Boundary cohomology of Shimura varieties. II. Hodge theory at the boundary.
Bounds for arithmetic multiplicities.
Bounds for frequencies of class groups of real quadratic genus 1 function fields
Bounds for the degrees of CM-fields of class number one
Bounds for the minimal solution of genus zero diophantine equations
Bounds for the order of the Tate-Shafarevich group
Bounds for the smallest integer point of a rational curve
Calcul du nombre de points sur une courbe elliptique dans un corps fini : aspects algorithmiques
Nous décrivons dans cet article les algorithmes nécessaires à une implantation efficace de la méthode de Schoof pour le calcul du nombre de points sur une courbe elliptique dans un corps fini. Nous tentons d’unifier pour cela les idées d’Atkin et d’Elkies. En particulier, nous décrivons le calcul d’équations pour , premier, ainsi que le calcul efficace de facteurs des polynômes de division d’une courbe elliptique.
Calculating Root Numbers of Elliptic curves over Q.
Canonical heights on the Jacobians of curves of genus 2 and the infinite descent
Canonical integral structures on the de Rham cohomology of curves
For a smooth and proper curve over the fraction field of a discrete valuation ring , we explain (under very mild hypotheses) how to equip the de Rham cohomology with a canonical integral structure: i.e., an -lattice which is functorial in finite (generically étale) -morphisms of and which is preserved by the cup-product auto-duality on . Our construction of this lattice uses a certain class of normal proper models of and relative dualizing sheaves. We show that our lattice naturally...
Canonical local heights and multiplication formulas for the Jacobians of curves of genus 2
Capacity theory on varieties
Certain maximal curves and Cartier operators
Characterization of the torsion of the Jacobian of y² = x⁵+Ax and some applications
Characterization of the torsion of the Jacobians of two families of hyperelliptic curves
Consider the families of curves and where A is a nonzero rational. Let and denote their respective Jacobian varieties. The torsion points of and are well known. We show that for any nonzero rational A the torsion subgroup of is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for (A ≠ 4) and . We also almost...
Class fields of abelian extensions of Q.
Class invariants and cyclotomic unit groups from special values of modular units
In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order -recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which...
Class Invariants for Quartic CM Fields
One can define class invariants for a quartic primitive CM field as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to . Such constructions were given by de Shalit-Goren and Lauter. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct -units in certain abelian extensions of a reflex field of , where is effectively determined by , and to bound the primes appearing...
Class invariants of Mordell-Weil groups.