scholarly journals Overpartition $M2$-rank differences, class number relations, and vector-valued mock Eisenstein series

2019 ◽  
Vol 189 (4) ◽  
pp. 347-365
Author(s):  
Brandon Williams
Keyword(s):  
Eisenstein Series ◽  
Class Number ◽  
Vector Valued ◽  
Rank Differences

ANALYTIC CONTINUATION OF NONANALYTIC VECTOR-VALUED EISENSTEIN SERIES

2009 ◽  
Vol 05 (05) ◽  
pp. 805-830
Author(s):  
KAREN TAYLOR
Keyword(s):  
Analytic Continuation ◽  
Eisenstein Series ◽  
Complex Plane ◽  
Cusp Form ◽  
Vector Valued

In this paper, we introduce a vector-valued nonanalytic Eisenstein series appearing naturally in the Rankin–Selberg convolution of a vector-valued modular cusp form associated to a monomial representation ρ of SL(2,ℤ). This vector-valued Eisenstein series transforms under a representation χρ associated to ρ. We use a method of Selberg to obtain an analytic continuation of this vector-valued nonanalytic Eisenstein series to the whole complex plane.

scholarly journals Products of vector valued Eisenstein series

2017 ◽  
Vol 29 (1) ◽  
Author(s):  
Martin Westerholt-Raum
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Congruence Subgroups ◽  
Level 1 ◽  
Vector Valued ◽  
Level Aspect

AbstractWe prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin–Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context their previously mentioned result.

scholarly journals Vector-valued Eisenstein series of small weight

2019 ◽  
Vol 15 (02) ◽  
pp. 265-287 ◽  
Author(s):  
Brandon Williams
Keyword(s):  
Eisenstein Series ◽  
Weil Representation ◽  
Small Weight ◽  
Vector Valued

We study the (mock) Eisenstein series [Formula: see text] of weight [Formula: see text] for the Weil representation on an even lattice, defined as the result of Bruinier and Kuss’s coefficient formula for the Eisenstein series naively evaluated at [Formula: see text]. We describe the transformation law of [Formula: see text] in general. Most of this paper is dedicated to collecting examples where the coefficients of [Formula: see text] contain interesting arithmetic information. Finally, we make a few remarks about the case [Formula: see text].

ANALYTIC PROPERTIES OF EISENSTEIN SERIES AND STANDARD -FUNCTIONS

2020 ◽  
pp. 1-36
Author(s):  
OLIVER STEIN
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Analytic Continuation ◽  
Eisenstein Series ◽  
Weil Representation ◽  
Jacobi Forms ◽  
Doubling Method ◽  
Real Analytic ◽  
Forms Of Higher Degree ◽  
Vector Valued

We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of $\operatorname{Sp}(n,\mathbb{Z})$ on $\mathbb{C}[(L^{\prime }/L)^{n}]$ . By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree $n$ , a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard $L$ -function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.

scholarly journals A note on the Fourier coefficients of a Cohen–Eisenstein series

2016 ◽  
Vol 12 (05) ◽  
pp. 1149-1161
Author(s):  
Srilakshmi Krishnamoorthy
Keyword(s):  
Elliptic Curve ◽  
Prime Number ◽  
Eisenstein Series ◽  
Class Number ◽  
Fourier Coefficients ◽  
Rank Zero ◽  
Tate Group ◽  
Class Number Formula ◽  
Free Level ◽  
Number Formula

We prove a formula for the coefficients of a weight [Formula: see text] Cohen–Eisenstein series of square-free level [Formula: see text]. This formula generalizes a result of Gross, and in particular, it proves a conjecture of Quattrini. Let [Formula: see text] be an odd prime number. For any elliptic curve [Formula: see text] defined over [Formula: see text] of rank zero and square-free conductor [Formula: see text], if [Formula: see text], under certain conditions on the Shafarevich–Tate group [Formula: see text], we show that [Formula: see text] divides [Formula: see text] if and only if [Formula: see text] divides the class number [Formula: see text] of [Formula: see text]

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