Tamagawa Numbers and Component Groups of Neron Models

Component Groups

Suppose \(A\) is a newform modular abelian variety over \({\mathbb{Q}}\) over level \(N\). For any prime \(p\) that exactly divides \(N\), the order of the component group of \(A\) over the algebraic closure of \({\bf F}_{p}\) can be computed. The nontrivial algorithm is described in [Conrad and Stein, 2001] and [Kohel and Stein, 2000]. It is an open problem to compute the structure of the component group or the order under more general hypothesis.

ComponentGroupOrder(A, p): ModAbVar, RngIntElt -> RngIntElt

The order of the component group of the special fiber of the Neron model of \(A\) over the algebraic closure of \({\bf F}_{p}\). The abelian variety \(A\) must be attached to a newform.

Example: Compgrp Component Groups (ex-535c76)

> J := JZero(65); J;
Modular abelian variety JZero(65) of dimension 5 and level 5*13 over Q
> A := Decomposition(J)[3];
> ComponentGroupOrder(A,13);
1
> ComponentGroupOrder(A,5);
7

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Tamagawa Numbers

Suppose \(A\) is an abelian variety over \({\mathbb{Q}}\) that is attached to a newform. A divisor and an integer some power of which is a multiple of the Tamagawa number of \(A\) at a prime \(p\) can be determined. When \(p^2\) divides the level, the reduction is additive, we use the Lenstra-Oort bound from [Lenstra and Oort, 1985].

TamagawaNumber(A, p): ModAbVar, RngIntElt -> RngIntElt, RngIntElt, BoolElt

A divisor of the Tamagawa number of the abelian variety \(A\) at the prime \(p\) and an integer some power of which is a multiple of the Tamagawa number of \(A\) at \(p\). Also return true if the divisor of the Tamagawa number is provably equal to the Tamagawa number of \(A\). The abelian variety \(A\) must be attached to a newform.

TamagawaNumber(A): ModAbVar -> RngIntElt, RngIntElt, BoolElt

Let \(c\) be the product of the Tamagawa numbers of \(A\) at primes of bad reduction, where \(A\) is an abelian variety over \({\mathbb{Q}}\) attached to a newform. This command returns a divisor of \(c\), an integer some power of which is a multiple of \(c\), and true if the divisor is provably equal to \(c\).

Example: Compgrp Tamagawa Numbers (ex-0a4c08)

> J := JZero(65);
> TamagawaNumber(J(2),5);
2 2 false
> TamagawaNumber(J(2),13);
3 3 true
> TamagawaNumber(J(3),5);
7 7 true
> TamagawaNumber(J(3),13);
2 2 false
>
> J := JZero(5^2*7);
> TamagawaNumber(J(1));
2 30 false
> TamagawaNumber(J(1),5);
1 30 false
> TamagawaNumber(J(1),7);
2 2 false

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