Tamagawa number of reductive algebraic groups
Tate duality and ramification of division algebras
Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication.
Terms in elliptic divisibility sequences divisible by their indices
Terne di quadrati consecutivi in un campo di Galois
Explicit formulae for the number of triplets of consecutive squares in a Galois field are given.
The absolute anabelian geometry of canonical curves.
The absolute Galois group of a pseudo real closed field
The algebraic groups leading to the Roth inequalities
We determine the algebraic groups which have a close relation to the Roth inequalities.
The Algebraic nature of the elementary theory of PRC fields.
The annihilation of divisor classes in abelian extensions of the rational function field
The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms
In this paper we extend the arithmetic Grothendieck-Riemann-Roch Theorem to projective morphisms between arithmetic varieties that are not necessarily smooth over the complex numbers. The main ingredient of this extension is the theory of generalized holomorphic analytic torsion classes previously developed by the authors.
The arithmetic of certain del Pezzo surfaces and K3 surfaces
We construct del Pezzo surfaces of degree violating the Hasse principle explained by the Brauer-Manin obstruction. Using these del Pezzo surfaces, we show that there are algebraic families of surfaces violating the Hasse principle explained by the Brauer-Manin obstruction. Various examples are given.
The arithmetic of curves defined by iteration
We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...
The Arithmetic of Elliptic Curves.
The arithmetic of Fermat curves.
The arithmetic of function fields
The automorphism group of the modular curve
The average rank of elliptic curves I. With Appendix "The Explicit formula for elliptic curves over function fields" by Oisín McGuinness.
The Bauer Group of a Rational Surface.
The Bloch-Kato conjecture on special values of -functions. A survey of known results
This paper contains an overview of the known cases of the Bloch-Kato conjecture. It does not attempt to overview the known cases of the Beilinson conjecture and also excludes the Birch and Swinnerton-Dyer point. The paper starts with a brief review of the formulation of the general conjecture. The final part gives a brief sketch of the proofs in the known cases.