Numerical verification of the Birch and Swinnerton-Dyer conjecture for hyperelliptic curves of higher genus over $\mathbb Q$ up to squares

Raymond van Bommel

Published 2017-11-28Version 1

The Birch and Swinnerton-Dyer conjecture has been numerically verified for the Jacobians of 32 modular hyperelliptic curves of genus 2 by Flynn, Lepr\'evost, Schaefer, Stein, Stoll and Wetherell, using modular methods. In the calculation of the real period, there is a slight inaccuracy, which might give problems for curves with non-reduced components in the special fibre of their N\'eron model. In this present paper we explain how the real period can be computed, and how the verification has been extended to many more hyperelliptic curves, some of genus 3, without using modular methods.