Varieties via their L-functions
Journal article, 2019

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, 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.


Degree 4

Hasse–Weil L-function

Functional equation


David W. Farmer

American Institute of Mathematics

Sally Koutsoliotas

Bucknell University

Stefan Lemurell

Chalmers, Mathematical Sciences, Algebra and geometry

Journal of Number Theory

0022-314X (ISSN) 1096-1658 (eISSN)

Vol. 196 364-380

Subject Categories

Algebra and Logic


Mathematical Analysis



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