The Functional Equation for L-Functions of Hyperelliptic Curves

ABSTRACT

We compute the L-functions of a large class of algebraic curves and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local L-factor and the conductor exponent at the primes of bad reduction. We mainly restrict to the case of hyperelliptic curves of genus g ⩾ 2 defined over that have semistable reduction at every prime p. We also discuss a few more general cases to illustrate the usefulness of our method for general superelliptic curves.

KEYWORDS:

• L-functions
• curves
• semistable reduction

2000 AMS SUBJECT CLASSIFICATION:

• Primary 11G40
•  Secondary 14G10, 11G20

Acknowledgments

We thank the referees for very useful comments and suggestions. We are grateful to one of the referees for pointing out the reference [Liu 96] to us, which contains many of the arguments from Section 3 to describe models of hyperelliptic curves in a similar form.