Selmer余秩至多一时2-primary Birch–Swinnerton-Dyer公式
The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one
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具体内容
稿件讨论有理数域上椭圆曲线的Birch–Swinnerton-Dyer主项公式的2-primary部分。作者声称，对2-power Selmer群余秩至多1的每条椭圆曲线，该公式成立；在此范围内代数秩、解析秩与Selmer余秩相等，且Tate–Shafarevich群有限。结合二次扭Selmer分布，作者进一步声称对每条固定曲线的密度一的有符号无平方扭，按绝对值排序，得到精确的2-primary公式，公共秩为0或1且各占一半密度。当前收录摘要未给出证明细节。
- 作者声称对2-power Selmer余秩≤1的有理椭圆曲线证明2-primary BSD主项公式。
- 该范围内代数秩、解析秩与Selmer余秩相等，Tate–Shafarevich群有限。
- 结合二次扭分布，声称密度一的有符号无平方扭满足精确公式。
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英文原始摘要
We prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.
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