Related papers: Ramanujan Complexes from Unitary Groups over Numbe…

In the first part of the paper we characterize certain systems of first order nonlinear differential equations whose space of solutions is an $\mathfrak{sl}_2(\mathbb{C})$-module. We prove that such systems, called Ramanujan systems of…

We consider the structures formed by isogenies of abelian varieties with polarizations that are not necessarily principal, specifically with the $[\ell]$-polarizations we have previously defined. Our primary interest is in superspecial…

In their seminal paper, Lubotzky, Phillips and Sarnak (LPS) defined the notion of regular Ramanujan graphs and gave an explicit construction of infinite families of $(p+1)$-regular Ramanujan Cayley graphs, for infinitely many primes $p$. In…

We present a generalization of the construction of graphs by Lubotzky, Phillips and Sarnak in their celebrated article "Ramanujan graphs". The new approach consists in using octonion algebras rather than quaternions. A key tool is the…

Recently, $4$-regular partitions into distinct parts are connected with a family of overpartitions. In this paper, we provide a uniform extension of two relations due to Andrews for the two types of partitions. Such an extension is made…

We use the representation theory of the quasisplit form G of SU(3) over a p-adic field to investigate whether certain quotients of the Bruhat--Tits tree associated to this form are Ramanujan bigraphs. We show that a quotient of the tree…

Let G be a semisimple quasi-split group defined over a global function field. We express the relative Tamagawa number of G in terms of local data including the number of types of special vertices in one orbit of the Bruhat-Tits building…

We describe higher dimensional generalizations of Ramanujan's classical differential relations satisfied by the Eisenstein series $E_2$, $E_4$, $E_6$. Such "higher Ramanujan equations" are given geometrically in terms of vector fields…

Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, the first author and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in…

While examples of Ramanujan-type congruences are amply available via their relation to Hecke operators, it remains unclear which of them should be considered of combinatorial origin and which of them are mere artifacts of the connection…

In a very recent work, G. E. Andrews defined the combinatorial objects which he called {\it singular overpartitions} with the goal of presenting a general theorem for overpartitions which is analogous to theorems of Rogers--Ramanujan type…

We construct an infinite family of bounded-degree bipartite unique-neighbour expander graphs with arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed by Capalbo et al., our construction is simpler and may…

Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last three decades with many applications in computer science, combinatorics and even pure mathematics. In these notes we describe various…

We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with…

It is observed that the conjugacy growth series of the infinite finitary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed,…

We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also…

We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes…

We give a generalization of subspace codes by means of codes of modules over finite commutative chain rings. We define a new class of Sperner codes and use results from extremal combinatorics to prove the optimality of such codes in…

The method of Murty and Cioab\u{a} shows how one can use results about gaps between primes to construct families of almost-Ramanujan graphs. In this paper we give a simpler construction which avoids the search for perfect matchings and thus…

We prove several Ramanujan-type congruences modulo powers of $5$ for partition $k$-tuples with $5$-cores, for $k=2, 3, 4$. We also prove some new infinite families of congruences modulo powers of primes for $k$-tuples with $p$-cores, where…