Related papers: Notes on Atkin-Lehner theory for Drinfeld modular …

We determine the group structure of the normalizer of $\Gamma_0(N)$ in $SL_2(\R)$ modulo $\Gamma_0(N)$. These results correct the Atkin-Lehner statement at the paper Hecke operators of $\Gamma_0(N)$.

Let $N$ be a prime and $\phi$ be a Hecke-Maass cuspidal newform for the Hecke congruence subgroup $\Gamma_0(N)$ in $\operatorname{SL}_n(\mathbb{R})$. Let $\Omega$ be an adelic compactum and let $\Omega_N$ be its projection to $\Gamma_0(N)…

This paper contains a proof of a conjecture of Breuil and Schneider, on the existence of an invariant norm on any locally algebraic representation of $\GL(n)$, with integral central character, whose smooth part is given by a generalized…

Let $\Gamma$ be a countable abelian semigroup and $A$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(\Gamma \times A)$-graded Lie superalgebra ${\frak L}=\bigoplus_{(\alpha,a)…

We propose a conjecture refining the Stark conjecture St(K/k,S) (Tate's formulation) in the function field case. Of course St(K/k,S) in this case is a theorem due to Deligne and independently to Hayes. The novel feature of our conjecture is…

Let $\frak{g} = \frak{k} +\frak{p}$ be a complexified Cartan decomposition of a complex semisimple Lie algebra $\frak{g}$ and let $K$ be the subgroup of the adjoint group of $\frak{g}$ corresponding to $\frak{k} $. If $H$ is an irreducible…

In this paper, we consider normalized newforms $f\in S_k(\Gamma_0(N),\varepsilon_f)$ whose non-constant term Fourier coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a rational prime $p$. In this…

An irreducible module for the parafermion vertex operator algebra $K(\mathfrak{sl}_2,k)$ is said to be of $\sigma$-type if an automorphism of the fusion algebra of $K(\mathfrak{sl}_2,k)$ of order $k$ is trivial on it. For any integer $k \ge…

We develop an explicit theory of formal modular forms over arbitrary number fields $K$, as functions of modular points. We define modular points for $\Gamma_0({\mathfrak n})$ and $\Gamma_1({\mathfrak n})$, where the level ${\mathfrak n}$ is…

We prove that the span of normalized characters of subprincipal admissible modules over an affine Lie algebra of subprincipal admissible level $k$ is $SL_2(\mathbf{Z})$-invariant and find the explicit modular transformation formula.

Let $V$ be a two-dimensional vector space over a field $\mathbb F$ of characteristic not $2$ or $3$. We show there is a canonical surjection $\nu$ from the set of suitably generic commutative algebra structures on $V$ modulo the action of…

For any n>1 we define an isotopy invariant, <Gamma>_n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the…

In this paper, we study the Drinfeld cusp forms for $\Gamma_1(T)$ and $\Gamma(T)$ using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the…

We prove a compatibility theorem between the Stark conjecture and the Harris-Venkatesh conjecture for imaginary dihedral modular forms of weight $1$. The key technical input is a general two-variable $\mathrm{PGL}_2$ Siegel-Weil formula…

Let $f$ be a newform of weight $2$ on $\Gamma_0(N)$ with Fourier $q$-expansion $f(q)=q+\sum_{n\geq 2} a_n q^n$, where $\Gamma_0(N)$ denotes the group of invertible matrices with integer coefficients, upper triangular mod $N$. Let $p$ be a…

Let K be an imaginary quadratic field of discriminant -D_K<0. We introduce a notion of an adelic Maass space S_{k, -k/2}^M for automorphic forms on the quasi-split unitary group U(2,2) associated with K and prove that it is stable under the…

Let $\mathfrak g$ be a finite dimensional Lie algebra over a field $\mathbf k$, $U\mathfrak g$ be its enveloping algebra and $S\mathfrak g$ be the symmetric algebra on $\mathfrak g$. Extending the work of Braverman and Gaitsgory on the…

By the unfolding method, Rankin-Selberg L-functions for ${\rm GL}(n)\times{\rm GL}(m)$ can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic…

We define, for each subset $S$ of primes, an $S_n$-module $Lie_n^S$ with interesting properties. When $S=\emptyset,$ this is the well-known representation $Lie_n$ of $S_n$ afforded by the free Lie algebra. The most intriguing case is…

Inspirations for this paper can be traced to Urbanik (1972) where convolution semigroups of multiple decomposable distributions were introduced. In particular, the classical gamma $\mathbb{G}_t$ and $\log \mathbb{G}_t$, $t>0$ variables are…