A lower bound for the canonical height on elliptic curves over abelian extensions

Let E/K be an ellptic curve defined over a number field, let h be the canonical height on E, and let K^ab be the maximal abelian extension of K. Extending work of M. Baker, we prove that there is a positive constant C(E/K) so that every nontorsion point P in E(K^ab) satisfies h(P) > C(E/K)

Necessary truths, evidence, and knowledge.

According to the knowledge view of evidence notoriously defended by Timothy Williamson (2000), for any subject, her evidence consists of all and only her propositional knowledge (E=K). Many have found (E=K) implausible. However, few have offered arguments against Williamson’s positive case for (E=K). In this paper, I propose an argument against Williamson’s positive case in favour of (E=K). Central to my argument is the possibility of the knowledge of necessary truths. I also draw some more general conclusions concerning theorizing about evidence

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.Comment: Final version. To appear in Mathematische Annalen