Tamagawa numbers for motives with (non-commutative) coefficients.
Tamely ramified Hida theory
Let be the Jacobian of the modular curve associated with and the one associated with . We study as a Hecke and Galois-module. We relate a certain matrix of -adic periods to the infinitesimal deformation of the -operator.
Tate duality and ramification of division algebras
Tate-Shafarevich groups of the congruent number elliptic curves
Tensor products of drinfeld modules and v-adic representations.
Terms in elliptic divisibility sequences divisible by their indices
Tetragonal modular curves
The absolute anabelian geometry of canonical curves.
The additive dilogarithm.
The Analytic Rank of a Family of Jacobians of Fermat Curves
We study the family of curves , where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves . As a corollary we conclude that the jacobians of the curves with even analytic rank and those with odd analytic rank are equally distributed.
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 Theory of Local Constants for abelian Varieties
The average rank of elliptic curves I. With Appendix "The Explicit formula for elliptic curves over function fields" by Oisín McGuinness.
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.
The Bloch-Wigner-Ramakrishnan polylogarithm function.
The boundary of the Eisenstein symbol.
The boundary of the Eisenstein symbol (Erratum).