On pairs of p-adic analogues of the conjectures of Birch and Swinnerton-Dyer

Abstract

For a weight two modular form and a good prime $p$, we construct a vector of Iwasawa functions $(L_p^\sharp,L_p^\flat)$. In the elliptic curve case, we use this vector to put the $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer for ordinary [MTT] and supersingular [BPR] primes on one footing. Looking at $L_p^\sharp$ and $L_p^\flat$ individually leads to a stronger conjecture containing an extra zero phenomenon. We also give an explicit upper bound for the analytic rank in the cyclotomic direction and an asymptotic formula for the $p$-part of the analytic size of the \v{S}afarevi\v{c}-Tate group in terms of the Iwasawa invariants of $L_p^\sharp$ and $L_p^\flat$. A very puzzling phenomenon occurs in the corresponding formulas for modular forms. When $p$ is supersingular, we prove that the two classical $p$-adic $L$-functions ([AV75],[VI76]) have finitely many common zeros, as conjectured by Greenberg.