Related papers: Abelian varieties with complex multiplication (for…

This is the final version of the 2007 preprint titled "On the derived category of 1-motives, I". It has been substantially expanded to contain a motivic proof of (two thirds of) Deligne's conjecture on 1-motives with rational coefficients,…

Deligne has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A.…

This paper gives a conjectural characterization of those elliptic curves over the field of complex numbers which "should" be covered by standard modular curves. The elliptic curves in question all have algebraic j-invariant, so they can be…

We study the eigenforms of the action of A. Baker's Hecke operators on the holomorphic elliptic homology of various topological spaces. We prove a multiplicity one theorem (i.e., one-dimensionality of the space of these "topological Hecke…

This is an addendum to our earlier paper on the defect of an ample divisor of an abelian variety. It modifies an argument of the original paper to handle one difficulty there. At the same time the modification improves the result in the…

This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of…

Let A be an abelian threefold defined over a number field K with potential multiplication by an imaginary quadratic field M. If A has signature (2,1) and the multiplication by M is defined over an at most quadratic extension, we attach to A…

The symplectic blob algebra is a physically motivated quotient of the Hecke algebra $H(\tilde{C}_n)$ with a diagram calculus. We find the blocks for the symplectic blob algebra for all specialisations of its parameters over the complex…

The aim of this paper is to show how differential characters of Abelian varieties can be used to construct differential modular forms of weight 0 and order 2 which are eigenvectors of Hecke operators. These differential modular forms have…

In combinatorics on words, a classical topic of study is the number of specific patterns appearing in infinite sequences. For instance, many works have been dedicated to studying the so-called factor complexity of infinite sequences, which…

In the paper we presnt a solution of problem 31 of chaptr 8 of the monograph of K. Denecke and S. L. Wismath: Hypridentities and clones, Gordon & Breach, 2000, by presnting two main theorems on hyperbasis for hypervarieties and…

This text is devoted to the theory of varieties, which provides an important tool, based in universal algebra, for the classification of regular languages. In the introductory section, we present a number of examples that illustrate and…

This is an expository article that concerns the various related notions of algebraic idele class characters, the Groessencharaktere of Hecke, and cohomological automorphic representations of GL(1), all under the general title of algebraic…

It is well-known that abelian varieties are projective, and so that there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational…

We prove that the existence of an automorphism of finite order on a (defined over a number field) variety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special…

Lecture notes at a conference on Arithmetic Geometry, Goettingen, July/August 2006: Density of ordinary Hecke orbits and a conjecture by Grothendieck on deformations of p-divisible groups.

Let X be a proper smooth variety over the complex numbers. We consider the generalized Albanese variety Alb(X,Y) of X of modulus Y, which is a higher dimensional analogue of the generalized Jacobian variety with modulus of Rosenlicht-Serre.…

In 2012, Zilber used model-theoretic techniques to show that a curve of high genus over an algebraically closed field is determined by its Jacobian (viewed only as an abstract group with a distinguished subset for an image of the curve). In…

We prove a variant of the reciprocity laws for CM abelian varieties, CM K3 surfaces, and CM points on Shimura varieties. Given a CM object over the complex numbers, our variation describes the set of all models over a given number field $F$…

These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this…