Related papers: Remarks on Eisenstein
We give a description of simple functors taking finitely generated values, from a small additive category to the category of vector spaces over a field. This result is analogous to Steinberg's tensor product theorems in group representation…
The Fourier coefficients of a negative weight eta-quotient, in many particular cases, and after Sussman in general, are known to be expressible by Hardy-Ramanujan-Rademacher type series. We show that the central terms of the coefficients of…
We introduce a coarse algebraic invariant for coarse groups and use it to differentiate various coarsifications of the group of integers. This lets us answer two questions posed by Leitner and the second author. The invariant is obtained by…
If $p\geq 5$ is prime and $k\geq 4$ is an even integer with $(p-1)\nmid k$ we consider the Eisenstein series $G_k$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo powers of $p$. It is classically known that for such $k$ we have $G_k\equiv…
In this article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight $2,4$ and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein…
In this note, we give a criteria whether given two Eisenstein polynomials over a padic field define the same extension (Proposition 1.6). In particular, we completely identify Eisenstein polynomials of degree p (Theorem 1.16). This note is…
We study power series of members of a class of positive linear operators reproducing linear function and constituting a link between genuine Bernstein-Durrmeyer and classical Bernstein operators. Using the eigenstructure of the operators we…
In this paper we present a method of characteristic sets for inversive difference polynomials and apply it to the analysis of systems of quasi-linear algebraic difference equations. We describe characteristic sets and compute difference…
We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the…
In this paper, we use counting theorems from the geometry of numbers to extend the Riemann-Roch theorem and the Riemann-Hurwitz formula to global fields of arbitrary characteristic.
We study the number of ramified prime numbers in finite Galois extensions of $\mathbb{Q}$ obtained by specializing a finite Galois extension of $\mathbb{Q}(T)$. Our main result is a central limit theorem for this number. We also give some…
In this paper we treat the Fourier coefficients of Siegel Eisenstein series of level $p$ with trivial or quadratic character, for an odd prime $p$. The Euler $p$-factor of the Fourier coefficient is called the ramified Siegel series. First…
In this paper, we introduce a class of functions that behave like classical Eisenstein series in many ways, but with a key distinction: only their non-holomorphic completions transform like (quasi)modular forms. We show how the partition…
This paper presents some refinements of the representation of the Fermat quotient of base 2 as an alternating series which was discovered by Eisenstein in 1850, including some evaluations that are believed to be new.
We offer some elementary characterisations of group and round quadratic forms. These characterisations are applied to establish new (and recover existing) characterisations of Pfister forms. We establish "going-up" results for group and…
We study sums over primes of trace functions of $\ell$-adic sheaves. Using an extension of our earlier results on algebraic twist of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of…
M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in…
Let A be a polynomial algebra with complex coefficients. Let B be a finite extension ring of A which is also a polynomial algebra. We describe the factorisation of the Jacobian J of the extension into irreducibles. We also introduce the…
In this paper, we show that incoherent Hilbert Eisenstein series for a real quadratic fields can be expressed as the Doi-Naganums lift of an incoherent Eisenstein series over $\mathbb{Q}$. As an application, we show when $N$ is odd and…
In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas-Eisenstein Series. Firstly, we show that the Pell-Lucas-Eisenstein series are convergent on their domain. Afterwards we…