A functional equation originating from elliptic curves.
A fundamental domain for the Fermat curves and their quotients.
A general framework for subexponential discrete logarithm algorithms
A generalization of the Chowla-Selberg formula.
A geometric proof of Kummer's reciprocity law for seventh powers
A Grunwald-Wang type theorem for abelian varieties
A “Hardy-Littlewood” approach to the -unit equation
A local-global criterion for dynamics on ℙ¹
A local-global principle for rational isogenies of prime degree
Let be a number field. We consider a local-global principle for elliptic curves that admit (or do not admit) a rational isogeny of prime degree . For suitable (including ), we prove that this principle holds for all , and for , but find a counterexample when for an elliptic curve with -invariant . For we show that, up to isomorphism, this is the only counterexample.
A lower bound for the rank of
A moduli approach to quadratic -curves realizing projective mod Galois representations.
A motivic Chebotarev density theorem.
A natural series for the natural logarithm.
A new record for the canonical height on an elliptic curve over .
A note on a paper of Franklin
A note on a product formula for the cubic Gauss sum
A note on arithmetic progressions on elliptic curves.
A note on arithmetic progressions on quartic elliptic curves.
A Note on heights in certain infinite extensions of
We study the behaviour of the absolute Weil height of algebraic numbers in certain infinite extensions of . In particular, we obtain a Northcott type property for infinite abelian extensions of finite exponent and also a Bogomolov type property for certain fields which are a -adic analog of totally real fields. Moreover, we obtain a non-archimedean analog of a uniform distribution theorem of Bilu in the archimedean case.
A note on integral points on elliptic curves
We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of Macsyma, with there being a unique rational nondegenerate solution. For the second case...