Varieties via their L-functions

Abstract

We describe a procedure for determining the existence, or nonexistence, 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, with 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.

Hypergeometric L-functions in average polynomial time

“…6 • For Euler factors and conductor exponents at wild primes, even the conjectural picture remains incomplete, but see [RRV22] for a partial description. Given enough Fourier coefficients at good primes, one can empirically verify a complete guess for the conductor, the global root number, and all bad Euler factors using the approximate functional equation, as in [FKL19].…”