Related papers: Theta operators, Goss polynomials, and v-adic modu…

We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial.…

This is an extended (factor 2.5) version of arXiv:math/0601371 and arXiv:0808.3486. We present new results in the theory of the classical $\theta$-functions of Jacobi: series expansions and defining ordinary differential equations (\odes).…

This paper grew out of the author's work on arXiv:2504.18460. Differential operators in the sense of Grothendieck acting between modules over a commutative ring can be interpreted as torsion elements in the bimodule of all operators with…

Let $k$ be a global function field with field of constants $\Fr$ and let $\infty$ be a fixed place of $k$. In his habilitation thesis \cite{boc2}, Gebhard B\"ockle attaches abelian Galois representations to characteristic $p$ valued cusp…

The Drinfeld module is a tool of the explicit class field theory for the function fields. We first observe a similarity of such modules with the noncommutative tori, and then use it to develop an explicit class field theory for the number…

We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on…

Since the work of F. Rankin and Swinnerton-Dyer on the zeros of Eisenstein series, many results have been obtained concerning the zeros of modular forms. In this paper, we study the zeros of Serre derivatives of modular forms. In…

We determine the action of the Hecke operators \(T_{\mathfrak{p},i}\) on the coefficient forms \(g_{1}, \dots, g_{r-1}, g_{r} = \Delta\), and \(h\), which together generate the ring of modular forms for \(\mathrm{GL}(r,…

We apply differential operators to modular forms on orthogonal groups $\mathrm{O}(2, \ell)$ to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are…

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear…

Let $\mathbb{F}_{q}$ be a finite field, and $A:=\mathbb{F}_{q}[T]$. In this article, we give explicit criteria, involving concrete valuations, on the coefficients of the Drinfeld $A$-modules of rank $r$ for $r=2,3$, which ensure the…

We prove an Ax-Lindemann-Weierstrass differential transcendence result for Euler's gamma function, namely that the functions $\Gamma(\nu-\zeta_1(\nu)),\dots,\Gamma(\nu-\zeta_n(\nu))$ are differentially independent over the field of rational…

Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the rational function field $K:=\mathbb{F}_q(\theta)$. For a Drinfeld module $\phi$ defined over $K$, we study the transcendence of special values of the Goss…

We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the…

We derive lattice invariants from the heat flux of a lattice. Using systems of harmonic polynomials, we obtain sums of products of spherical theta functions which give new invariants of integer lattices which are modular forms. In…

In this paper, we consider degenerate Carlitz's type q-Euler polynmials and numbers and we investigate some identities arising from the fermionic p-adic integral equations and the generating function of thoe polynomials.

Let $(A,\Theta)$ be a complex principally polarized abelian variety of dimension $g\geq 4$. Based on vanishing theorems, differentiation techniques and intersection theory, we show that whenever the theta divisor $\Theta$ is irreducible,…

We introduce a Grassmannian structure for a class of quotient Hilbert modules and attack the polydisc version of Arveson-Douglas conjecture associated to distinguished varieties. More interestingly, we obtain an operator-theoretic…

We construct a differential operator on sheaves of $p$-adic modular forms defined over the locus of $p$-rank $\ge 1$ of the Siegel threefold, by applying a revisited version of the approach that Sean Howe recently introduced in his paper "A…

This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define…