A Banach space determined by the Weil height
A Bogomolov property for curves modulo algebraic subgroups
Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.
A bound for the average rank of a family of abelian varieties
In this note, we consider a one-parameter family of Abelian varieties , and find an upper bound for the average rank in terms of the generic rank. This bound is based on Michel's estimates for the average rank in a one-parameter family of Abelian varieties, and extends previous work of Silverman for elliptic surfaces.
A characterization of integral elliptic automorphic forms
A Classical Diophantine Problem and Modular Forms of Weight 3/2.
A comparison of elliptic units in certain prime power conductor cases
The aim of this paper is to compare two modules of elliptic units, which arise in the study of elliptic curves E over quadratic imaginary fields K with complex multiplication by , good ordinary reduction above a split prime p and prime power conductor (over K). One of the modules is a special case of those modules of elliptic units studied by K. Rubin in his paper [Invent. Math. 103 (1991)] on the two-variable main conjecture (without p-adic L-functions), and the other module is a smaller one,...
A condition for the rationality of certain elliptic modular forms over primes dividing the level
Let be a weight holomorphic automorphic form with respect to . We prove a sufficient condition for the integrality of over primes dividing . This condition is expressed in terms of the values at particular curves of the forms obtained by iterated application of the weight Maaß operator to and extends previous results of the Author.
A construction of curves over finite fields
A descent map for curves with totally degenerate semi-stable reduction
Let be a local field of residue characteristic . Let be a curve over whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to- rational torsion subgroup on the Jacobian of . We also determine divisibility of line bundles on , including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of .
A dimension formula for Ekedahl-Oort strata
We study the Ekedahl-Oort stratification on moduli spaces of PEL type. The strata are indexed by the classes in a Weyl group modulo a subgroup, and each class has a distinguished representative of minimal length. The main result of this paper is that the dimension of a stratum equals the length of the corresponding Weyl group element. We also discuss some explicit examples.
A dynamical interpretation of the global canonical height on an elliptic curve.
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 formula for the supersingular polynomial
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