A family of elliptic ℚ-curves defined over biquadratic fields and their modularity
A formula for the Selmer group of a rational three-isogeny
A functional equation originating from elliptic curves.
A generalization of the Chowla-Selberg formula.
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 moduli approach to quadratic -curves realizing projective mod Galois representations.
A new record for the canonical height on an elliptic curve over .
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 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...
A Note on squares in arithmetic progressions, II
We show that the number of squares in an arithmetic progression of length is at most , for certain absolute positive constants , . This improves the previous result of Bombieri, Granville and Pintz [1], where one had the exponent in place of our . The proof uses the same ideas as in [1], but introduces a substantial simplification by working only with elliptic curves rather than curves of genus as in [1].
A parametric family of elliptic curves
A procedure to calcute torsion of elliptic curves over Q.
À propos de la conjecture de Manin pour les courbes elliptiques modulaires
A quantitative primitive divisor result for points on elliptic curves
Let be an elliptic curve defined over a number field, and let be a point of infinite order. It is natural to ask how many integers fail to occur as the order of modulo a prime of . For , a quadratic twist of , and as above, we show that there is at most one such .
A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.
A refined cojecture of Mazur-Tate type for Heegner points.
A remark on certain simultaneous divisibility sequences
We investigate possible orders of reductions of a point in the Mordell-Weil groups of certain abelian varieties and in direct products of the multiplicative group of a number field. We express the result obtained in terms of divisibility sequences.
A search for high rank congruent number elliptic curves.