Related papers: A rank inequality for the Tate Conjecture over glo…

We show that the algebraic rank of divisors on certain graphs is related to the realizability problem of matroids. As a consequence, we produce a series of examples in which the algebraic rank depends on the ground field. We use the theory…

We formulate for function fields an analog of Serre's conjecture on the modularity of 2-dimensional irreducible mod l representations of the absolute Galois group of Q: our analog is not restricted to 2-dimensional represntations. While the…

We propose a formulation of the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras; something which seems to have been heretofore missing because the complexes of…

We show that the Tate conjecture for divisors over a finite field $\mathbb F$ is equivalent to an explicit algebraic problem about the third Milnor K-group of the function field $\bar {\mathbb F}(x,y,z)$ in three variables over $\bar…

This paper investigates a Tate algebra version of the Jacobian conjecture, referred to as the Tate-Jacobian conjecture, for commutative rings $R$ equipped with an $I$-adic topology. We show that if the $I$-adic topology on $R$ is Hausdorff…

Let $k$ be a field and $X$ a smooth projective variety over $k$. When $k$ is a number field, the Beilinson-Bloch conjecture relates the ranks of the Chow groups of $X$ to the order of vanishing of certain $L$-functions. We consider the same…

We prove that the Tate conjecture for divisors is ''generically true'' for mod p reductions of complex projective varieties with $h^{2, 0} = 1$, under a mild assumption on moduli. By refining this general result, we establish a new case of…

We generalize Soergel's tilting algorithm to singular weights and deduce from this the validity of the Lascoux-Leclerc-Thibon conjecture on the connection between the canonical basis of the basic submodule of the Fock module and the…

The Total Rank Conjecture is a coarser version of the Buchsbaum-Eisenbud-Horrocks conjecture which, loosely stated, predicts that modules with large annihilators must also have ``large'' syzygies. This conjecture was proved by the second…

In this paper we study fundamental model-theoretic questions for free associative algebras, namely, first-order classification, decidability of the first-order theory, and definability of the set of free bases. We show that two free…

The local equivariant Tamagawa number conjecture (local ETNC) for a motive predicts a precise relationship between the local arithmetic complex and the root numbers which appear in the (conjectural) functional equations of the…

We consider various regular graphs defined on the set of elements of given rank of a finite polar space. It is likely that no two such graphs, of the same kind but defined for different ranks, can have the same degree. We shall prove this…

In [Frobenius1896] it was shown that many important properties of a finite group could be examined using formulas involving the character ratios of group elements, i.e., the trace of the element acting in a given irreducible representation,…

We give the theorem of coincidence of a class of functions defined by a generalised modulus of smoothness with a class of functions defined by the order of the best approximation by algebraic polynomials. We also prove the appropriate…

Let $T_P$ be the theory of beautiful pairs of algebraically closed fields of fixed characteristic. It is known that for real tuples in models of $T_P$, SU-rank coincides with Morley rank and can be computed effectively. Building on Pillay's…

In 1966, Tate proposed the Artin--Tate conjectures, which expresses special values of zeta function associated to surfaces over finite fields. Conditional on the Tate conjecture, Milne--Ramachandran formulated and proved similar conjectures…

In this paper, we establish two main results concerning the Mumford-Tate conjecture for hyper-K\"ahler varieties. First, we prove the conjecture for the semisimplified $\ell$-adic Galois representations attached to hyper-K\"ahler varieties…

For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field is topologically dense in the set of its points with…

Given a field with a set of discrete valuations $V$, we show how the genus of a division algebra over the field is related to the genus of the residue algebras at various valuations in $V$ and the ramification data. When the division…

Ahmadi-Shparlinski conjectured that every ordinary, geometrically simple Jacobian over a finite field has maximal angle rank. Using the L-Functions and Modular Forms Database, we provide two counterexamples to this conjecture in dimension…