The eisenstein-series tag has no wiki summary.

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### 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 $\Gamma_0(N)$ ...

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### 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 ...

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### 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 ...

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### 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}
...

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### 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) / ...

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### 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 ...

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### Eisenstein series and 163?

Given $q = e^{2\pi i \tau}$ and the Eisenstein series $E_{2k}(\tau)$, i.e.,
$$E_2(\tau) = 1-24\sum_{n=1}^\infty \frac{n q^n}{1-q^n}$$
$$E_4(\tau) = 1+240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n}$$
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...