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 $(L_p^{♯},L_p^{♭})$. 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 $L_p^{♯}$ and $L_p^{♭}$ 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 Šafarevič-Tate group in terms of the Iwasawa invariants of $L_p^{♯}$ and $L_p^{♭}$. 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.

41 pages, one figure This paper is being withdrawn as it is being subsumed by two new papers

References in corpus (1)

• A p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties

Cited by in corpus (1)

• Factorisation of two-variable p-adic L-functions