Related papers: The Andr\'e-Oort Conjecture for Drinfeld Modular V…

The modular invariant of rank 1 Drinfeld modules is introduced and used to formulate and prove an exact analog of the Weber-Fueter theorem for global function fields. The main ingredient in the proof is a version of Shimura's Main Theorem…

We introduce a certain family of Drinfeld modules that we propose as analogues of the Legendre normal form elliptic curves. We exhibit explicit formulas for a certain period of such Drinfeld modules as well as formulas for the supersingular…

In the paper we complete a case by case proof of Reeder's Conjecture started in our previous work, proving the conjecture for simple Lie algebras of type $D$ and for the exceptional cases.

Let $F$ be a function field over $\mathbb{F}_q$, $A$ its ring of regular functions outside a place $\infty$ and $\mathfrak{p}$ a prime ideal of $A$. First, we develop Hida theory for Drinfeld modular forms of rank $r$ which are of slope…

In this text, we develop the theory of vectorial modular forms with values in Tate algebras introduced by the first author, in a very special case (dimension two, for a very particular representation of {\Gamma} := GL 2 (Fq[$theta$])).…

We prove the long standing conjecture in the theory of two-point boundary value problems that completeness and Dunford's spectrality imply Birkhoff regularity. In addition we establish the even order part of S.G.Krein's conjecture that…

The classical Brauer-Siegel conjecture describes the asymptotic behaviour of the product of the class number and the regulator in families of number fields. All known cases of the conjecture rely on reducing the problem, via group theoretic…

We study the $v$-adic distance from the torsion of a Drinfeld module to an affine variety.

We establish a finite-dimensional version of the Arveson-Stinespring dilation theorem for unital completely positive maps on operator systems. This result can be seen as a general principle to deduce finite-dimensional dilation theorems…

Deligne's conjecture that $\ell$-adic sheaves on normal schemes over a finite field admit $\ell'$-companions was proved by L. Lafforgue in the case of curves and by Drinfeld in the case of smooth schemes. In this paper, we extend Drinfeld's…

Let $C$ be a smooth irreducible irreducible projective curve of genus $g \ge 2$. Let $\mathcal{M}_C(n, \delta)$ be the moduli space of semi-stable vector bundles on $C$ of rank $n$ and fixed determinant $\delta$ of degree $d$. Then the…

The primary objective of this paper is to derive explicit formulas for rank one and rank two Drinfeld modules over a specific domain denoted by A. This domain corresponds to the projective line associated with an infinite place of degree…

In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as…

In this paper we prove a special case of the Lehmer inequality for Drinfeld modules. Also, based on this inequality, we prove certain Mordell-Weil type of theorems for certain infinitely generated fields.

Let $p$ be a rational prime and $q$ a power of $p$. Let $\wp$ be a monic irreducible polynomial of degree $d$ in $\mathbf{F}_q[t]$. In this paper, we define an analogue of the Hodge-Tate map which is suitable for the study of Drinfeld…

We prove the 'hybrid conjecture' which is a common generalisation of the Andre\'e-Oort conjecture and the Andr\'e-Pink-Zannier conjecture, in the case of Shimura varieties of abelian type.

In the present paper, we introduce the notion of nearly holomorphic Drinfeld modular forms and study an analogue of Maass-Shimura operators in this context. Furthermore, for a given nearly holomorphic Drinfeld modular form, we show that its…

In the seventies, V. G. Drinfeld proved that a moduli problem of deformations by quasi-isogenies of certain $p$-divisible groups with extra actions is representable by an explicit semi-stable model of the $p$-adic symmetric space. This…

The authors previously formulated the hybrid conjecture, unifying Andr\'e-Pink-Zannier and Andr\'e-Oort conjectures, and proved it in Shimura varieties of abelian type. We study its analogue for mixed Shimura varieties, and consider the…

Using class field theory we prove an explicit result of Andr\'e-Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C})$ are the singular moduli, while the…