Related papers: Explicit Large Image Theorems for Modular Forms

Let $\Gamma$ be a finite group, let $\theta$ be an involution of $\Gamma$, and let $\rho$ be an irreducible complex representation of $\Gamma$. We bound $\dim \rho^{\Gamma^{\theta}}$ in terms of the smallest dimension of a faithful…

If $\rho$ denotes a finite dimensional complex representation of $\textbf{SL}_2(\textbf{Z})$, then it is known that the module $M(\rho)$ of vector valued modular forms for $\rho$ is free and of finite rank over the ring $M$ of scalar…

In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all…

For a rational prime $p \geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\geq 2$ with associated $p$-adic Galois representations $\rho_f$ and $\mod{p^n}$ reductions $\rho_{f,n}$. Under suitable hypotheses on the…

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent…

Consider the affine Lie algebra $\hat{s\ell}(n)$ with null root $\delta$, weight lattice $P$ and set of dominant weights $P^+$. Let $V(k\Lambda_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\hat{s\ell}(n)$-module…

We present several upper bounds for the height of global residues of rational forms on an affine variety. As a consequence, we deduce upper bounds for the height of the coefficients in the Bergman-Weil trace formula. We also present upper…

We classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities. For the natural…

We give a new interpretation of representation theory of the finite-dimensional half-integer weight modules over the queer Lie superalgebra $\mathfrak{q}(n)$. It is given in terms of Brundan's work of finite-dimensional integer weight…

We show some preservation results of amenably extending strongly Ulam stable groups under mild decay assumptions, including quantitative preservation of asymptotic bounds under the assumption that the modulus of stability is H\"older…

Fix a prime $p > 2$. Let $\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{I})$ be the Galois representation coming from a non-CM irreducible component $\mathbb{I}$ of Hida's $p$-ordinary Hecke algebra. Assume the…

In this paper, we generalize a result by Berman and Billig on weight modules over Lie algebras with polynomial multiplication. More precisely, we show that a highest weight module with an exp-polynomial ``highest weight'' has finite…

Basic modules of McLain groups $M=M(\Lambda,\leq, R)$ are defined and investigated. These are (possibly infinite dimensional) analogues of Andr\'e's supercharacters of $U_n(q)$. The ring $R$ need not be finite or commutative and the field…

Let G be an almost simple, simply connected algebraic group over an algebraically closed field of characteristic p>0. In this paper we restate our conjecture from 1979 on the characters of irreducible modular representations of G so that it…

Let $F$ be a non-archimedean local field of residual characteristic $p>3$ and residue degree $f>1$. We study a certain type of diagram, called \emph{cyclic diagrams}, and use them to show that the universal supersingular modules of…

In this paper it is proved that an irreducible weight module with finite-dimensional weight spaces over the Schr\"{o}dinger-Virasoro algebras is a highest/lowest weight module or a uniformly bounded module. Furthermore, indecomposable…

Let $(\mathbb{T}_f,\mathfrak{m}_f)$ denote the mod $p$ local Hecke algebra attached to a normalised Hecke eigenform $f$, which is a commutative algebra over some finite field $\mathbb{F}_q$ of characteristic $p$ and with residue field…

For a field $F$ of characteristic zero and an additive subgroup $G$ of $F$, a Lie algebra $B(G)$ of lock type is defined with basis $\{L_{a,i},c|a \in G, i>-2\}$ and relations…

We define regular Kac-Moody superalgebras and classify them using integrable modules. We give conditions for irreducible highest weight modules of regular Kac-Moody superalgebras to be integrable. This paper is a major part of the proof for…

By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In…