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We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing "invariant versions" of iterated integrals of modular forms. The construction will be based on an extension of…

Computation of the extended gcd of two quadratic integers. The ring of integers considered is principal but could be euclidean or not euclidean ring. This method rely on principal ideal ring and reduction of binary quadratic forms.

For a prime number $p$, we consider its primorial $P:=p\#$ and $U(P):={\left(\mathbb{Z}/P\mathbb{Z}\right)}^\times$ the set of elements of the multiplicative group of integers modulo $P$ which we represent as points anticlockwise on a…

In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules.

Let $N$ and $p$ be primes such that $p$ divides the numerator of $\frac{N-1}{12}$. In this paper, we study the rank $g_p$ of the completion of the Hecke algebra acting on cuspidal modular forms of weight $2$ and level $\Gamma_0(N)$ at the…

What are all rings $R$ for which $R^*$ (the group of invertible elements of $R$ under multiplication) is an elementary abelian $p$-group? We answer this question for finite-dimensional commutative $k$-algebras, finite commutative rings,…

In this paper, based on techniques of Newton polygons, a result which allows the computation of a p integral basis of every quartic number field is given. For each prime integer p, this result allows to compute a p-integral basis of a…

We show the 3 by 3 magic square of squares problem equivalent to solving quartic polynomials with certain factorization constraints over an abelian extension of the rationals. We analyze a particular case in which said extension is assumed…

In this paper, we introduce the concept of the over-Mahonian number, which counts the overlined permutations of length $n$ with $k$ inversions, allowing the first elements associated with the inversions to be independently overlined or not.…

For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we…

In this article, we present a concise combinatorial formula for efficiently determining the Wedderburn decomposition of rational group algebra associated with a split metacyclic $p$-group $G$, where $p$ is an odd prime. We also provide a…

It is known that, for any positive non-square integer multiplier $k$, there is an infinity of multiples of triangular numbers which are triangular numbers. We analyze the congruence properties of the indices $\xi$ of triangular numbers that…

Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible…

In this paper, we investigate subnormal subgroups of the multiplicative group of an almost locally simple artinian algebra with involution. In particular, we show that if either the set of traces or the set of norms of such a subgroup with…

In this article, we present a combinatorial formula for computing the Wedderburn decomposition of the rational group algebra associated with an ordinary metacyclic $p$-group $G$, where $p$ is any prime. We also provide a formula for…

Heisenberg groups over algebras with central involution and their automorphism groups are constructed. The complex quaternion group algebra over a prime field is used as an example. Its subspaces provide finite models for each of the real…

In this note I give a formula for calculating the number of orbits of irreducible binary forms of degree $n$ over GF$(p)$ under the action of GL$(2,p)$. This formula has applications to the classification of class two groups of exponent $p$…

In this article, we give the most genaral form of the quaternions algebra depending on 3-parameters. We define 3-parameter generalized quaternions (3PGQs) and study on various properties and applications. Firstly we present the definiton,…

We continue the analysis of the Modular Isomorphism Problem for $2$-generated $p$-groups with cyclic derived subgroup, $p>2$, started in [D. Garc\'ia-Lucas, \'A. del R\'io, and M. Stanojkovski. On group invariants determined by modular…

The main purpose of this paper is to develop new algorithms for computing invariant rings in a general setting. This includes invariants of nonreductive groups but also of groups acting on algebras over certain rings. In particular, we…