Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves
Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves
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2 格曲线模雅克比行列式的 Birch 和 Swinnerton-Dyer 猜想的经验证据
DOI：
10.1090/s0025-5718-01-01320-5
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发表时间：
2001
期刊：
影响因子：
--
通讯作者：
J. L. Wetherell
中科院分区：
文献类型：
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作者：
E. V. Flynn;Franck Leprévost;Edward F. Schaefer;W. Stein;M. Stoll;J. L. Wetherell
本文给出亏格为2的模雅可比曲线的Birch和Swinnerton-Dyer定理的经验证据。第二个公式涉及与有理数上的雅可比矩阵相关的六个量。这六个量之一是Shafarevich-Tate群的大小。无法计算，我们计算了其他五个量，并解决了最后一个。在所有32种情况下，结果都非常接近2的幂的整数。此外，这个2的幂与我们可以计算的Shafarevich-Tate群的2-挠的大小一致。
This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute.