21
votes

Accepted

### How can one understand the Eisenstein series E2 in terms of automorphic representation?

Excellent question indeed. The quick answer is that $E_2(z)$ is an almost holomorphic modular form of weight $2$ and level $1$, so the automorphic representation generated by it is not irreducible. ...

17
votes

Accepted

### Primer on Eisenstein series

I think few people would argue claims that Langlands' SLN 544 was written "in a different time" (e.g., pre-rigidity-theorems), and never really edited carefully (although the retyping is a relief), ...

14
votes

Accepted

### Properties of coefficients in expansion of $E_6/E_4$ and $E_8/E_6$

The congruence is true for all $m,n$, and Manyama's calculations for
$n \leq 500$ are more than enough to prove it. Similar congruences
hold for the coefficients of other quotients of modular forms; ...

11
votes

Accepted

### History of spectral methods to the study of real analytic $GL_2$-Eisenstein series

R. Rankin's 1939 paper giving a non-trivial estimate on Ramanujan's $\tau$ function used the "real-analytic Eisenstein series" for $SL_2(\mathbb Z)$, at least. Selberg's related paper just-slightly ...

10
votes

Accepted

### Atkin-Lehner theory for nonholomorphic Eisenstein series

I don't believe such a theory exists anywhere in the literature. To the best of my knowledge, there are a couple of results that deal closely with what you are asking.
There exists a theory of ...

10
votes

Accepted

### How much can an Eisenstein series be truncated?

I think the answer truly does depend on whether you use the naive truncation, as in Iwaniec's book, or the Arthur truncation, as in Paul Garrett's note. In particular, the method of proof for the Maaß-...

10
votes

Accepted

### Algebraic independence of $P,Q,R$ or $E_2,E_4,E_6$ over $\mathbb C(z)$

I think the follwing is the answer of this question. It is not depend on the above comments because I didn't understand them. This is just my approach.
(But not fully my idea because it depends on a ...

9
votes

Accepted

### Eisenstein series for quadratic number fields

When $k$ is a real quadratic field (or more generally a totally real number field) the short answer is Hilbert modular forms. The corresponding Eisenstein series are called Hecke-Eisenstein series and ...

9
votes

### Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?

All these identities can indeed be proved essentially trivially using modular forms and quasi-modular forms (those involving $E_2$), and the fact that the dimension
of such spaces is $1$ for weight 4,...

8
votes

Accepted

### How to compute Coefficients in Chudnovsky's Formula?

Let $\tau$ be any CM point. By basic theorems of complex multiplication, if you choose a suitable period
$\omega(\tau)$, $E_4(\tau)/\omega(\tau)^4$, $E_6(\tau)/\omega(\tau)^6$,
and $\sqrt{D}E_2^*(\tau)...

8
votes

Accepted

### Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?

Numerical experiments suggest that
$$A_2(n) := \sum_{k=1}^{n-1} k^2\sigma(k)\sigma(n-k) = \frac{n^2}{8}\sigma_3(n) - \frac{4n^3-n^2}{24}\sigma(n).$$
PS. In fact, it directly follows from the quoted ...

7
votes

### An explicit formula for a cuspidal form of weight $2$ and arbitrarily large prime level

You ask for the explicit Fourier expansion of a weight $2$ cusp form of level $p$ for $p$ arbitrarily large. This suggests you're OK with only certain primes $p$. If $p \equiv 11 \pmod{12}$ is prime, ...

7
votes

Accepted

### History of points of view on Eisenstein series

Q: What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?
A: Yes, he did, see Elliptic Functions According to
Eisenstein and Kronecker.
The reference is ...

6
votes

Accepted

### Special values of real analytic Eisenstein series

Combining paul garret's comments (see also his wonderful notes, Standard compact periods for Eisenstein series) with the class number formula, the functional equation of the Dedekind zeta function and ...

6
votes

Accepted

### Analytic properties of Eisenstein series

The main properties of these Eisenstein series (meromorphic continuation, functional equation, poles and residues) are discussed and derived in Chapter 6 of Iwaniec: Spectral methods of automorphic ...

6
votes

### Critical values of L-functions and weights of Eisenstein Series

(I disclaim expertise on p-adic L-function and p-adic automorphic forms things, though I did make some earlier contributions to rationality statements and local archimedean computations that helped p-...

6
votes

### How can one understand the Eisenstein series E2 in terms of automorphic representation?

(I took @LSpice's inquiry as encouragement to add some remarks to @GH from MO's good answer... But/and one of the issues that helped me overcome my skepticism about the utility of representation ...

6
votes

Accepted

### Why does this quasi-modular function have integral values?

Algebraicity is proven in appendix one of [1]. The function considered there is
$$ \psi(\tau) = \frac{3E_4(\tau)}{2E_6(\tau)} (E_2(\tau) - \frac{3}{\pi \rm{Im} \tau}) = \frac{3}{2} s_2(\tau).$$
The ...

6
votes

### An explicit formula for a cuspidal form of weight $2$ and arbitrarily large prime level

There are various ways of constructing explicit cuspforms, but unfortunately none of them really qualify as "explicit formulae". This is probably how it should be: cusp forms are deep objects and you ...

5
votes

### An easier reference than "On the Functional Equations Satisfied by Eisenstein Series"?

Professor Paul Garrett's book
http://www-users.math.umn.edu/~garrett/m/v/current_version.pdf
treats many cases treated in Langlands' book although the comparison between them will not be easy. Also,...

5
votes

### Simplest case of Langlands-Shahidi method

I think that the best introduction to the Langlands-Shahidi method still is the following book by Shahidi and Gelbart:
Gelbart & Shahidi, "Analytic Properties of Automorphic L-Functions" (1988)
...

5
votes

### An explicit formula for a cuspidal form of weight $2$ and arbitrarily large prime level

Borisov and Gunnells have defined and studied the so-called toric modular forms. The idea is to take products of Eisenstein series of lower weight. So in your case, (linear combinations of) products ...

5
votes

Accepted

### References for the construction of Beilinson's motivic Eisenstein classes

The Eisenstein classes $\mathrm{Eis}^k_\phi$ live in the motivic cohomology $H^{k+1}_{\mathcal{M}}(E^k, \mathbf{Q}(k+1))$, where $E \to Y(N)$ is the universal elliptic curve over the open modular ...

4
votes

Accepted

### Eisenstein series of weight $2$ for $\Gamma_0(N)$ : where am I wrong?

The first part of Theorem 4.6.2 in the book says that $\left\{E_2^{\psi,\varphi,t}:(\psi,\varphi,t)\in A_{N,2}\right\}$ represents a basis of the Eisenstein space of weight $2$ with respect to $\...

4
votes

Accepted

### Intertwining Operators Associated to Simple Reflections

W. Casselman's 1980 paper in Comp. Math. ‘The unramified principal series of p-adic groups, I: the spherical function’, does this.

4
votes

### Fourier expansion of Eisenstein Series

The relationship has actually motivated several studies:
Eisenstein series and the Riemann zeta function (1981)
Moments of the Riemann zeta function and Eisenstein series part I and part II (2004)
...

4
votes

### Analogues of Hecke relations for Maass forms

Two remarks supplementing the excellent comments below your post.
The relation $(\star)$ holds for every $m$ and $n$ if your restrict the sum to $(d,q)=1$. This is because $f$ is a newform (and we ...

4
votes

Accepted

### Analytic continuation of the Eisenstein series defined over Hecke and Fricke subgroups

One natural generalization of $E_s$ to $\Gamma_0(N) \backslash \mathbb{H}$ is the family of Eisenstein series, $$E_{s; \chi_1, \chi_2}(\tau) = \sum_{\gamma \in \Gamma_{\infty} \backslash \mathrm{SL}...

3
votes

### Bound on an expression involving J-function coefficients

Let us define $j=J+744=\frac{1728E_4^3}{E_4^3-E_6^2}$. Then, as noted by OP, the question is equivalent to positivity of coefficients of
$$
f=q\frac{d}{dq}J+E_2(J+24)=q\frac{d}{dq}j+E_2j-720E_2.
$$
...

3
votes

### Quantum ergodicity of Eisenstein series on arithmetic quotients of hyperbolic space

If $K$ is a real quadratic field of degree $n$ Truelsen (see https://arxiv.org/abs/0706.4239) showed QUE for for Eisenstein series on the arithmetic quotient $\text{PSL}_2(O_K)\backslash (H^2)^n$. I ...

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