The Manin index of optimal new quotients

William A. Stein

October 1, 1999

Contents

1. The Birch and Swinnerton-Dyer conjecture

2. Motivation

3. The Manin index

4. Bounding the Manin index

5. Proof (continued)

6. Proof (continued)

§1. The Birch and Swinnerton-Dyer conjecture

Consider the optimal quotient A = A_{f} 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.

Theorem 1 (Agashé,—).

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

Here • A := Néron model of A over Z.

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

• c_{∞} := number of connected components of A(R), a power of 2.

• Manin index

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

Conjecture 2 (Birch, Swinnerton-Dyer).

\frac{L(A, 1)}{Ω_{A}} = \frac{#� (A/Q) · ∏ c_{p}}{#A(Q) · #A^{∨}(Q)} .

§2. Motivation

Problem. Verify the BSD conjecture, up to a power of 2, for specific A_{f} at square-free level. Example 3. p = 2333

J_{0}(2333) ≃ A × B × C

dim A = 4, dim B = 89, dim C = 101.

L(A, 1) = L(B, 1) = 0.

\frac{L(C, 1)}{Ω_{C}} · c_{C} = \frac{83341^{2}}{11 · 53} · 2^{?}

c_{p} = 11 · 53 = #C(Q) = #C^{∨}(Q) = numer((p − 1)/12).

If we know that c_{C} is a power of two then, to verify BSD up to 2-powers, must show that #� = 83341^{2} · 2^{?}. In this case we are lucky: 83341 | #(A ∩ C), which can (probably) be used to show that 83341 | #� . Example 4. p = 2111 is prime.

J_{0}(2111) ≃ A × B

with dim A = 64, dim B = 112. L(A, 1) = 0

\frac{L(B, 1)}{Ω_{B}} · c_{B} = \frac{211}{5} · 2^{?}.

c_{p} = 5 · 211 = #B(Q) = #B^{∨}(Q) = numer((p − 1)/12).

If we know that c_{B} is a power of 2, then to verify BSD, must show that #� = 211^{2} · 2^{?}/c_{B}.

§3. The Manin index

Conjecture 5 (Agashe, —, Manin when dim A = 1).

c_{A} = 1.

Caution: Without the condition that A is new, the conjecture is false. The Manin index of J_{0}(33) is 3, because of a mod 3 congruence.

A. Logan is skeptical of Conjecture 5.

Evidence: See Amod’s talk from Wednesday.

Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Flynn, Leprévost, Schaeffer, —, Stoll, Wetherell. We considered 28 optimal quotients of dimension 2 that happened to be Jacobians.

• We computed Ω_{A} directly from a minimal proper regular models.

• We then computed the volume of A(R) with respect to S_{2}(N; Z)[I], and got the same answer, to several decimal places.

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.

§4. Bounding the Manin index

We present a slight generalization of a theorem of Mazur.

Theorem 6 (Mazur when dim A = 1, —). Let m be the largest square dividing N. Let A be an optimal quotient of J_{0}(N)^{new}. 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. We give the proof assuming that A = A_{f}.

• 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).

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]].

W. A. Stein, The Manin index of optimal new quotients V

§5. Proof (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 [Mazur, Cor. 1.1] gives an exact sequence

0 → H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) → H^{0}(J /Z_{ℓ}, Ω_{J /Z_{ℓ}}) → H^{0}(B/Z_{ℓ}, Ω_{B/Z_{ℓ}}) → 0.

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}}

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}})

§6. Proof (continued)

\xymatrix{H^{0}(A, Ω_{A}) ⊗ F_{ℓ} \ar@{^{(}->}[rr] &  & H^{0}(J , Ω_{J}) ⊗ F_{ℓ} \ar[d]^{≅} \\  &  & H^{0}(X, Ω_{X}) ⊗ F_{ℓ} \ar[dd]^{≅} \\  & injective? \ar[ddr] \\  &  & H^{0}(X_{F_{ℓ}}, Ω_{X_{F_{ℓ}}}) \ar[d]^{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 is 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.