Schinzel Hypothesis with probability 1 and rational points

Abstract: Schinzel's Hypothesis states that there are infinitely many primes represented by any integer polynomial satisfying the necessary congruence assumptions. Equivalently, there exists at least one prime represented by any such polynomial. The problem is completely open, except in the very special case of polynomials of degree 1. We shall describe our recent proof of the existence version of Schinzel's Hypothesis for almost all polynomials, preprint: arxiv.org/abs/2005.02998. We apply our result to showing that generalised Châtelet surfaces have a rational point with positive probability. These surfaces play an important role in the Brauer-Manin obstruction in arithmetic geometry, however, very little is known about their arithmetic. The talk is based on joint work with Alexei Skorobogatov.

number theory

Audience: researchers in the topic

Number theory during lockdown

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