Related papers: A note on the dimension of the largest simple Heck…

For all odd primes N up to 500000, we compute the action of the Hecke operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then…

Let $G$ be a graph of minimum degree at least $k$ and let $G_p$ be the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. We are interested in the size of the largest complete minor that $G_p$ contains…

In this article, we prove an omega-result for the Hecke eigenvalues $\lambda_F(n)$ of Maass forms $F$ which are Hecke eigenforms in the space of Siegel modular forms of weight $k$, genus two for the Siegel modular group $Sp_2(\Z)$. In…

For any graph $G$ of order $n$ with degree sequence $d_{1}\geq\cdots\geq d_{n}$, we define the double Slater number $s\ell_{\times2}(G)$ as the smallest integer $t$ such that $t+d_{1}+\cdots+d_{t-e}\geq2n-p$ in which $e$ and $p$ are the…

Let $\bbk$ be an algebraically closed field of prime characteristic $p$. If $p$ does not divide $n$, irreducible modules over $\frak {sl}_n$ for regular and subregular nilpotent representations have already known(see \cite{Jan2} and…

Using the modularity technique of Wiles, we study the Hecke algebra of weight 2 and prime level N localized at the Eisenstein primes. On the way, we recover some results of Mazur ("Modular Curves and the Eisenstein Ideal") from a…

Let $K$ be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions of $n$, where $\lambda=(\lambda_1,...,\lambda_n)$ and $\mu=(\mu_1,..,\mu_n)$. By $S^{\lambda}$ we denote the Specht module corresponding to $\lambda$…

Let $\gg$ be a complex reductive Lie algebra and $\kk\subset\gg$ be any reductive in $\gg$ subalgebra. We call a $(\gg,\kk)$-module $M$ bounded if the $\kk$-multiplicities of $M$ are uniformly bounded. In this paper we initiate a general…

Let $F$ be a p-adic local field and $G=GL_2(F)$. Let $\mathcal{H}^{(1)}$ be the pro-p Iwahori-Hecke algebra of $G$ with coefficients in an algebraic closure of $\mathbb{F}_p$. We show that the supersingular irreducible…

For $2\le k\le t<s$, the Erd\H{o}s-Rogers function $f^{(k)}_{t,s}(N)$ denotes the largest $m$ such that every $K^{(k)}_s$-free $k$-graph on $N$ vertices contains a $K^{(k)}_t$-free induced subgraph on $m$ vertices. Mubayi and Suk (J. London…

A set of natural numbers is primitive if no element of the set divides another. Erd\H{o}s conjectured that if S is any primitive set, then \sum_{n\in S} 1/(n log n) \le \sum_{n\in \P} 1/(p log p), where \P denotes the set of primes. In this…

New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part alpha(24z) of a certain weak Maass form M(z) of weight 3/2. We show that a normalized…

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $\Gamma_0(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$…

In this paper we deduce some new supercongruences modulo powers of a prime $p>3$. Let $d\in\{0,1,\ldots,(p-1)/2\}$. We show that $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{2k}{k+d}}{8^k}\equiv 0\ (\mbox{mod}\ p)\ \ \ \mbox{if}\ d\equiv…

The main results of this article concern the definition of a compactly supported cohomology class for the congruence group $\Gamma_0(p^n)$ with values in the second Milnor $K$-group (modulo 2-torsion) of the ring of $p$-integers of the…

By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii)…

Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and…

We give a decomposition as a direct sum of indecomposable modules of several types of Specht modules in characteristic $2$. These include the Specht modules labelled by hooks, whose decomposability was considered by Murphy. Since the main…

For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy…

Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a non-trivial semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ can be constructed either from group algebras and their duals by…