Component groups of quotients of J_{0}(N)^{∗}

William A. Stein Friday, 7 July 2000

∗Joint work with David Kohel.

Component groups of modular abelian varieties

Let f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N); C) be a newform.

f ⇝ optimal quotient A_{f} = J_{0}(N)/IJ_{0}(N), where I is annihilator of f in T = Z[… T_{n} …].

\xymatrix{A_{f}^{∨} \ar@{^{(}->}[r] & J_{0}(N) \ar@{->>}[d] \\  & A_{f}}

Let A_{f} be the Néron model of A_{f}. This is a smooth commutative group scheme over Z.

Component group:

Φ_{A_{f},p} := (A_{f,F_{p}}/A_{f,F_{p}}^{∘})(\overline{F}_{p}).

Goal: Find #Φ_{A_{f},p} as function of f and p.

Computing component groups

Suppose p || N.

T = maximal torus of J_{0}(N)_{F_{p}} X = Hom _{\overline{F}_{p}}(T, G_{m}) (character group) ⟨ , ⟩ : X × X → Z (natural pairing) α : X → Hom (X[I], Z) (induced map)

H = H_{1}(X_{0}(N), Z) (modular symbols) S = S_{2}(Γ_{0}(N); C) (cusp forms) β : H → Hom (S[I], C) (integration pairing)

Theorem. (Stein)

#Φ_{A_{f},p} = \frac{√#β(H)/β(H[I])}{#α(X)/α(X[I])} · #coker (α).

Image of Φ_{J_{0}(N),p} in Φ_{A_{f},p} is isomorphic to coker (α).

Remark. Every quantity in the formula can be computed, e.g., in MAGMA.

Example

Biggest numerator of L-value: N = 1154 = 2 · 577 dim A_{f} = 20 L(A_{f}, 1)/Ω_{A_{f}} = 2^{?}·85495047371/17^{2} BSD ⟹ Big component group! Our formula gives:

#Φ_{A_{f},2} = 2^{?}·17^{2}·85495047371

Abelian variety B = 577E (dim 22) has a_{2} with 85495047371 | Norm (a_{2}^{2} − (2 + 1)^{2}), so Ribet’s level raising theorem implies that A_{f} has component group of order divisible by 85495047371.

Conjectures and Questions 1. Find group structure of Φ_{A_{f},p}.

2. Find expression for

\frac{√#β(H)/β(H[I])}{#α(X)/α(X[I])}

purely in terms of homology.

E.g., when N prime quotient appears to be 1 (Emerton seems to have proved this).

3. Formula in the additive reduction case.