The Arithmetic of Quadratic Twists of Elliptic Curves

Explore a 46-minute lecture on the arithmetic of quadratic twists of elliptic curves, presented by Ye Tian for the International Mathematical Union. Delve into the behavior of Selmer groups and L-values of elliptic curves under quadratic twists, using the congruent number problem as a fundamental example. Gain insights into recent progress towards the BSD conjecture. Examine topics such as the Goldfeld Conjecture for CM Families, equivalence relations for quadratic twist families, and the distribution of 2-Selmer groups. Investigate Kolyvagin's Question, modularity of Heegner cycles, and the Arithmetic Rallis inner product formula. Access accompanying slides for visual support and deeper understanding of this advanced mathematical discourse.

Intro
The Congruent Number Problem
Quadratic twists of eliptic curves over 0
Conjectures on Leading terms of L-series under Quadratic Twist
Goldfeld Conjecture for CM Families
Equivalence relation for quadratic twist families
Elliptic curves with full rational 2-torsion
Distribution of 2-Selmer groups for type (A)
Distribution of 2-Selmer groups for type (B) and (C)
Kolyvagin's Question Now we are in the situation that
Modularity of Heegner cycles
Arithmetic Rallis inner product formula
Application to Kolyvagin's Problem