The Manin index of optimal new quotients

William A. Stein

September 27, 1999

Contents

1. The Manin index

2. Main theorem

3. Main theorem (continued)

4. Main theorem (continued)

5. Theoretical evidence

6. Numerical evidence

§1. The Manin index

Consider the optimal quotient A associated to a newform f:

[figure: IJ_{0}(N); J_{0}(N); π; f = ∑ a_{n}q^{n}; A; ι; A]

Here I := ker(T → → Z[… a_{n} …]) is a prime ideal.

Problem: Study the arithmetic of A.

Theorem 1 (Agashe,—).

\frac{L(A, 1)}{Ω_{A}} · c_{A} = [π_{∗}(H_{1}(J_{0}(N), Z)^{+} : π_{∗}(Te)]

Here • A := Neron model of A over Z.

• Ω_{A} := volume of A(R)^{0} wrt basis for H^{0}(A, Ω_{A}).

• π_{∗} : H_{1}(J_{0}(N), Q) → H^{1}(A, Q) (tangent spaces).

• Manin index:

c_{A} := [S_{2}(N; Z)[I] : π^{∗}ι^{∗}H^{0}(A, Ω_{A})]

Conjecture 2 (Agashe, —, Manin when dim A = 1).

c_{A} = 1

The purpose of this talk is to present evidence for this conjecture.

Caution: Without the condition that A is new, the conjecture is false. For example, the Manin index of J_{0}(33) is 3.

§2. Main theorem

We present a slight generalization of a theorem of Mazur’s.

Theorem 3 (Mazur when dim A = 1, —). Let m be the largest square dividing N. Then c_{A} is a unit in Z[\frac{1}{2m}].

Many ideas of the proof are in Mazur’s Rational isogenies of prime degree.

Proof.

• R := Z[\frac{1}{2m}].

• J := Neron model of J_{0}(N) over R.

• A := ditto for A.

• X := smooth locus of the minimal proper regular model of X_{0}(N)/R.

Consider

\xymatrix{H^{0}(A, Ω_{A/R}) \ar@{^{(}->}[r]^{π^{∗}}_{pullback} & H^{0}(J , Ω_{J /R}) \ar[r]^{≅} & H^{0}(X, Ω_{X}) \ar[r]^{q-exp} & R[[q]].}

• π^{∗} is an inclusion because H^{0}(A, Ω_{A/R}) is free (A/R is smooth) and π_{C}^{∗} is an inclusion.

• The middle isomorphism uses Michéle Raynaud’s result:

Pic^{0}(X) ≅ J^{0},

induces H^{1}(X, O_{X}) ⟶^{≅} H^{1}(J , O_{J}) on tangent spaces. Now apply Grothendieck duality.

• q-exp comes from Tate curve:

τ : Spec R[[q]] → X.

Formal completion of X at ∞ ∞ ∞_{/R} is (Spec R)[[q]].

§3. Main theorem (continued)

To show: Image of H^{0}(A, Ω_{A}) saturated. Then c_{A} ∈ R^{∗} since S_{2}(N, R)[I] ⊂ R[[q]] is saturated.

Consider

\xymatrix{0 \ar[r] & H^{0}(A, Ω_{A}) \ar[r]^{q-exp} & R[[q]] \ar[r] & D \ar[r] & 0}

Is D torsion free? Let ℓ ∤ 2m be a prime; tensor with F_{ℓ}:

[figure: Tor^{1}(D, F_{ℓ}); 0; D[ℓ]; H^{0}(A, Ω_{A}) ⊗ F_{ℓ}; F_{ℓ}[[q]]; D ⊗ F_{ℓ}; 0]

Thus

D[ℓ] = 0 ⟺ H^{0}(A, Ω_{A}) ⊗ F_{ℓ} → F_{ℓ}[[q]] injective

Since ℓ ∤ 2m, we have e(Q_{ℓ}) = 1 < ℓ − 1, so an extension of a theorem of Raynaud gives an exact sequence:

\xymatrix{0 \ar[r] & H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) \ar[r] & H^{0}(J /Z_{ℓ}, Ω_{J /Z_{ℓ}}) \ar[r] & H^{0}(B/Z_{ℓ}, Ω_{B/Z_{ℓ}})}

Here B = ker(π : J → A). Since H^{0}(B/Z_{ℓ}, Ω_{B/Z_{ℓ}}) is torsion free,

\xymatrix{H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) ⊗ F_{ℓ} \ar@{^{(}->}[r] & H^{0}(J /Z_{ℓ}, Ω_{J /Z_{ℓ}}) ⊗ F_{ℓ}}

Nontrivial fact (see e.g., pg 183 of Eisenbud’s book): Z_{ℓ}/R is flat. “Cohomology commutes with flat base change”:

H^{0}(A, Ω_{A}) ⊗ Z_{ℓ} ≅ H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}})

§4. Main theorem (continued)

[figure: H^{0}(A, Ω_{A}) ⊗ F_{ℓ}; injective?; H^{0}(J , Ω_{J}) ⊗ F_{ℓ}; ≅; H^{0}(X, Ω_{X}) ⊗ F_{ℓ}; ≅; H^{0}(X_{F_{ℓ}}, Ω_{X_{Fℓ}}); q-exp; F_{ℓ}[[q]].]

The isomorphism H^{0}(X, Ω_{X}) ⊗ F_{ℓ} ≅ H^{0}(X_{F_{ℓ}}, Ω_{X_{F_{ℓ}}}) uses cohomological trickery and that k = 2, N ≥ 5.

• If ℓ ∤ N, then q-exp is injective (see Deligne-Rapoport, Theorem 3.9). Thus H^{0}(A, Ω_{A}) ⊗ F_{ℓ} ↪ F_{ℓ}[[q]], so D[ℓ] = 0, as required.

• If ℓ ∤ N then q-exp not necessarily injective. However, ℓ exactly divides N. Deligne-Rapoport model of X_{F_{ℓ}}:

X_{0}(N/ℓ) X_{0}(N/ℓ)  ^{∞ ∞ ∞} •_{000}•

The intersections are supersingular points; the Atkin-Lehner involution w_{ℓ} swaps 000 and ∞ ∞ ∞. Consider g ∈ H^{0}(A, Ω_{A})⊗F_{ℓ} such that q-exp(g) = 0. Then g vanishes on the component of X_{F_{ℓ}} containing ∞. Since A is new and corresponds to a single eigenform, w_{ℓ}(g) = ±g, so q-exp(w_{ℓ}(g)) = 0; hence g vanishes on the other component of X_{F_{ℓ}}. This forces g = 0, and hence D[ℓ] = 0.

The proof is complete.

§5. Theoretical evidence

— Edixhoven’s thesis, O. Gabber: c_{A} is an integer (uses Katz-Mazur).

— Edixhoven: Much more when dim A = 1. For example, he constructs a “stable” model of X_{0}(p^{2}N) when (p, N) = 1. If dim A = 1 and p ≥ 5, then p^{3} ∤ cond(A). Thus he obtains information about all elliptic curves. Maybe a lot more??

— Raynaud: If 4 ∤ N, then

0 ≤ ord_{2}(c_{A}) ≤ dim A

This is a straightforward generalization of Proposition 3.1 of Abbes-Ullmo; this proposition uses a deep theorem of Raynaud on semiabelian reduction.

— Abbes-Ullmo: If dim A = 1, then c_{A} is a unit in Z[\frac{1}{m}].

§6. Numerical evidence

— dim A = 1: Cremona.

— dim A = 2: Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Flynn, Leprévost, Schaeffer, —, Stoll, Wetherell. We considered optimal quotients of dimension 2.

• Computed Ω_{A} directly by finding minimal proper regular models.

• Computed Ω_{A} using modular symbols.

• Find

c_{A} = \frac{Ω_{A}}{Ω_{A}}.

The non square-free levels treated:

N = 3^{2} · 7, 3^{2} · 13, 5^{3}, 3^{3} · 5, 3 · 7^{2}, 5^{2} · 7, 2^{2} · 47, 3^{3} · 7.

In every case, c_{A} = 1.