The Birch and Swinnerton-Dyer conjecture – which is one of the seven million-dollar Clay Mathematics Institute Millennium Prize Problems – and its generalizations are a significant focus of 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. We 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 containing a Galois-stable lattice T, let $W=V/T$ and let K be an imaginary quadratic field in which the prime p splits. Our main result is that under some conditions, the p-index of the size of the Shafarevich-Tate group of W with respect to the Galois group of K 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 we make to the 2017 arguments are (1) a generalized version of a previous calculation of the size of the cokernel of a localization-modulo-torsion map, and (2) a comparison of different Heegner cycles.

Birch 和 Swinnerton-Dyer 猜想——这是价值 700 万美元的克莱数学研究所千年奖问题之一——及其推广是数论研究的一个重要焦点。

Jetchev、Skinner 和 Wan 2017 年的一篇文章证明了 Birch 和 Swinnerton-Dyer 公式在素数p处适用于某些等级为 1 的有理椭圆曲线。我们推广并调整该文章的论点以证明某些模形式的类似公式。对于偶数权重大于 2 的新形式f ，其伽罗瓦表示V包含一个伽罗瓦稳定格T，让$W=五/吨$令K为素数p在其中分裂的假想二次域。我们的主要结果是，在某些条件下，W的 Shafarevich-Tate 群的大小相对于K的 Galois 群的p指数是Heegner 的 Abel-Jacobi 映射的对数的p指数的两倍由 Bertolini、Darmon 和 Prasanna 定义的循环。

我们对 2017 年的论点做出的重要原始改编是（1）先前计算定位模扭转图的核心大小的通用版本，以及（2）不同 Heegner 循环的比较。