arXiv subjects

Adebisi Agboola

Publications and source records attributed to Adebisi Agboola.

4 recordsLinked to original sources

arXiv2019

On anticyclotomic variants of the $p$-adic Birch and Swinnerton-Dyer conjecture

We formulate analogues of the Birch and Swinnerton-Dyer conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna attached to elliptic curves $E/\mathbf{Q}$ at primes $p$ of good ordinary reduction. Using Iwasawa theory, we then prove under mild hypotheses one of the inequalities predicted by the rank part of our conjectures, as well as the predicted leading coefficient formula up to a $p$-adic unit. Our conjectures are very closely related to conjectures of Birch and Swinnerton-Dyer type formulated by Bertolini-Darmon in 1996 for certain Heegner distributions, and as application of our results we also obtain the proof of an inequality in the rank part of their conjectures.

math.NT↗

arXiv2010

On Rubin's variant of the p-adic Birch and Swinnerton-Dyer conjecture II

Let E be an elliptic curve over Q with complex multiplication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz two-variable p-adic L-function lying outside the range of $p$-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton-Dyer conjecture when $E(K)$ is infinite, and he proved that his conjecture is true for E(K) of rank one. When E(K) is finite, however, the statement of Rubin's original conjecture no longer applies, and the relevant special value of the appropriate $p$-adic L-function is equal to zero. In this paper we extend our earlier work and give an unconditional proof of an analogue of Rubin's conjecture when E(K) is finite.

math.NT↗

arXiv2006

Anticyclotomic Iwasawa theory of CM elliptic curves

We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic $\mathbf{Z}_p$-extension of the CM field, where $p$ is a prime of good, ordinary reduction for $E$. When the complex $L$-function of $E$ vanishes to even order, the two variable main conjecture of Rubin implies that the Pontryagin dual of the $p$-power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg shows that it is not a torsion module. In this paper we show that in the case of odd order of vanishing the dual of the Selmer group has rank exactly one, and we prove a form of the Iwasawa main conjecture for the torsion submodule.

math.NT↗

arXiv2005

Anticyclotomic Iwasawa theory of CM elliptic curves II

We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic $\mathbf{Z}_p$-extension $D_\infty$ of the CM field $K$, where $p$ is a prime of good, supersingular reduction for $E$. Our main result yields an asymptotic formula for the corank of the $p$-primary Selmer group of $E$ along the extension $D_\infty/K$.

math.NT↗