arXiv subjects

Piermarco Milione

Publications and source records attributed to Piermarco Milione.

3 recordsLinked to original sources

arXiv2017

CM points on Shimura curves and $p$-adic binary quadratic forms

We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion $\mathbb{Q}$-algebra, is in bijection with the set of certain classes of $p$-adic binary quadratic forms, where $p$ is a prime dividing the discriminant of the quaternion algebra. The classes of $p$-adic binary quadratic forms are obtain by the action of a discrete and cocompact subgroup of $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$ arising from the $p$-adic uniformization of the Shimura curve. We finally compute families of $p$-adic binary quadratic forms associated to an infinite family of Shimura curves studied in a previous paper of Amorós-Milione. This extends results of Alsina-Bayer to the $p$-adic context.

math.NT↗

arXiv2017

Mumford curves covering p-adic Shimura curves and their fundamental domains

We give an explicit description of fundamental domains associated to the $p$-adic uniformisation of families of Shimura curves of discriminant $Dp$ and level $N\geq 1$, for which the one-sided ideal class number $h(D,N)$ is $1$. The obtained results generalise those in \cite[Ch. IX]{Gerritzen_vanderPut1980} for Shimura curves of discriminant $2p$ and level $N=1$. The method we present here enables us to find Mumford curves covering Shimura curves, together with a free system of generators for the associated Schottky groups, $p$-adic good fundamental domains and their stable reduction-graphs. This is based on a detailed study of the modular arithmetic of an Eichler order of level $N$ inside the definite quaternion algebra of discriminant $D$, for which we generalise classical results of Hurwitz \cite{Hurwitz1896}. As an application, we prove general formulas for the reduction-graphs with lengths at $p$ of the considered families of Shimura curves.

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arXiv2015

Statistics for biquadratic covers of the projective line over finite fields

We study the distribution of the traces of the Frobenius endomorphism of genus $g$ curves which are quartic non-cyclic covers of $\mathbb{P}^{1}_{\mathbb{F}_{q}}$, 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 $\mathbb{P}^{1}_{\mathbb{F}_{q}}$ with Galois group isomorphic to $r$ copies of $\mathbb{Z}/2\mathbb{Z}$. For $r = 1$, we recover the already known hyperelliptic case. We also include an appendix by Alina Bucur giving the heuristic of these distributions.

math.NT↗