Related papers: Can a Drinfeld module be modular?

In this paper we study some local and global regularity properties of Fourier series obtained as fractional integrals of modular forms. In particular we characterize the differentiability at rational points, determine their H\"older…

Let $k$ be a perfect field of characteristic $p >0$, $U$ be a variety over $k$ and $F$ be a power of Frobenius. We construct the category of overholonomic arithmetical ($F$-)$\D$-modules over $U$ and the category of overholonomic…

Let $p$ be a rational prime, let $q>1$ be a $p$-power integer, let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A$ be a nonzero element and let…

Let $p$ be a prime number and $F$ a totally real number field unramified at places above $p$. Let $\bar{r}:\operatorname{Gal}(\bar F/F)\rightarrow\operatorname{GL}_2(\bar{\mathbb{F}_p})$ be a modular Galois representation which satisfies…

We give a geometric perspective on the algebra of Drinfeld modular forms for congruence subgroups $\Gamma\leq \GL_2(\bbF_q[T]).$ In particular, we describe an isomorphism between the section ring of a line bundle on the stacky modular curve…

The standard notion of the non-Abelian duality in string theory is generalized to the class of $\si$-models admitting `non-commutative conserved charges'. Such $\si$-models can be associated with every Lie bialgebra $(\cg ,\cgt)$ and they…

We study the gerbal representations of a finite group $G$ or, equivalently, module categories over Ostrik's category $Vec_G^\alpha$ for a 3-cocycle $\alpha$. We adapt Bartlett's string diagram formalism to this situation to prove that the…

This article is divided in two parts. In the first part we endow a certain ring of ``Drinfeld quasi-modular forms'' for $\GL_2(\FF_q[T])$ (where $q$ is a power of a prime) with a system of "divided derivatives" (or hyperderivations). This…

For each positive integer $r$, we construct a nowhere-vanishing, single-cuspidal Drinfeld modular form for $\GL_r(\FF_q[\theta])$, necessarily of least possible weight, via determinants using rigid analytic trivializations of the universal…

Elliptic sheaves (which are related to Drinfeld modules) were introduced by Drinfeld and further studied by Laumon--Rapoport--Stuhler and others. They can be viewed as function field analogues of elliptic curves and hence are objects "of…

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard…

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…

The conjecture of Serre referred in the title is the one about modularity of odd Galois representations into GL(2,F) where F is a finite field of characteristic p. We present an analogous conjecture where GL(2) is replaced by GL(n). We…

We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, and relate their geometry to the weight part of Serre's conjecture for GL(2).

We construct a category of Breuil-Kisin $G_K$-modules to classify integral semi-stable Galois representations. Our theory uses Breuil-Kisin modules and Breuil-Kisin-Fargues modules with Galois actions, and can be regarded as the algebraic…

Let $\overline{\rho}: G_{\mathbf{Q}} \rightarrow {\rm GSp}_4(\mathbf{F}_3)$ be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equivalent, the Galois representation associated to the…

We define a module that is an extension of the diagonal harmonics and whose graded Frobenius characteristic is conjectured to be the symmetric function expression which appears in `the Delta conjecture' of Haglund, Remmel and Wilson…

This lecture series is based on joint work in progress with Shaul Barkan, as well as work in progress of the author. The five sections of these notes correspond to the five lectures, but more details have been added. $2$-dimensional…

We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role of upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms…

We show that an elliptic modular form with integral Fourier coefficients in a number field $K$, for which all but finitely many coefficients are divisible by a prime ideal $\frak{p}$ of $K$, is a constant modulo $\frak{p}$. A similar…