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 Vf containing a Galois-stable lattice Tf, let Wf=Vf/Tf and let K be an imaginary quadratic field in which the prime p splits as v0v―0. My main result is that under some conditions, the p-index of the size of the Shafarevich-Tate group \Sha(K,Wf) 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 Hf1(K,Tf)→Hf1(Kv0,Tf)/tor, and (2) a comparison of different Heegner cycles.