Date:

Mon, 06/12/202114:30-16:00

Location:

Ross 70

Title: Prime torsion in the Tate-Shafarevich groups of abelian varieties over $\mathbb{Q}$

Abstract:

Very little is known about the Tate-Shafarevich groups of abelian varieties. On the one hand, the BSD conjecture predicts that they are finite. On the other hand, heuristics suggest that, for each prime $p$, a positive proportion of elliptic curves $E/\mathbb{Q}$ have $\Sha(E)[p]

e 0$, and one expects something similar for higher dimensional abelian varieties as well. Yet, despite these expectations, it seems to be an open question whether, for each prime $p$, there exists even a

e 0$. In this talk, I will show that, for each prime $p$, there exists a geometrically simple abelian variety $A/\mathbb{Q}$ with $\Sha(A)[p]

e 0$. Our examples arise from modular forms with Eisenstein congruences. This is joint work with Ari Shnidman.

Abstract:

Very little is known about the Tate-Shafarevich groups of abelian varieties. On the one hand, the BSD conjecture predicts that they are finite. On the other hand, heuristics suggest that, for each prime $p$, a positive proportion of elliptic curves $E/\mathbb{Q}$ have $\Sha(E)[p]

e 0$, and one expects something similar for higher dimensional abelian varieties as well. Yet, despite these expectations, it seems to be an open question whether, for each prime $p$, there exists even a

*single*elliptic curve over $\mathbb{Q}$ with $\Sha(E)[p]e 0$. In this talk, I will show that, for each prime $p$, there exists a geometrically simple abelian variety $A/\mathbb{Q}$ with $\Sha(A)[p]

e 0$. Our examples arise from modular forms with Eisenstein congruences. This is joint work with Ari Shnidman.