Simultaneous resolution of rational singularities
Singular Bott-Chern classes and the arithmetic Grothendieck Riemann Roch theorem for closed immersions.
Singularities of Quotients by Vector Fields in Characteristic p.
Slope filtration of quasi-unipotent overconvergent -isocrystals
We study local properties of quasi-unipotent overconvergent -isocrystals on a curve over a perfect field of positive characteristic . For a --module over the Robba ring , we define the slope filtration for Frobenius structures. We prove that an overconvergent -isocrystal is quasi-unipotent if and only if it has the slope filtration for Frobenius structures locally at every point on the complement of the curve.
Slope filtrations revisited.
Small points and Arakelov theory.
Small points on a multiplicative group and class number problem
Let be an algebraic subvariety of a torus and denote by the complement in of the Zariski closure of the set of torsion points of . By a theorem of Zhang, is discrete for the metric induced by the normalized height . We describe some quantitative versions of this result, close to the conjectural bounds, and we discuss some applications to study of the class group of some number fields.
Smooth degeneration of complete intersection curves in positive characteristc.
Smooth ponts of p-adic subanalytic sets.
Solomon's Zeta Functions over Algebraic Function Fields.
Solving algebraic equations in roots of unity.
Solving an indeterminate third degree equation in rational numbers. Sylvester and Lucas
This article concerns the problem of solving diophantine equations in rational numbers. It traces the way in which the 19th century broke from the centuries-old tradition of the purely algebraic treatment of this problem. Special attention is paid to Sylvester’s work “On Certain Ternary Cubic-Form Equations” (1879–1880), in which the algebraico-geometrical approach was applied to the study of an indeterminate equation of third degree.
Some abelian threefolds with nontrivial Griffiths group
Some analogies between number theory and dynamical systems on foliated spaces.
Some Consequences of the Riemann Hypothesis for Varieties over Finite Fields.
Some elliptic function identities
Some estimates from étale cohomology.
Some examples of 5 and 7 descent for elliptic curves over
We perform descent calculations for the families of elliptic curves over with a rational point of order or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over may become arbitrarily large.
Some families of non-congruent numbers
Some functions related to the derivatives of the L-series of an elliptic curve at s=1