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If $X$ is a scheme of finite type over $\mathbb{Z}$, and $X_0$ denotes its set of closed points, then one can define its zeta function on the half plane $Re(s)>\text{dim}(X)$: \begin{equation} \zeta_X(s)=\prod_{x\in X_0}\frac{1}{1-|k(x)|^{-s}}, \end{equation} where $k(x)$ denotes the (finite) residue field at $x$.

The zeta function has a completion, which includes the conductor and a gamma factor. I want to ask about the latter, in a special case, namely when $X$ is a (proper, regular) model of an abelian variety over a number field.

For example, if $X$ is a model of an elliptic curve $E$ over a number field $k$, then the zeta function of $X$ can be written: \begin{equation} \zeta_X(s)=n_{X}(s)\frac{\zeta_k(s)\zeta_k(s-1)}{L_E(s)}, \end{equation} where $n_X(s)$ depends only on the bad primes. Indeed, an identical expression holds for a model of any smooth projective curve over a number field. The quotient on the right hand side of the above expression is what some call the Hasse--Weil $\zeta$-function of $E$. This notion can be extended to any smooth projective variety $V$ of dimension $n$: \begin{equation} \frac{\prod_{i=0}^nL(H^{2i}_{et}(V),s)}{\prod_{i=0}^{n-1}L(H^{2i+1}_{et}(V),s)}. \end{equation} The gamma factor for the zeta function of a model of $V$ is the alternating product of the gamma factors for each Hasse--Weil L-factor (coming from the Hodge decomposition of each cohomology group). In the case of elliptic curves, direct computation shows that the gamma factor for a model is in fact a rational function. I have been lead to believe that this is true for any abelian variety, but I cannot prove it.

Firstly, is it true that the gamma factor of a model $\mathcal{A}\rightarrow\text{Spec}(\mathcal{O}_k)$ of an abelian variety $A$ over a number field $k$ is rational? Secondly, is there a profound reason that this should be true?

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up vote 8 down vote accepted

This amounts to an Euler characteristic argument. A ratio of products of Gamma functions such as $$ \frac{\prod_{i=1}^r \Gamma(s - a_i)}{\prod_{j=1}^t \Gamma(s - b_j)} $$ is rational if and only if $r = t$ (i.e. there are as many functions in the numerator as in the denominator). But the number of $\Gamma$ factors in the numerator of the zeta function is the sum of the dimensions of the even-degree cohomology groups, and the number of factors in the denominator is the sum of the odd-degree ones. So the ratio is a rational function of $s$ if and only if the Euler characteristic is 0.

However, all Abelian varieties have Euler characteristic 0 (since over $\mathbf{C}$ they are products of circles) and that does it.

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Thank you, that is exactly the sort of answer I was hoping for. To be slightly more precise, for a smooth curve of genus g the computation yields a transcendental function (product of gamma for real places multiplied by product of gamma for complex places) raised to the power g-1, which I now understand as half the Euler characteristic. So, I can make an educated guess at the general expression for nonsingular V of dimension n. – Tom163 Oct 4 '13 at 10:55

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