Selmer companion curves

Selmer Companion Curves

We say that two elliptic curves E1, E2 over a number field K are n-Selmer companions for a positive integer n if for every quadratic character χ of K, there is an isomorphism Seln(E χ 1 /K) ∼= Seln(E 2 /K) between the nSelmer groups of the quadratic twists E 1 , E χ 2 . We give sufficient conditions for two elliptic curves to be n-Selmer companions, and give a number of examples of non-isogenou...

Selmer varieties for curves with CM Jacobians

We study the Selmer variety associated to a canonical quotient of the Qp-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over Q whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multivariable Iwasawa theory is used to prove dimension bounds, which, in turn, lead to a new proof of Diophantine finitenes...

The unipotent Albanese map and Selmer varieties for curves

We discuss p-adic unipotent Albanese maps for curves of positive genus, extending the theory of p-adic multiple polylogarithms. This construction is then used to relate linear Diophantine conjectures of ‘Birch and Swinnerton-Dyer type’ to non-linear theorems of Faltings-Siegel type. In a letter to Faltings [14] dated June, 1983, Grothendieck proposed several striking conjectural connections bet...

Symmetric Squares of Elliptic Curves: Rational Points and Selmer Groups

Average Size of 2-selmer Groups of Elliptic Curves, I

In this paper, we study a class of elliptic curves over Q with Qtorsion group Z2×Z2, and prove that the average order of the 2-Selmer groups is bounded.