Related papers: Moduli relations between l-adic representations an…

In this work, the following conjectures are proven in the case of a Riemann surface with abelian group of symmetry: a) The $b-c$ systems on a Riemann surface $M$ are equivalent to a multivalued field theory on the complex plane if $M$ is…

Consider the moduli space of pairs (C,w) where C is a smooth compact complex curve of a given genus and w is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This…

We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian…

Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials…

Let A be a basic connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. Under a hypothesis on T, we establish a correspondence…

As a consequence of the Riemann-Roch theorem, a closed Riemann surface $S$ can be described by a non-singular complex projective algebraic curve $C$. A field of definition for $S$ is any subfield $D$ of $\mathbb{C}$ so that we may choose…

For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes $\ell$ for which the Galois action…

Under some assumptions, we compute the Picard group of moduli of stable sheaves on Abelian surfaces. Considering relative moduli spaces, it is sufficient to consider the moduli of stable sheaves on the product of elliptic curves. By using…

We study several integrable Hamiltonian systems on the moduli spaces of meromorphic functions on Riemann surfaces (the Riemann sphere, a cylinder and a torus). The action-angle variables and the separated variables (in Sklyanin's sense) are…

Let $(X,x_0)$ be any one--pointed compact connected Riemann surface of genus $g$, with $g\geq 3$. Fix two mutually coprime integers $r>1$ and $d$. Let ${\mathcal M}_X$ denote the moduli space parametrizing all logarithmic…

The Seiberg-Witten map links noncommutative gauge theories to ordinary gauge theories, and allows to express the noncommutative variables in terms of the commutative ones. Its explicit form can be found order by order in the noncommutative…

We study the moduli space of solutions to the Seiberg-Witten equations with $N$ spinors on a compact Riemann surface. These moduli spaces arise in a program to define a new enumerative invariant of 3-manifolds. They are also of independent…

We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system…

The moduli space of stable relative maps to the projective line combines features of stable maps and admissible covers. We prove all standard Gromov-Witten classes on these moduli spaces of stable relative maps have tautological…

We look into a construction of principal abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to place it in a broader framework. This is related to Weil intermediate Jacobians but it also…

For a crystalline p-adic representation of the absolute Galois group of Qp, we define a family of Coleman maps (linear maps from the Iwasawa cohomology of the representation to the Iwasawa algebra), using the theory of Wach modules. Let f =…

We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the…

Conformal field theory and its axiomatisation in terms of vertex operator algebras or chiral algebras are most commonly considered on the Riemann sphere. However, an important constraint in physics and an interesting source of mathematics…

For a non-CM elliptic curve $E$ defined over $\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $\rho_E: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to…

We construct infinitely many abelian surfaces A defined over the rational numbers such that, for a prime ell <= 7, the ell-torsion subgroup of A is not isomorphic as a Galois module to the ell-torsion subgroup of its dual. We do this by…