The eisenstein-series tag has no wiki summary.
1
vote
0answers
124 views
about lemma 5.9 of Mazur's famous Eisenstein ideal paper
In Lemma 5.9 of Chapter II of his famous Eisenstein ideal paper, Mazur proved that
when $1/N$ is invertible in the ring $R$, if $\phi$ is a holomorphic modular form in $\omega^k$ over $\Gamma_0(N)$ ...
1
vote
1answer
137 views
Real cusp forms
Most literature on modular functions (invariant or covariant with weight k under the full modular PSL_2(Z) group) treats holomorphic functions and introduce the notion of cusp forms (modular functions ...
1
vote
0answers
94 views
Twists in “Eisenstein property” in Geometric Langlands
I am trying to read and understand (parts of) Gaitsgory's “Outline of the proof of the Geometric Langlands conjecture for GL(2)” [arXiv link]. In Section 6.4.8 he states "Property Ei", which basically ...
3
votes
0answers
214 views
On nonholomorphic Eisenstein series
Could you suggest me a reference where the following non-holomorphic generalization of the Eisenstein series is discussed?
$$ G_{k,l}(\tau,z) = \sum_{m,n} (z+m+n\tau)^{-k}(\bar z+m+n\bar \tau)^{-l}
...
1
vote
0answers
228 views
Numerical methods for Eisenstein series
Are there any existing numerical libraries for Eisenstein series? In particular I am interested in calculating values of parabolic Eisenstein series on $ SL(n,\mathbb Z) \setminus GL(n,\mathbb R) / ...
4
votes
2answers
479 views
Fourier expansion of Eisenstein series at various cusps
Here is my setting: Let $E\in\mathcal{M}_k(\Gamma_0(N))$ be an Eisenstein series (of trivial Nebentypus) that is a normalized eigenform for all the Hecke operators at level $\Gamma_0(N)$. Assume that ...
6
votes
1answer
490 views
Eisenstein series and 163?
Given $q = e^{2\pi i \tau}$ and the Eisenstein series $E_{2k}(\tau)$, i.e.,
$$E_2(\tau) = 1-24\sum_{n=1}^\infty \frac{n q^n}{1-q^n}$$
$$E_4(\tau) = 1+240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n}$$
...
2
votes
0answers
234 views
What is different about the Residual Spectrum
In the context of spectral decomposition of functions in $L^2(\Gamma \backslash \mathfrak{h})$, or Selberg trace formula, we come across three different types of spectrum.
First off there is the ...
11
votes
4answers
1k views
Where do the real analytic Eisenstein series live?
In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a ...
7
votes
2answers
957 views
Relation between Theta series and Eisensteinseries
In "Mackey - Unitary Group Representation in Physics, Probability and Number Theory" on page 326, George Mackey mentions a result of Ludwig Siegel, which was later generalized to semi-simple Lie ...
10
votes
3answers
1k views
How do modular forms of half-integral weight relate to the (quasi-modular) Eisenstein series?
The Eisenstein series
$$
G_{2k} = \sum_{(m,n) \neq (0,0)} \frac{1}{(m + n\tau)^{2k}}
$$
are modular forms (if $k>1$) of weight $2k$ and quasi-modular if $k=1$. It is clear that given modular forms ...
2
votes
1answer
555 views
Eisenstein series and the Kronecker limit theorem
It is well known that the first Kronecker limit theorem gives the Laurent expansion of the Eisenstein series $E(z,s)$ over $SL(2,Z)$ at $s=1$; see, for example, Serge Lang's book Elliptic Curves, ...
14
votes
4answers
2k views
Unitary representations of SL(2, R)
I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group G being ...
3
votes
3answers
293 views
Functions on hyperbolic space and modular curves
The decomposition of L^2(S^2) under SO(3; R) is well-known.
Focus now on the hyperbolic plane H presented as the quotient ...
