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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

Γ0(N) =

• a

b c d

• ∈ GL2(Z) : c ≡ 0 (mod N)
• X0(N) - the modular curve Γ0(N) \ H ∪ cusps

J0(N) - Jacobian of X0(N) E - a modular elliptic curve over Q of conductor N,

with E = C/Λ

fE - the modular form in S2(N) associated to E with Fourier

coefficients an.

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 : X0(N) → E given by τ ∈ X0(N)(C) π(τ) = −2πi

i∞

τ

f (τ ′)dτ ′ =

∞

• n=1

an n e2πinτ ∈ C/Λ.

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 4 / 34

Modular Degree

Let π : X0(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 J0(N) → E has connected kernel.

Definition

If E is an optimal quotient of J0(N), π : J0(N) → E, π∨ : E → J0(N) π ◦ π∨ ∈ End(E) is multiplication by an integer mE. This integer mE 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 ΓD

0 (M) to be the group of norm-1 units in O.

Our Shimura curve is X D

0 (M) = ΓD 0 (M) \ H.

We denote its Jacobian by JD

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 ΓD

0 (M) is a holomorphic

function f on H such that f (γτ) = (cγ + d)kf (τ) for all γ =

• a

b c d

• ∈ ΓD

0 (M).

The space of such forms is denoted by MD

k (M), and cusp forms by

SD

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 (rτ) ∈ S¯

2(N). Thus we have maps S¯ 2(M) → S¯ 2(N) for each r | D.

Combining these maps gives φM : ⊕r|DS¯

2(M) → S¯ 2(N).

Definition

The image of φM is called the D-old subspace S¯

2(N)D-old. The

• rthogonal 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

Jacquet-Langlands correspondance

Theorem (Eichler-Shimura-Jacquet-Langlands)

There is an injective map of Hecke modules SD

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 SD

2 (M) ≈ S¯ 2(N)D−new.

Working over Q, let JD-new (N) be the D-new part of J0(N).

Corollary

The Jacobians JD-new (N) and JD

0 (M) are isogenous.

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 JD

0 (M) (or JD-new

(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), δD(M) (or δD-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

• f 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:

Ap - Néron model Φp(A) = Ap/A0

p - Component Group

Tp(A) - Toric part of Ap Xp(A) = Hom(Tp(A), Gm) - Character Group

Theorem (Grothendieck)

There is a natural exact sequence 0 → Xp(A) α − → Hom(Xp(A), Z) → Φp(A) → 0 in which α is obtained from the monodromy pairing uA,p by (α(x))(y) = uA,p(x, y).

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 12 / 34

Alternate Description of Shimura Degree

A → Xp(A) is functorial, so induces maps: π∗ : Xp(E) → Xp(J) π∗ : Xp(J) → Xp(E) then π∗ ◦ π∗ : Xp(E) → Xp(E) is multiplication by δD(M) on Xp(E).

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 13 / 34

Diagram Chasing

In particular we have → Xp(J) → Hom(Xp(J), Z) → φp(J) → ↓↑ ↓ ↓ → Xp(E) → Hom(Xp(E), Z) → φp(E) → As Xp(E) injects into Xp(J), let Lp(E) denote the saturation of π∗Xp(E). Alternatively, Lp(E) = {x ∈ Xp(J) : Tnx = an(fE)x for all n coprime to N} Note: Lp(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 gp be a generator of Lp(E) and π∗ : Φp(J) → Φp(E). Define the following notation: hp = uJ,p(gp, gp), ¯ cp = #Φp(E), ip = #image(π∗), jp = #coker(π∗).

Theorem (F = Q due to Takahashi)

The #image(π∗) divides uJ(gp, gp) and δD(M) = uJ,p(gp, gp) #image(π∗) · #coker(π∗) = hpjp ip = hp¯ cp i2

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[ClR(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]ωI/2 where δ[I],[J] = 1 if [I] = [J] and 0 otherwise and ωI = #OL(I)×/Z×

F .

Hecke operators are matrices with entries:

T(p)i,j = #

• x ∈ IiI−1

j

: nrm(xIiI−1

j

) = (p)

• .

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, Xp(JD

0 (pM)) = X(Dp, M).

Theorem (Ribet, Buzzard over Q)

We have the following short exact sequence of Hecke modules 0 → Xp(JDpq (M)) → Xq(JD

0 (Mpq)) → Xq(JD 0 (Mq))×Xq(JD 0 (Mq)) → 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 Xp(JD

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 → Xp(JDpq (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

Let J′ = JDpq (M) and J = JD

0 (Mpq). Denote invariants of J′ with ′s.

Corollary

h′

p = hq and iq | i′ p.

Corollary

We have the following relationship between Shimura degrees: δ′ = δ ¯ c′

p¯

cq i2

qj

′2

p .

Corollary

If we instead let J′ = JD-new (N) and J = J0(N), then hp = h′

p and

ip | i′

• p. Further, mD-new

E

| mE.

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 19 / 34

How to Compute hp and ip

The following are now straight forward:

hp: compute the monodromy paring on the generator for L(f )

using the action of Hecke operators on Xp.

ip: compute the generator of the ideal Ip of Z by computing the

monodromy pairings with hp. Oddly enough, in most cases this is enough to compute δ and ¯ cp’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

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 mE Labels δD(M) Labels 14a 14 1 1 a1 1 a2 30a 15 2 2 a1 2 a2 30a 6 5 2 a1 1 a7 30a 10 3 2 a1 1 a3 39a 39 1 2 a1 2 a1 55a 55 1 2 a1 2 a1 65a 65 1 2 a1 2 a1 66b 6 11 4 b1 2 b2 66b 22 3 4 b1 2 b2 84a 21 4 6 a1 6 a1

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 21 / 34

Question of Takahashi

Question (Takahashi)

If p | D, is the map Φp(J) → Φp(E) surjective? If p | M, this is not true.

Corollary (Takahashi)

Asuming the conjecture, for p | D, δD(M) = uJ(gp, gp) #image(π∗) = hp ip . Note: When working over Q this is always enough to compute δD(M) and find the optimal quotient!

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 22 / 34

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 JD

0 (M).

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 23 / 34

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 fE ∈ S2(ΓD

0 (M)).

Compute generators for the fundamental domain of ΓD

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

Q( √ 5) Example

Let F = Q( √ 5), a = 1+

√ 5 2

and E : y2 + xy + ay = x3 + (−a − 1) x2, 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 : y2 + xy + ay = x3 − (a + 1)x2 − (30a + 45)x − (111a + 117)

Only one curve in the isogeny class with ordN(∆) = 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

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 X D

0 (M) be

the Shimura curve with D = 2 and M = −4a + 1. Case p | D, so p = 2. Computing Brandt Modules: 2 = uJ(gp, gp), ip = 1 so δ = 2¯

• c2. Two choices for ¯

c2, 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 ¯ c2 = ¯ c−4a+1.

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 26 / 34

Modular Degree and Congruence Numbers

Let S = S2(Γ0(N), Z) be the space of weight 2, level N, cuspforms with integral Fourier coefficients. Let L = (fE)⊥ ∩ S.

Definition

The congruence number rE 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)

mE | rE.

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 27 / 34

Modular Degree and Congruence Numbers

Zagier computed mE for N = p. In all of these examples mE = rE. This lead to Frey and Muller asking if it is always the case that mE = rE. Stein, Agashe, investigate and found, no, not even close. Example: The elliptic curve with Cremona label 54b1 has mE = 2 and rE = 6.

Theorem (Agashe,Ribet,Stein-2009)

mE | rE and if ordp(N) ≤ 1 then ordp(rE) = ordp(mE). For Γ1(N) they find examples where mE ∤ rE, 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

Q( √ 5) Degrees and Congruence Primes

Iso. N gen D M LMFDB Label δD(M) rE a 31b 5a − 2 5a − 2 1 a5 1 1 a 36b 6 2 3 a3 1 1 a 36b 6 3 2 a4 1 1 a 41b a + 6 a + 6 1 a2 1 1 a 45a −6a + 3 3 −2a + 1 a5 1 1 a 45a −6a + 3 −2a + 1 3 a4 1 1 a 49a 7 7 1 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 + 8 1 a4 1 1 a 76a −8a + 2 2 −4a + 1 a1∗ 2∗ 2 a 76a −8a + 2 −4a + 1 2 a2 2 2

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 29 / 34

D-new parts

Let S = S2(Γ0(N), Z)D−new and L = (f )⊥ ∩ S.

Definition

The D-new congruence number r D-new

E

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

Isogeny Class D M Cremona Label δD(M) mD-new

E

r D-new

E

14a 1 14 a1 1 − − 14a 14 1 a2 1 1 1 15a 1 15 a1 1 − − 15a 15 1 a1 1 1 1 21a 1 21 a1 1 − − 21a 21 1 a2 1 1 1 26a 1 26 a1 2 − − 26a 26 1 a1 2 2 2 26b 1 26 b1 2 − − 26b 26 1 b2 2 2 2 30a 1 30 a1 2 − − 30a 15 2 a2 2 2 2 30a 6 5 a7 1 1 1 30a 10 3 a3 1 1 1

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 31 / 34

Conjectures

Computing mD-new

E

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: δD(M) = mD-new

E

= r D-new

E

. If this is true, it gives more evidence of Takahashi’s conjecture:

Conjecture

When p | D, φp(J) → φp(E) is surjective. And, as mD-new

E

| mE:

Conjecture

δD(M) | mE.

Alyson Deines (CCR La Jolla) Shimura Degrees, New Modular Degrees, and Congruence Primes 32 / 34

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 δD(M) | rE? 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

Thank you!

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