# 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](../../references.md#cite-conrad-stein-compgroup)] and [[Kohel and Stein, 2000](../../references.md#cite-kohel-stein-ants4)]. 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)`

```magma
> 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

```

## 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](../../references.md#cite-lenstra-oort-additive)].

### `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)`

```magma
> 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

```