A Birch and Swinnerton-Dyer formula for high-weight modular forms

Abstract

The Birch and Swinnerton-Dyer conjecture -- which is one of the seven million-dollar Clay Mathematics Institute Millennium Prize Problems -- and its generalizations to modular forms, motives, etc. are a significant focus of current number theory research.

A 2017 article of Jetchev, Skinner and Wan proved a Birch and Swinnerton-Dyer formula at a prime $p$ for certain rational elliptic curves of rank 1. I generalize and adapt that article's arguments to prove an analogous formula for certain modular forms. For newforms $f$ of even weight higher than 2 with Galois representation $V_{f}$ containing a Galois-stable lattice $T_{f}$, let $W_{f} = V_{f}/T_{f}$ and let $K$ be an imaginary quadratic field in which the prime $p$ splits as $v_{0}\overline{v}_{0}$. My main result is that under some conditions, the $p$-index of the size of the Shafarevich-Tate group $\Sha(K,W_{f})$ is twice the $p$-index of a logarithm of the Abel-Jacobi map of a Heegner cycle defined by Bertolini, Darmon and Prasanna.

Significant original adaptations I make to the 2017 arguments are (1) a generalized version of a previous calculation of the size of the cokernel of the localization-modulo-torsion map $H^{1}_{f}(K,T_{f}) \rightarrow H^{1}_{f}(K_{v_{0}},T_{f})/$tor, and (2) a comparison of different Heegner cycles.