Varieties via their L-functions

TL;DR

This paper introduces an analytic method to determine the existence of algebraic varieties of a given conductor by analyzing their L-functions, without relying on automorphic or Galois representations.

Contribution

It presents a novel procedure for identifying algebraic varieties through L-functions, applicable even without known automorphic or Galois structures.

Findings

Successfully identified L-functions of all hyperelliptic curves under conductor 500

Demonstrated the method's effectiveness in variety existence determination

Provided a new approach for studying algebraic varieties via L-functions

Abstract

We describe a procedure for determining the existence, or non-existence, of an algebraic variety of a given conductor via an analytic calculation involving L-functions. The procedure assumes that the Hasse-Weil L-function of the variety satisfies its conjectured functional equation, but there is no assumption of an associated automorphic object or Galois representation. We demonstrate the method by finding the Hasse-Weil L-functions of all hyperelliptic curves of conductor less than 500.