Third Italian Number Theory Meeting

Statistics for biquadratic covers of the projective line over finite fields (with E. Lorenzo and P. Milione)

speaker: Giulio Meleleo (Università degli Studi "Roma Tre")

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.

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