Studying the growth of Mordell-Weil.
Ranks of elliptic curves in families of quadratic twists.
Rank frequencies for quadratic twists of elliptic curves.
The "main conjectures" of Iwasawa theory for imaginary quadratic fields.
p-adic L-functions and rational points on elliptic curves with complex multiplication.
Elliptic curves and -extensions
Stark units and Kolyvagin's "Euler systems".
Congruences for Special Values of L-functions of Elliptic Curves with Complex Multiplication.
Elliptic Curves With Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer.
On the main cojecture of Iwasawa theory for imaginary quadratic fields.
A Stark conjecture “over ” for abelian -functions with multiple zeros
Suppose is an abelian extension of number fields. Stark’s conjecture predicts, under suitable hypotheses, the existence of a global unit of such that the special values for all characters of can be expressed as simple linear combinations of the logarithms of the different absolute values of . In this paper we formulate an extension of this conjecture, to attempt to understand the values when the order of vanishing may be greater than one. This conjecture no longer predicts...
Anticyclotomic Iwasawa theory of CM elliptic curves
We study the Iwasawa theory of a CM elliptic curve in the anticyclotomic -extension of the CM field, where is a prime of good, ordinary reduction for . When the complex -function of vanishes to even order, Rubin’s proof of the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the -power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg show that it is not a torsion module. In...
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