**abstract:**
We study the distribution of the traces of the Frobenius endomorphism of genus \( g\) curves which are quartic non-cyclic covers of \( \mathbb{P}_{\mathbb{F}_q}^{1} \), as the curve varies in an irreducible component of the moduli space. We show that for \( q \) fixed, the limiting distribution of the trace of Frobenius equals the sum of \( q + 1 \) independent random discrete variables. We also show that when both \( g \) and \( q \) go to infinity, the normalized trace has a standard complex Gaussian distribution. Finally, we extend these computations to the general case of arbitrary covers of \( \mathbb{P}_{\mathbb{F}_q}^{1} \) with Galois group isomorphic to \( r \) copies of \( \mathbb{Z}/2\mathbb{Z} \). For \( r=1 \), we recover the already known hyperelliptic case. This is a joint work with Elisa Lorenzo and Piermarco Milione.

Wed 23 Sep, 17:00 - 17:25, Aula Russo

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