Multiplicities of poles/zeros of local zeta function

Question

The following question arises when I tried to collect some information about local zeta function of Weil and global zeta function.

Question: There is a "folklore" conjecture that all the zeros of a primitive global L-function are simple zeros apart from a possible zero with multiplicity $>1$ lying in the center of the critical strip. The multiplicities of such zeros encode certain important information (e.g., rank of an elliptic curve, etc.). Of course, one cannot prove anything about such conjectures at this time.

If one considers the local zeta function $Z(X,u)$(adapting the notation in the Wikipedia) of a smooth projective variety $X$ over $\mathbb{Q}$ instead:

1. what do we know about the multiplicities of poles and zeros of the rational function $Z(X,u)$?

2. What kind of information does a pole/zero of $Z(X,u)$ with multiplicity $>1$ carry?

Thank you for your answer.

Answer 1

There is a conjecture called the Linear Independence hypothesis that states, in its most general form, that not only are the nontrivial zeroes of a primitive $L$-function simple outside of the point $s = 1/2$, but that the imaginary ordinates of the zeroes in the upper half-plane are linearly independent over $\mathbb{Q}$.

For a nonsingular projective curve $\mathcal{C}$ of genus $2g$ over a finite field $\mathbb{F}_q$, the zeta function has the form \[Z_{\mathcal{C}/\mathbb{F}_q}(u) = \frac{P_{\mathcal{C}/\mathbb{F}_q}(u)}{(1 - u)(1 - qu)},\] where $u = q^{-s}$ and \[P_{\mathcal{C}/\mathbb{F}_q}(u) = \prod_{j = 1}^{g} \left(1 - \gamma_j u\right) \left(1 - \overline{\gamma_j} u\right)\] for some complex numbers $\gamma_j$ of the form $\sqrt{q} e^{i\theta_j}$ with $\theta_j \in [0,\pi]$. Then the analogue of the Linear Independence hypothesis is that the collection $\{\theta_1,\ldots,\theta_g,\pi\}$ is linearly independent over $\mathbb{Q}$.

Unfortunately, there are counterexamples, even for $g = 1$ (elliptic curves). These are certain supersingular elliptic curves over $\mathbb{F}_q$ for which $\theta = \theta_1$ is a rational multiple of $\pi$. Even worse, there are supersingular elliptic curves for which $\theta = 0$, so that there is a double zero of $P_{\mathcal{C}/\mathbb{F}_q}(u)$ at $u = 1/\sqrt{q}$; these curves always exist if $q = p^{2m}$ for some prime $p$ and some $m \geq 1$.

For more details, see Theorem 4.1 of this paper for the genus one case, and Theorem 1.2 of this paper for the genus two case. For higher genus, little is known.