An identity for sums of polylogarithm functions.
An infinite family of elliptic curves over Q with large rank via Néron's method.
An integrality criterion for elliptic modular forms
Let be an elliptic modular form level of N. We present a criterion for the integrality of at primes not dividing N. The result is in terms of the values at CM points of the forms obtained applying to the iterates of the Maaß differential operators.
An interesting family of curves of genus 1.
An upper bound for the minimum genus of a curve without points of small degree
We prove that for any prime p there is a constant Cₚ > 0 such that for any n > 0 and for any p-power q there is a smooth, projective, absolutely irreducible curve over of genus g ≤ Cₚqⁿ without points of degree smaller than n.
Analogs of Δ(z) for triangular Shimura curves
We construct analogs of the classical Δ-function for quotients of the upper half plane 𝓗 by certain arithmetic triangle groups Γ coming from quaternion division algebras B. We also establish a relative integrality result concerning modular functions of the form Δ(αz)/Δ(z) for α in B⁺. We give two explicit examples at the end.
Analytical construction of Weil curves over function fields
Annihilators of the class group of a compositum of quadratic fields
This paper is devoted to a construction of new annihilators of the ideal class group of a tamely ramified compositum of quadratic fields. These annihilators are produced by a modified Rubin’s machinery. The aim of this modification is to give a stronger annihilation statement for this specific type of fields.
Anticyclotomic Iwasawa theory of CM elliptic curves
We study the Iwasawa theory of a CM elliptic curve in the anticyclotomic -extension of the CM field, where is a prime of good, ordinary reduction for . When the complex -function of vanishes to even order, Rubin’s proof of the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the -power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg show that it is not a torsion module. In...
Anticyclotomic main conjectures.
Approximating algebraic numbers by j-invariants of elliptic curves
Approximation algébrique en caractéristique p
Approximation diophantienne dans les groupes algébriques commutatifs. (I): Une version effective du théorème du sous-group algébrique.
Approximation diophantienne sur les variétés semi-abéliennes
Arakelov computations in genus curves
Arakelov invariants of arithmetic surfaces are well known for genus 1 and 2 ([4], [2]). In this note, we study the modular height and the Arakelov self-intersection for a family of curves of genus 3 with many automorphisms:Arakelov calculus involves both analytic and arithmetic computations. We express the periods of the curve in terms of elliptic integrals. The substitutions used in these integrals provide a splitting of the jacobian of as a product of three elliptic curves. Using the corresponding...
Arakelov's Theorem for Abelian Varieties.
Arithmetic differential equations in several variables
We survey recent work on arithmetic analogues of ordinary and partial differential equations.
Arithmetic Distance Functions and Height Fuctions in Diophantine Geometry.
Arithmetic hyperbolic surface bundles.
Arithmetic infinite Grassmannians and the induced central extensions.