Fourier-Mukai partners of canonical covers of bielliptic and Enriques surfaces
Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree
For each integer s ≥ 1, we present a family of curves that are -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be -Frobenius nonclassical with respect to the linear system of conics. In the -Frobenius nonclassical cases, we determine the exact number of -rational points. In the remaining cases, an upper bound for the number of -rational points will follow from Stöhr-Voloch...
From the algebras of Riemann polyzetas to the algebra of Hurwitz polyzetas. (De l'algèbre des de Riemann multivariées à l'algèbre des de Hurwitz multivariées.)
From the elliptic regulator to exotic relations.
From the Taniyama-Shimura conjecture to Fermat's last theorem
Function fields of certain arithmetic curves and application
Fundamental domains for Shimura curves
We describe a process for defining and computing a fundamental domain in the upper half plane of a Shimura curve associated with an order in a quaternion algebra . A fundamental domain for realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves . The first example is a classical example of a triangle group and the second is a corrected version of that appearing in the book of Vignéras [13], due to Michon....
Fundamental groups and Diophantine geometry
This is a brief exposition on the uses of non-commutative fundamental groups in the study of Diophantine problems.