Questions about modular forms and related areas

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-1
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0answers
41 views

$GL(2, \mathbb{Z})$ modular form [on hold]

Recall that an ordinary modular form (of weight $k$) is a holomorphic function on the upper half plane $\mathbb{H}^+$ satisfying $$ f\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{k} \, f(\tau) . ...
1
vote
1answer
118 views

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

Let $A_{N,2}$ be the set of triples $(\psi,\varphi,t)$ such that $\psi$ and $\varphi$ are primitive Dirichlet characters modulo $u$ and $v$ with $(\psi\varphi)(-1)=1$, and $t$ is an integer such ...
1
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0answers
121 views

Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism

MOTIVATION Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
7
votes
1answer
323 views

Is this theorem on $L$-functions known?

Notations For $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written ...
5
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1answer
110 views

Bounding a Sum of Adjoint L-Function Values

Fix integers $k\geq2$ and $N>1$, and let $S(k,N)$ denote the normalized new Hecke eigenforms in $S_k(\Gamma_1(N))$. [If it makes my question easier to answer, feel free to replace this with ...
2
votes
1answer
69 views

A question about decomposing mod 2 modular forms of level p^2

Fix an odd prime $p$. Each $f \in \mathbb{Z}/2[[x]]$ can be written as $f_{+} + f_{-} + f_0$ where each exponent k of $x$ appearing in $f_{+}$ (resp. $f_{-}$, $f_0$) has Legendre symbol $(k/p)$ equal ...
10
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0answers
116 views

Nekrasov-Okounkov hook length formula

I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls ...
5
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1answer
159 views

Dimensions of a vector space akin to modular symbols

The group $\operatorname{SL}_2(\mathbb Z)$ acts on polynomials in two variables $\mathbb C[x,y]$ via $A\cdot f(x,y)\mapsto f(A^{-1}.(x,y))$ where $(x,y)$ is regarded as a column vector. There are two ...
6
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1answer
206 views

Complete L-function and FE of Rankin-Selberg on GL(2)?

Let $f$ be a Maass cusp form of $\Gamma_0(N)$ on the upper half plane with character $\chi$ mod $N$ and eigenvalue $1/4+\mu^2$. What is the complete $L$-function of the Rankin-Selberg product ...
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0answers
43 views

Fourier-Jacobi expansion for Siegel modular forms of non parallel weight

For Siegel modular forms of parallel weight there is a good theory of Fourier-Jacobi expansions (as for example in Faltings-Chai) and I suspect that there is a similar theory for the general case, but ...
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18 views

Submodular Monimization with Cardinality Constraint

I am new to submodular functions and I am reading the introductions to submodular functions and applications ( https://www.ima.umn.edu/optimization/seminar/queyranne.pdf ). In this introduction, it ...
2
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0answers
136 views

A question about the paper “The Main Conjecture for $GL(2)$” by Skinner and Urban

I am studying the paper "The Main Conjecture for $GL(2)$" by Skinner and Urban (available here). In this paper, in Section 5.5.3, the authors define a filtration $M_{\underline k}^{n,q}$ on the space ...
7
votes
2answers
360 views

Averages over integer points of the sphere

A paper of William Duke proves that integer points on the sphere are equidistributed: $$ V_n = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = n \}. $$ Up to reflections across the $x$, $y$ and $z$ ...
6
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1answer
199 views

Is Scholl construction of modular motives related to Deligne's construction of $\ell$-adic representations?

first of all, I need to declare my extreme ignorance on the topic of modular forms, so , please, does not assume that I know Deligne's construction in details. In ...
5
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0answers
241 views

The $\ell = p$ case of Ihara's lemma

Let $N \ge 1$ and let $\ell$ and $p$ be primes not dividing $N$. The classical Ihara lemma says that if $Y_1(N, \ell)$ is the modular curve attached to the subgroup $\Gamma_1(N) \cap \Gamma_0(\ell)$, ...
1
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0answers
141 views

Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$

I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is striclty greater than ...
3
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1answer
219 views

New series for $1/\pi$ based on Ramanujan's ideas

In his classic paper "Modular Equations and Approximations to $\pi$ (1914)", Ramanujan gives a standard technique to obtain a general family of series for $1/\pi$ based on series for $(2K/\pi)^{2}$ in ...
3
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1answer
169 views

Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms

I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ...
2
votes
2answers
180 views

Relations of eisenstein series with eta quotient

Theorem 1.67 On page 19 of Ken Ono's book The Web of Modularity says: Every modular form on $SL_2(\mathbb{Z})$ may be expressed as a rational function in $\eta(z)$, $\eta(2z)$ and $\eta(4z)$. The ...
11
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0answers
185 views

Power series which are $p$-adic modular forms for all $p$; a local-to-global principle?

Suppose that, for some integer $k$, a series $f(q) \in \mathbb Q \otimes \mathbb Z[[q]]$ has the property that for every prime $p$, $f(q)$ is the $q$-expansion of a $p$-adic modular form of weight $k$ ...
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0answers
158 views

Fourier expansions at the cusps of $\Gamma_0(N)$

My question may be basic but I can't find any answer. Let $N$ be a positive integer. I need to find the constant term (of the Fourier series) at each cusps of a modular form ...
6
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0answers
254 views

Twisted equivariant modular forms?

I'd like to know where I can find information about a class of objects which I think deserve to be called twisted equivariant modular forms. Let me guess a definition, indicate how it can be made more ...
0
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1answer
118 views

Fixed field of the Nebentypus of a newform for $\Gamma_1(N)$

Let $f=\sum_{n\geq 1}\in S_2(\Gamma_1(N),\varepsilon)$ be a normalized newform without CM and with Nebentypus $\varepsilon$. Let $L=\mathbb Q(a_n\colon n\in \mathbb N)$ be the number field generated ...
6
votes
1answer
166 views

Mean value of Maass forms

Let $X = SL_2(\mathbb{Z}) \backslash \mathbb{H}$ be the modular surface. Consider a basis of $L^2$-normalized Hecke-Maass cusps forms $\phi_j$ on $X$ with $-\Delta$-eigenvalue $\lambda_j$. ...
5
votes
1answer
323 views

An old conjecture of M.Newman

M.Newman raised several questions in his 1957 paper on modular forms. Definition: $H_n$ is the subclass of all zero-free weakly modular forms of weight 0 on $\Gamma_0(n)$, where $n$ is a composite ...
2
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1answer
116 views

Proving the Eichler-Shimura Isomorphism defines a global section

Let $ f \in S_k(\Gamma)$ be a weight k modular cusp form of level $\Gamma$, with modular curve $Y_{\Gamma}$. Let $V^{k-2}$ be the homogenous polynomials in X and Y of degree k-2 with complex ...
3
votes
2answers
172 views

Examples of component crossing between families of modular forms

Is there a reference that contains explicit examples of component crossing of Hida families at height one primes? The paper of Emerton, Pollack, and Weston addresses component crossing obtained ...
12
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3answers
1k views

On $e^{\pi\sqrt{4\cdot163}}$ and unusual connections

We are familiar with the expansion of the j-function, $$j(\tau) = \tfrac{1}{q}+744+ 196884{q} + 21493760{q}^2 + \dots\tag1$$ and maybe with the approximation, $$e^{\pi\sqrt{652}} = ...
6
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0answers
106 views

Logarithmic view on the classical Eichler–Shimura isomorphism

[Apologies ahead of time that this question is (1) quite heavy in laying out notations, and (2) of the can-you-spot-my-error variety.] I am trying to recast the Eichler–Shimura isomorphism as a ...
3
votes
2answers
145 views

Can the pre-image of the real points in the complex upper-half plane of a modular elliptic curve under the modular parametrization be identified?

Consider an elliptic curve $E/\Bbb Q$ and let $\Phi:\Gamma_0(N)\backslash\overline{\mathfrak{H}}\rightarrow E(\Bbb C)$ be the analytical description of its modular parametrization. We know that this ...
1
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0answers
50 views

Intersection of ordinary subspaces at different primes

Choose two distinct primes $\ell$ and $\ell'$, and embeddings $\iota_\ell : \overline{\mathbb Q} \to \overline{\mathbb Q_\ell}$, $\iota_{\ell'} : \overline{\mathbb Q} \to \overline{\mathbb ...
5
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1answer
500 views

Modular forms and “too many symmetries”

How do we interpret Barry Mazur's quote of Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence ...
10
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1answer
219 views

String Orientation and Level Structures

Atiyah, Bott and Shapiro defined orientations of real and complex K-theory that were later refined to maps of ($E_\infty$-ring) spectra $$MSpin \to KO$$ and $$MSpin^c \to KU.$$ Likewise, but more ...
28
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3answers
1k views

Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula. I. Given the fundamental unit, ...
0
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0answers
138 views

Is the Jacobi theta function invertible?

Let $\theta$ denote the Jacobi theta function: $$\theta=\sum_{k=0}^{\infty}{(-1)^kq^{k(k+1)}sin((2k+1)\frac{2\pi}{\omega_1}Re(z))},$$ and we have a complex number $t$. Suppose that we know there ...
5
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0answers
68 views

(Eichler-Shimura Isomorphism) Proving c(f) is not a boundary

I have seen a couple of questions related to the Eichler-Shimura Isomorphism, but almost all of them have to do with hodge theory (things I am unfamiliar with) and seem, to me, different/unrelated. ...
1
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0answers
116 views

Automorphism group of a modular curve and its action on the set of cusps

Let $X$ be a modular curve, that is the compact Riemann surface obtained by adding cusps to a quotient $Y=Y_\Gamma=\mathbb H/\Gamma$ of Poincaré upper half-plane $\mathbb H$ by a congruence subgroup ...
9
votes
1answer
359 views

Eichler-Shimura congruence

I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$. Two possible ways to compute $T_p$ mod $p$ seem to be: A) ...
10
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1answer
598 views

Questions about the “universal elliptic curve” over the affine $j$-line punctured at 0 and 1728

So my question refers to families of elliptic curves over the $\mathbb{A}^1_\mathbb{C}\setminus\{0,1728\}$ whose fiber above a point $j$ has $j$-invariant equal to $j$ (I understand it's not ...
4
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1answer
176 views

Voronoi formula and twists by additive characters

I was wondering if there are any references for the error term in the problem $$\sum_{n\leq x} r(n) \exp(2\pi i\frac{a}{q}n)$$ where $r(n)$ is the number of representations of $n$ as a sum of two ...
4
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1answer
58 views

Petersson-product of the cusp part of the theta series

Who can help me solving this problem: $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the thetaseries ...
2
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0answers
107 views

Hodge Bundles on Tropical Spaces

I am not sure that this question even makes sense, which I suppose is part of the questions itself. In any case, I attended a talk recently wherin there was some discussion about a "tropical ...
4
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0answers
42 views

Counting cosets of matrices of determinant > 1 under the action of a congruence subgroup

I tried asking this on math exchange, but no luck, so thought I'd try here. Let $M_2(m,\mathbb{Z}) $ be the $2\times 2$ matrices with integer entries and determinant $m$. Let $\Gamma^0(N)$ be the ...
12
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0answers
193 views

What's the dimension of the space of CM cusp forms?

I would guess that the following is very well known, but I don't know the answer and I couldn't find anything with some googling. Let $\Gamma \subset \mathrm{SL}(2,\mathbf Z)$ be a congruence ...
6
votes
3answers
400 views

Asymptotic formulas for Monster-related modular functions?

Define the following, $$j(\tau) = \Big(\tfrac{E_4(\tau)}{\eta^8(\tau)}\Big)^3 = {1 \over q} + 744 + \color{blue}{196884} q + 21493760 q^2 + 864299970 q^3 + \cdots \tag{1}$$ $$j_{2A}(\tau) ...
14
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1answer
402 views

Special fiber of $X(p)$ in characteristic $p$

Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_1(p)$ be the fine moduli space representing ...
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0answers
126 views

A curious property of Ramanujan's function $\tau(n)$

As it is well known, Ramanujan's $\tau(n)$ function can be defined through the power series expansion of the modular discriminant: $$\Delta(q)=q\prod\limits_{n=1}^\infty (1-q^n)^{24}=\sum ...
1
vote
1answer
67 views

Slope decomposition of a product of operators

I'm trying to relate the slope decomposition of a product of linear operators to the slope decompositions with regard to each of the operators in the product. First I'll give some background, for ...
6
votes
1answer
305 views

Cusps forms for $\Gamma (N)$

I know how to build a basis of the vector space of cusp forms for the congruence subgroups $\Gamma_1 (N)$ and $\Gamma_0 (N)$, but I couldn't find in the literature how to build a basis for ...
6
votes
2answers
486 views

Characterizing the real analytic Eisenstein series

Consider the classical real analytic Eisenstein series $$ E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}}, $$ where $z=x+iy$. We think of $E(z,s)$ as a ...