Mk D (M), and cusp forms by Sk D(M). Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 7 / 34

8 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 (rτ) 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 The image of φ M is called the D-old subspace S 2 (N)D-old. The orthogonal complement of S 2 (N)D-old in S 2 (N) with respect to the Petersson inner product is called the D-new subspace S 2 (N)D-new. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 8 / 34

9 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 (M) S 2 (N)D new. Working over Q, let J D-new 0 (N) be the D-new part of J 0 (N). Corollary The Jacobians J0 D-new (N) and J0 D (M) are isogenous. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 9 / 34

10 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 J0 D (M) (or J D-new 0 (N).) 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

11 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

12 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

13 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

14 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

15 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 ) #image(π ) #coker(π ) = h pj p i p = h p c p ip 2. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 15 / 34

16 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: { T (p) i,j = # x I i Ij 1 } : nrm(xi i Ij 1 ) = (p). Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 16 / 34

17 Level Lowering Sequence Theorem (Buzzard over Q) When N = DMp, X p (J0 D (pm)) = X(Dp, M). Theorem (Ribet, Buzzard over Q) We have the following short exact sequence of Hecke modules 0 X p (J Dpq 0 (M)) X q (J D 0 (Mpq)) X q (J D 0 (Mq)) X q (J D 0 (Mq)) 0. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 17 / 34

18 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 X p (J D 0 (M)) = 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 0 (M)) X(Dq, Mp) X(Dq, M) X(Dq, M) 0. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 18 / 34

19 Let J = J Dpq 0 (M) and J = J D 0 (Mpq). Denote invariants of J with s. Corollary h p = h q and i q i p. Corollary We have the following relationship between Shimura degrees: δ = δ c p c q i 2 qj 2 p. Corollary If we instead let J = J0 D-new (N) and J = J 0 (N), then h p = h p and i p i p. Further, me D-new m E. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 19 / 34

20 How to Compute h p and i p The following are now straight forward: h p : compute the monodromy paring on the generator for L(f ) using the action of Hecke operators on X p. i p : compute the generator of the ideal I p of Z by computing the monodromy pairings with h p. Oddly enough, in most cases this is enough to compute δ and c p s. Note: If you can compute the optimal quotient, there is an algorithm for finding the Shimura degree. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 20 / 34

21 Data In fact, for all semistable elliptic curves over Q with conductor N < 100 I can determine both the degree and the optimal quotient: Isog. Class D M m E Labels δ D (M) Labels 14a a1 1 a2 30a a1 2 a2 30a a1 1 a7 30a a1 1 a3 39a a1 2 a1 55a a1 2 a1 65a a1 2 a1 66b b1 2 b2 66b b1 2 b2 84a a1 6 a1 Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 21 / 34

22 Question of Takahashi Question (Takahashi) If p D, is the map Φ p (J) Φ p (E) surjective? If p M, this is not true. Corollary (Takahashi) Asuming the conjecture, for p D, δ D (M) = u J(g p, g p ) #image(π ) = h p i p. Note: When working over Q this is always enough to compute δ D (M) and find the optimal quotient! Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 22 / 34

23 Definition We say E and E are discriminant twins if E and E if N(E) = N(E ) and (E) = (E ), i.e., E and E have the same conductor and the same discriminant. Theorem (D. - Lundell) Over Q there are only finitely many pairs of semistable, isogenous discriminant twins. They occur for conductors 11, 17, 19, and 37. Corollary Assuming Takahashi s question, over Q there is an algorithm for finding the Shimura degree and the optimal quotient of J D 0 (M). Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 23 / 34

24 Using power series expansions of quaternion modular forms Zagier computed the complex periods of the optimal quotient directly using the Fourier series expansion of the modular form. Quaternionic modular forms don t have cusps, so don t have Fourier series expansions. Voight and Willis use power series expansions instead! Compute the power series expansion of the quaternionic modular form f E S 2 (Γ D 0 (M)). Compute generators for the fundamental domain of Γ D 0 (M). Use the generators to identify vertices of the fundamental domain. Integrate over vertices to find independent periods. Compute the j-invariant and match with curve in the isogeny class. This curve is the optimal quotient. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 24 / 34

25 Q( 5) Example Let F = Q( 5), a = and E : y 2 + xy + ay = x 3 + ( a 1) x 2, N = ( 5a + 3) of norm 31. X N 0 (1) is a genus one curve, so the modular degree is trivially 1. There are 6 curves in the isogeny class. Using the method of Voight and Willis, compute the j-invariant j(e) = ( a)( 51a + 37) 3 ( 39a + 25) 3 (5a 3) 8 and find: E : y 2 + xy + ay = x 3 (a + 1)x 2 (30a + 45)x (111a + 117) Only one curve in the isogeny class with ord N ( ) = 8, so we find this curve computing Hecke modules as well. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 25 / 34

26 Q( 5) Example Take N = 8a + 2, then dimm (2,2) ( 8a + 2) = 2 and N( 8a + 2) = 76. Let E be the elliptic curve 76a.a1. Let X0 D the Shimura curve with D = 2 and M = 4a + 1. (M) be Case p D, so p = 2. Computing Brandt Modules: 2 = u J (g p, g p ), i p = 1 so δ = 2 c 2. Two choices for c 2, 1 and 5, so δ = 2 or 10. Try p M, so p = 4a + 1 Use the Hecke module correspondence to get again get c 4a+1 = 1 or 5 and again δ = 2 or 10. Problem: For both curves in the isogeny class c 2 = c 4a+1. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 26 / 34

27 Modular Degree and Congruence Numbers Let S = S 2 (Γ 0 (N), Z) be the space of weight 2, level N, cuspforms with integral Fourier coefficients. Let L = (f E ) S. Definition The congruence number r E is the integer that satisifes the following equivalent conditions: r is the largest integer such that there exists g L with f g (mod r). {(f, h) h S} = r 1 (f, f )Z. r is the order of the finite group S/(Zf + L). Theorem (Ribet) m E r E. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 27 / 34

28 Modular Degree and Congruence Numbers Zagier computed m E for N = p. In all of these examples m E = r E. This lead to Frey and Muller asking if it is always the case that m E = r E. Stein, Agashe, investigate and found, no, not even close. Example: The elliptic curve with Cremona label 54b1 has m E = 2 and r E = 6. Theorem (Agashe,Ribet,Stein-2009) m E r E and if ord p (N) 1 then ord p (r E ) = ord p (m E ). For Γ 1 (N) they find examples where m E r E, in particular 54b1 and also for a curve of squarefree conductor, N = 38. They also note that the analogous statment does not hold for modular abelian varieties, but get a different statement in terms of the expontents of the groups. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 28 / 34

29 Q( 5) Degrees and Congruence Primes Iso. N gen D M LMFDB Label δ D (M) r E a 31b 5a 2 5a 2 1 a5 1 1 a 36b a3 1 1 a 36b a4 1 1 a 41b a + 6 a a2 1 1 a 45a 6a a + 1 a5 1 1 a 45a 6a + 3 2a a4 1 1 a 49a a2 1 1 a 55a a + 8 2a + 1 3a + 2 a5 1 1 a 55a a + 8 3a + 2 2a + 1 a5 1 1 a 71b a + 8 a a4 1 1 a 76a 8a a + 1 a1 2 2 a 76a 8a + 2 4a a2 2 2 Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 29 / 34

30 D-new parts Let S = S 2 (Γ 0 (N), Z) D new and L = (f ) S. Definition The D-new congruence number re D-new is the integer that satisifes the following equivalent conditions: r is the largest integer such that their exists g L with f g (mod r). {(f, h) h S} = r 1 (f, f )Z. r is the exponent of the finite group S/(Zf + L). Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 30 / 34

31 Isogeny Class D M Cremona Label δ D (M) m D-new E r D-new E 14a 1 14 a1 1 14a 14 1 a a 1 15 a1 1 15a 15 1 a a 1 21 a1 1 21a 21 1 a a 1 26 a1 2 26a 26 1 a b 1 26 b1 2 26b 26 1 b a 1 30 a1 2 30a 15 2 a a 6 5 a a 10 3 a Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 31 / 34

32 Conjectures Computing me D-new is just a few lines using modular symbols and is very fast compared to computing Brandt modules. Conjecture For semistable elliptic curves the following invariants are equal: δ D (M) = m D-new E = r D-new E. If this is true, it gives more evidence of Takahashi s conjecture: Conjecture When p D, φ p (J) φ p (E) is surjective. And, as m D-new E m E : Conjecture δ D (M) m E. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 32 / 34

33 Open Questions We can use the work of Voight and Willis to find the j-invariant of the optimal quotient of the Shimura curve parameterization up to some precision. Is there an algebraic way to find the optimal quotient? This would give a provable agorithm for computing the Shimura degree. For totally real number fields, do we get the same analogues? Does δ D (M) r E? When p D is the map on component groups surjective? Are there only finitely many semistable, isogenous discriminant twins over totally real number fields? Data indicates yes, but the proof over Q does not generalize. Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 33 / 34

34 Thank you! Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 34 / 34