OSU Mathematics Seminars and Colloquia
Calendar
Tue, Apr 27, 2021
Number Theory Seminar
2:00 PM
Online (Zoom)
Twists of hyperelliptic curves by integers in progressions modulo $p$
David Krumm, Reed College
Host: John Doyle
Contact John Doyle (john.r.doyle@okstate.edu) for the Zoom link.
[Abstract] [PDF]
Abstract: Let $C$ be a hyperelliptic curve defined over the rational numbers, and consider the set $S$ of all squarefree integers $d$ such that the quadratic twist of $C$ by $d$ has a rational point. In this talk we will discuss the question of whether, given a prime number $p$, the set $S$ contains a representative of every residue class modulo $p$. If $C$ has genus 0, this question can be resolved using elementary methods. If $C$ is given by $y^2=f(x)$, where $f(x)$ factors into polynomials of small degree, the question can be successfully approached using known results on squarefree values of binary forms. The general case, however, seems to require the use of the ABC Conjecture.
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