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Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes

组织者

胡悦科 , 埃马纽埃尔·勒库图里耶 , 凯撒·鲁普

演讲者

Lazar Radicevic

时间

2021年12月15日 16:00 至 17:00

地点

1118

线上

Zoom 388 528 9728 (BIMSA)

摘要

We consider the Kolyvagin cohomology classes associated to an elliptic curve E defined over ℚ from a computational point of view. We explain how to go from a model of a class as an element of (E(L)/pE(L))^Gal(L/ℚ), where p is prime and L is a dihedral extension of ℚ of degree 2p, to a geometric model as a genus one curve embedded in ℙ^(p−1). We adapt the existing methods to compute Heegner points to our situation, and explicitly compute them as elements of E(L). Finally, we compute explicit equations for several genus one curves that represent non-trivial elements of the p-torsion part of the Tate-Shafarevich group of E, for p≤11, and hence are counterexamples to the Hasse principle.