Related papers: On function field Mordell-Lang and Manin-Mumford

The main result of the work ``The nilpotence conjecture in K-theory of toric varieties'' is extended to all coefficient fields of characteristic 0, thus covering the class of genuine toric varieties.

We prove a function field version of Chowla's conjecture on the autocorrelation of the M\"obius function in the limit of a large finite field of characteristic 2.

We prove that under suitable graded and local hypothesis, a formally unramified algebra over a field must be reduced. We detail examples, including one due to Gabber, to show that it is not possible to generalize these results further.

Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$,…

A novel reduction procedure for covariant classical field theories, reflecting the generalized symplectic reduction theory of Hamiltonian systems, is presented. The departure point of this reduction procedure consists in the choice of a…

We develop the theory of generalized bi-Hamiltonian reduction. Applying this theory to a suitable loop algebra we recover a generalized Drinfeld-Sokolov reduction. This gives a way to construct new examples of algebraic Frobenius manifolds.

A method to construct free field realizations for the form factors of diagonal factorized scattering theories is described. Form factors are constructed from linear functionals over an associative `form factor algebra', which in particular…

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of $\GL_2(\A_F)$, $F$ a totally real…

We prove that the standard conjecture of Hodge type holds for powers of abelian threefolds. Along the way, we also prove the conjecture for powers of simple abelian variety of prime dimension over finite fields, and in other related cases…

We show that Lang's hyperbolic and function version conjectures hold for surfaces $S$ of general type having a fibration of general type onto a curve $C$. The notion of multiplicity used is natural, but not classical, which leds to orbifold…

We prove that the vanishing of the functoriality morphism for the \'etale fundamental group between smooth projective varieties over an algebraically closed field of characteristic $p>0$ forces the same property for the fundamental groups…

This is mainly a small exposition on extensions of valuation rings as a filtered union of smooth algebras.

Let $K$ be an algebraically closed field of characteristic zero, $\delta$ a nonzero $\mathcal{E}$-derivation of $K[x]$. We first prove that $\operatorname{Im}\delta$ is a Mathieu-Zhao space of $K[x]$ in some cases. Then we prove that LFED…

We propose a conjectural extension to positive characteristic case of a well known Deligne's theorem on the existence of super fiber functors. We prove our conjecture in the special case of semisimple categories with finitely many…

Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$-conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in…

We show that the category of motivic spaces with transfers along finite flat morphisms, over a perfect field, satisfies all the properties we have come to expect of good categories of motives. In particular we establish the analog of…

We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the…

For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V, we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant…

We prove that if a time-periodic Tonelli Lagrangian on a closed manifold $M$ satisfies a strong version of the Differentiability Problem for Mather's $\beta$-function, then the Legendre transforms of rational homology classes are dense in…

Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue…