# Understanding of Tamagawa numbers of hyperelliptic curve

One's can find following definition of tamagawa numbers in Dino Lorenzini paper "Torsion and Tamagawa numbers":

Let $K$ be any discrete valuation field with ring of integers $O_K$ , uniformizer $\pi$, and residue field $k$ of characteristic $p \ge 0$. Let $A/K$ be an abelian variety of dimension $g$. Let $A/O_K$ denote the Neron model of $A/K$. The special fiber $A_k/k$ of $A$ is the extension $$(0) \rightarrow A^0_k \rightarrow A_k \rightarrow Φ \rightarrow (0)$$ of a finite etale group scheme $Φ/k$, called the group of components, by a connected smooth group scheme $A^0_k/k$, the connected component of 0. The order of the finite abelian group $Φ(k)$ is called the Tamagawa number of A/K. Let now $K$ be a global field, and $v$ a non-archimedean place of $K$, with completion $K_v$ and residue field $k_v$. Let $c_v$ denote the Tamagawa number of $A_{K_v}/{K_v}$, and let $c = c(A/K) := \prod_v c_v$.

I have hyperelliptic curve of genus 2 over $\mathbb Q$. The questions are:

1. Is it true that I can extract Tamagawa numbers ($c_v$ in definition) from Sage function genus2reduction (see below) output and calculate Tamagawa product (called $c$ in definition) from them? Is it ok that group of connected components is over an algebraic closure of $\mathbb F_p$?
2. Is it true that I can use Magma's RegularModel (see below) and get tamagawa numbers $c_p$ ($=c_v$) as orders of Magma's ComponentGroup for corresponding $p$? I'm confused by group of components of the Neron model of the Jacobian of C over the **completion** at P. There is no mentioned of completition in Sage function. I'm not sure that I understand all things right.
3. Is it true that in second case if I want to get Tamagawa product I should just calculate product of $c_p$ for all $p$ that divide discriminant of my curve?

There are Necessary documentation notes about corresponding Sage and Magma functions.

Sage genus2reduction function ducumentation says following:

Use $R = genus2reduction(Q, P)$ to obtain reduction information about the Jacobian of the projective smooth curve defined by $y^2+Q(x)y=P(x)$. $$\dots$$ The second datum is the GROUP OF CONNECTED COMPONENTS (over an ALGEBRAIC CLOSURE (!) of $\mathbb F_p$) of the Neron model of $J(C)$.

From the other side Stain in his note "What are Neron Models?" writes:

When $A$ is the Jacobian of a curve $X$, there is an alternative approach that involves the "minimal proper regular model" of $X$. For example, when $A$ is an elliptic curve, it is the Jacobian of itself, and the Neron model can be constructed in terms of the minimal proper regular model $X$ of $A$ as follows. In general, the model $X → R$ is not also smooth. Let $X'$ be the smooth locus of $X → R$, which is obtained by removing from each closed fiber $X_{F_p} = \sum n_i C_i$ all irreducible components with multiplicity $n_i \ge 2$ and all singular points on each $C_i$, and all points where at least two $C_i$ intersect each other. Then the group structure on $A$ extends to a group structure on $X'$ , and $X'$ equipped with this group structure is the Neron model of A.

And Magma documentation on RegularModel says:

RegularModel(C, P) : Crv, Any -> CrvRegModel

This computes a regular model of the curve $C$ at the prime $P$. Here $C$ is a curve over a field $F$ (the rationals, a number field or a univariate rational function field), and $P$ is a prime of the maximal order $O_F$ of $F$ (given as an element or as an ideal).

ComponentGroup(M) : CrvRegModel -> GrpAb

Given a regular model of a curve $C$ at a prime $P$, this returns (as an abstract abelian group) the group of components of the Neron model of the Jacobian of $C$ over the completion at $P$. (This is computed from the IntersectionMatrix of the model.)

I'm quite new in this field and I'm sorry if my questions are silly.

(1) The answer is no for the first part of this question. This is explained in Sage's documentation you cited. The program genus2reduction only outputs the order of $\Phi(\overline{\mathbb F}_p)$, while $c_p$ is the order of $\Phi(\mathbb F_p)$. The later is a subgroup of $\Phi(\overline{\mathbb F}_p)$. I don't understand the second part of the question.

(2) I do not have access to Magma. But try the following example $$y^2=2(x(x-1)(x-2))^2+3$$ at $p=3$. According to Example 1.17 of this paper of S. Bosch and Q. Liu, $\Phi(\mathbb F_3)=\{0\}$ and $\Phi(\overline{\mathbb F}_3)=\mathbb Z/3\mathbb Z$. So if Magma gives you $c_3=1$, then it is capable of computing the Tamagawa number at least for this curve. But after reading the documentation of Magma, I would not be surprised that it will actually give an error message. For the second part of the question, it is not necessary to go to the completion of $K_v$ to compute $c_v$.

(3) Yes, at any $p$ prime to a discriminant of the curve, the later has good reduction at $p$. So its Jacobian has good reduction at $p$. This implies that $\Phi$ is trivial as algebraic group and $c_p=1$.

• I tried both Magma and Sage on example that you pointed. So, in both cases it is $\Bbb Z/3\Bbb Z$. In his paper (math.uga.edu/~lorenz/PaperTorsionTamagawaNumbers.pdf) Dino Lorenzini almost linked at genus2reduction as method of calculation of tamagawa number (he sad about Liu's algorithm implemented in Sage, I've found only one). May it be mistake in his approach? Jun 30, 2014 at 21:36
• @Dr.van: I couldn't find a place in the paper where he says the program gives the Tamagawa number in general. The only case this happens is when genus2reduction gives the trivial group. Then of course the Tamagawa number is $1$. Jul 1, 2014 at 7:32
• Yes, you are right! I missed it, but he always use Liu alghorithm's Sage implementation only in cases where corresponding $c_p$ is $1$. Thanks for your answer. Do you know any computational methods that allow to compute Tamagawa numbers for jacobians of hyperelliptic curves of genus 2? Jul 1, 2014 at 13:25
• I guess you read the discussions in the first link of mathoverflow.net/a/171720/39387. I don't know more, sorry. Jul 2, 2014 at 6:28
• @Dr.van: I've added a couple of comments to mathoverflow.net/a/171767/26576 which you may find useful.
– jsm
Jul 16, 2014 at 9:54