arXiv subjects

Michel Börner

Publications and source records attributed to Michel Börner.

2 recordsLinked to original sources

arXiv2017

Picard curves with small conductor

We study the conductor of Picard curves over $\mathbb{Q}$, which is a product of local factors. Our results are based on previous results on stable reduction of superelliptic curves that allow to compute the conductor exponent $f_p$ at the primes $p$ of bad reduction. A careful analysis of the possibilities of the stable reduction at $p$ yields restrictions on the conductor exponent $f_p$. We prove that Picard curves over $\mathbb{Q}$ always have bad reduction at $p=3$, with $f_3\geq 4$. As an application we discuss the question of finding Picard curves with small conductor.

math.AG↗

arXiv2015

The functional equation for L-functions of hyperelliptic curves

We compute the $L$-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local $L$-factor and the conductor exponent at the primes of bad reduction. Most of our examples are hyperelliptic curves of genus $g\geq 2$ defined over $\mathbb{Q}$ which have semistable reduction at every prime $p$. We also treat a few more general examples of superelliptic curves.

math.NT↗