8 papers
Rank frequencies for quadratic twists of elliptic curves
Karl Rubin, Alice Silverberg
We give explicit examples of infinite families of elliptic curves E over Q with (nonconstant) quadratic twists over Q(t) of rank at least 2 and 3. We recover some results announced…
Symplectic representations of inertia groups
A. Silverberg, Yu. G. Zarhin
Suppose is a prime number, , is a field that is an unramified finite extension of the field $\Q_\ell$ of -adic numbers, and is a finite group that is…
Polarizations on abelian varieties
A. Silverberg, Yu. G. Zarhin
The results in this paper imply that for every number field F and positive integer r, there exists an F-isogeny class of abelian varieties such that r divides the degree of every F…
Ranks of elliptic curves in families of quadratic twists
Karl Rubin, Alice Silverberg
We show that the unboundedness of the ranks of the quadratic twists of an elliptic curve is equivalent to the divergence of certain infinite series.
Polarizations on abelian varieties and self-dual ell-adic representations of inertia groups
Alice Silverberg, Yuri G. Zarhin
It is well-known that every finite subgroup of GL_d(Q_{\ell}) is conjugate to a subgroup of GL_d(Z_{\ell}). However, this does not remain true if we replace general linear groups b…
Mod 2 representations of elliptic curves
Karl Rubin, Alice Silverberg
This note gives explicit equations for the elliptic curves (in characteristic not 2 or 3) with mod 2 representation isomorphic to that of a given one.