Questions tagged [modular-forms]

A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group.

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A simple question in the proof that Hecke operators commute

So I am reading these lecture notes by Don Zagier and in his proof of theorem 2.1 he first shows that the Fourier expansion of T_n f is given by \begin{align*} T_{n}f(z)=\sum_{m=0}^{\infty}\sum_{r|n,m}...
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The Norm of a Differential Form on $\Gamma\backslash\mathbb{H}$

Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$. Let $\mathbb{H}$ be the upper half-plane. Let $\phi$ be a cusp form of weight $2$ for $\Gamma$. Then $\omega=...
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Functional equation for Rankin-Selberg L functions in the imprimitive case

If $f$ and $g$ are primitive modular forms of characters $\chi$ and $\psi$, such that $\chi, \psi$ and $\chi * \psi$ are all primitive, then we have an explicit functional equation. This is proven in ...
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Modular forms of weight 0 mod p-1

I'm studying Serre's "Formes modulaires et fonctions zêta p-adiques". There is a point which is not at all clear to me. He says that the algebra $\widetilde{M}^0$ of modular forms mod $p$ of ...
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Meaning of $\Psi$ will 'go out' for points on $\mathbf R \cup \{i\infty\}$

$SL_2(\mathbf Z)$ has a fundamental domain $\Omega$ for which it “acts’’ on the upper half-plane $\mathbf H$. Suppose that $f (z)$ is a modular form for a fixed congruence group $\Gamma$. We only know ...
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Modular forms with rational and $p$-integer coefficients (mod $p$)

I'm studying $p$-adic modular forms, and in particular I was trying to understand Swinnerton-Dyer's proof on the structure of the algebra of modular forms (of level 1) modulo $p\geq 5$. In his proof, ...
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Level of Hecke eigenforms

I have been reading some notes and books on modular forms and frequently meet the sententence "Let $f$ be a weight 2, level $N$ Hecke eigenform". But some of them have not make it clear ...
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Modular forms with finitely many non-zero Fourier coefficients

Is there a congruence subgroup $\Gamma \leq \mathrm{SL}_2(\Bbb Z)$, an integer $k > 0$ and a non-zero modular form $f \in M_k(\Gamma)$ such that $f$ has only finitely many non-zero Fourier ...
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Number of modular square root of an integer mod a square-free number.

The question came from H.Iwaniec's "Topics in Classical Automorphic Forms" Lemma 4.8 Let $q$ be an odd square-free number, and let $(\frac{c}{d})$ be the extended Jacobi symbol: If $c,d>...
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Base change of $H^n(\Gamma, \mathrm{Sym}^{k}(R^2))$ - A small step in Eichler-Shimura

I'm currently learning Eichler-Shimura mapping and found the note by Gabor Wiese is quite helpful. Yet I have come up with a quite detailed question. Let $R$ be a ring (all rings in this post are ...
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On Fourier coefficients of Bianchi modular forms, l-ordinary

Let $f\in S_2(\Gamma_1(N))$ be a Hecke eigenform and $\ell$ a prime number does not divide $N$. Let $a_f(\ell)$ be the $\ell$-th Fourier coefficient of $f$. Then $a_f(\ell)$ is is called $\ell$-...
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Inclusion operator on half-integer weight modular forms and its adjoint

We have an inclusion $\iota: S_{k+1/2}(8N) \hookrightarrow S_{k+1/2}(16N)$, whose adjoint with respect to the Petersson scalar product is apparently given by $$\iota^{*} = Tr: \begin{pmatrix} 1 & ...
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Evaluate $\sum_{\ell=1}^\infty \frac{1}{\ell(e^{2\pi \ell}-1)}$

I am trying to find if there is a simple expression for the following series $$ S=\frac{1}{2}\sum_{\ell=1}^\infty\frac{\coth(\pi\ell)-1}{\ell}=\sum_{\ell=1}^\infty \frac{1}{\ell(e^{2\pi \ell}-1)}\,. $$...
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Harmonic $1-$ form on the upper half-plane $\mathbb{H}$

Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$. A function $f:\mathbb{H}\rightarrow \mathbb{C}$ is called an entire modular form for the subgroup $\Gamma$ ...
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The Space of Modular Forms and Riemann - Roch Theorem

Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$. I think that it is well-known that a function $f:\mathbb{H}\rightarrow \mathbb{C}$ is called an entire ...
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What are some specific values of Eisenstein Series?

Just as the person who made this question, I am looking for specific values of Eisenstein series on certain values. Specifically, it would be nice to have the values of $E_k(\tau)$ for $k = 2,4$ and $\...
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The vanishing order of meromorphic modular forms at special points

Let $\Gamma \subset \operatorname{SL}_2(\mathbb Z)$ be a subgroup of finite index and denote by $X_\Gamma$ the associated modular curve. Given a meromorphic modular form $f$ of weight $k$ for $\Gamma$,...
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Differential and Modular Form on a Compact Riemann Surface

Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$. Let $\mathbb{H}$ be the upper half-plane. Let $C$ be the set of all cusps of $\Gamma$. Let $R = (\mathbb{H}\...
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The local Variable at a point of a compact Riemann surface

Elliptic Modular Functions, An Introduction, B. Schoeneberg, Chapter 5, Page 120. Let $R$ be a compact Riemann surface, $p$ a point of $R$ and $t_p$ the local variable at $p$. What does the local ...
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Sum of Kloosterman sums bound by $O(n^{\epsilon})$

I've been studying the book "Topics in Classical Automorphic forms" by Henryk Iwaniec and I am stuck at a bound. For even $k$ and any $3/4 < \sigma < 1$ the book claims the following ...
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Clarification about a notation $O_{\epsilon}(c^{1/2+ \epsilon})$

I've been reading the book "Some applications of modular forms" by Peter Sarnak. On page 25, where he is trying to find a bound for the Fourier coefficients of the Poincaré series the ...
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The modular and entire modular forms for a subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$

Let $PSL(2,\mathbb{Z})$ be the modular group and $\Gamma$ be a subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$. I think that it is well-known that a function $f:\mathbb{H}\rightarrow ...
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Evaluating an $L$-function of a cusp form at half its weight

For a non-zero cusp form $f=\sum_{n=1}^{\infty}a_{n}q^{n}$ of weight $k$ for $\text{SL}_{2}(\mathbb{Z})$, its (Hecke) $L$-function is $$L(f,s):= \sum_{n=1}^{\infty}a_{n}n^{-s}.$$ Why do we have that $...
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Highly Recommended References for the Theory of Modular Forms for Normal Subgroups of Finite Index of $PSL(2, \mathbb{Z})$

I am looking for a highly recommended reference for the Theory of Modular Forms for Normal Subgroups of Finite Index of $PSL(2, \mathbb{Z})$. Many references, that have been mentioned before on this ...
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Regarding the definition of the entire modular forms

I was reading the Wiki page of the Modular Forms https://en.m.wikipedia.org/wiki/Modular_form In the definition, the function is assumed to be holomorphic at all cusps, then the entire modular form is ...
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Completing at different ideals and obtaining isomorphic completed rings

Let $$MF_0(2) := \mathbb{Z}_{(3)}[x,y,\Delta^{-1}] / (\Delta = y^2(16x^2 - 64y)),$$ Where $\Delta = y^2(16x^2-64y)$. (So our ring is localized at the $\Delta$-generated multiplicative set.) Now, ...
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Cusps of the modular forms

According to the definition of the cusps attached above the set of cusps is infinite and to be more precise it is $Q \cup \infty$ since $G$ is a subgroup of $SL(2,Z)$ so the identity matrix (that is a ...
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The Riemann Surface $G/H$

Let $G$ be a subgroup of $SL(2,Z)$ that is of finite index and $H$ be the upper half-plane. How is the quotient topological space $G/H$ defined (understood)?
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Highly recommended references on Modular Forms

I was reading this following attached below part of the asymptotic winding of the geodesic flow on modular surfaces and continuous fractions article, Y. Guivarc'h and Y. Le Jan. To be honest, I know ...
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Is it possible for $f(z)$ and $f(\sigma z)$ to both be Hecke eigenfunctions?

Let $f(z)$ be an automorphic/modular form on $\Gamma_0(p)/\mathbb{H}$, where $p$ is some prime. I know that the Hecke operators $T_n$ act on this space of functions whenever $(n,p)=1$. Assume that $f(...
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Some simple questions regarding Modular Spaces, Modular Forms, Hyperbolic Geometry, Projective Special Linear and Modular Groups

Asymptotic Winding of the Geodesic Flow on Modular Surfaces and Continuous Fractions. Y. Guivarc'h and Y. Le Jan. I have been struggling with the first section (1. Framework and notations) for a long ...
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A Hecke-Maass eigenbasis for the space of Maass cusp newforms

I heard that the space of Maass cusp newforms on $\Gamma_0(N)/\mathbb{H}$ has a basis of Hecke eigenforms. Would anyone happen to know of a reference of this fact? Or, even better, how to prove it?
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Hecke operators for automorphic forms on $SL_2(\mathbb{Z})/SL_2(\mathbb{R})$

I am curious whether there is an analogue of the Hecke operators acting on modular forms, but instead acting on the space of automorphic forms on $SL_2(\mathbb{Z})/SL_2(\mathbb{R})$. Thanks in advance,...
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Application of Poisson summation formula [duplicate]

I am currently self reading " Spectral theory of Riemann zeta function" by Yoichi Motohashi. The example is in first chapter and of poincare series. I want to know how the Poisson sum ...
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Confusion in proof for writing Poincares series in terms of Kloosterman sum

I am currently self reading " Spectral theory of Riemann zeta function" by Yoichi Motohashi. The example is in first chapter and of poincare series. I am not able to justify from 1.1.5 to ...
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Showing that the order of the zero of the $j$ function is $3$

I'm aware that the $j$-invariant can be defined as a function on the upper half-plane $\Pi^{+}$(as seen in the following wikipedia article: https://en.wikipedia.org/wiki/J-invariant): $$ j(\tau)=1728\...
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Modular form Fourier series calculations

Let $f(z)$ be a modular form of weight $k$ for $\text{SL}_{2}(\mathbb{Z})$ and set $g(z) := f(Nz)$ for some $N>1$, $g$ is a modular form of weight $k$ for $\Gamma_{0}(N)$. I am trying to evaluate $...
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Computing Eisenstein series terms

I have to compute a few terms of the normalized Eisenstein series: $\xi = e^{2 \pi i \tau}$, where $\tau$ belongs to the upper-half plane. In particular, I have to show that: $$ E_{4} (\tau) =1+240(\...
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Computing Fourier expansion of modular forms

Say one knows the Fourier expansion $\displaystyle f(\tau) = \sum_{n=0}^{\infty}a_{n}q^{n}$, $q=e^{2\pi i\tau}$, of some modular form $f$ of weight $k$ for $SL_{2}(\mathbb{Z})$. How does one compute ...
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Conformal equivalance and modular invariance

I'm currently in an introductory complex analysis class. I'm working through the following problem: Let $\Bbb{H}$ be the upper half-plane and $B$ be left half of the fundamental region: $$ B = \{\tau \...
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Regarding $\lim_{z\rightarrow 0} z \sum_{m \geq 1} (mz)^a \exp(-mz)$

The lemma (9.3.13) in the book (Cohen-Stromberg: Modular forms: A classical approach) says that the limit $$\lim_{z\rightarrow 0} z \sum_{m \geq 1} (mz)^a e^{-mz} = \int_{0}^{\infty} x^ae^{-x}dx.$$ ...
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Definition of the ring of weakly modular forms over $\mathbb{Z}_{(p)}$

I am an undergraduate in a small mathematics course focused on a single project. Right now we are in the preliminary stage, the part where our advisor has given us a high-level reference, referred us ...
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Why does the derivative of the j-Invariant have only two zeros?

In Tom Apostols Proof of Picards Theorem [Modular Functions and Dirichlet Series in Number Theory, Thm. 2.10] he uses the Propertie of J' [Derivative of the J-Invariant] being only zero at $\rho$ and ...
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What is the connection between different definitions for theta functions?

In his 2002 thesis, Zwegers defines theta functions associated with definite and indefinite quadratic forms. For example, if $Q:\mathbb{R}^r\to \mathbb{R}$ is a positive definite quadratic form with ...
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L-function of Hilbert modular forms of non-parallel weight

I found that in many literature, they only define the L-function of Hilbert modular forms of parallel weight. Can we define the following L-function: \begin{equation} L(f;s_1,\cdots,s_n)=\int_0^{i\...
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Why is the kernel of any nonzero isogeny generated by 2 elements?

So I am wondering why the kernel of any nonzero isogeny φ : C/Λ1 → C/Λ2 is generated by 2 elements. I understand why the kernel is finite, and I can think of certain lattices that this is true, but I ...
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Identification of quotient involving the Hilbert Modular Group

$K$ is a real quadratic number field with ring of integers $\mathcal O_K$. In Zagiers Modular Forms Associated to Real Quadratic Fields from 1975 at page 6 he introduces a surjective map $$m : S\to T$$...
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Is this some sort of modular form? $f(\tau)=\eta(a\tau)\eta(b\tau)$

I recently found the following. Let $a,b\in\Bbb N$ with $24|(a+b)$ and consider the function $$f(\tau)=\eta(a\tau)\eta(b\tau).$$ We have the following symmetry relations: $$\begin{align} f(\tau+1)&...
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Modified weight 2 Eisenstein series is a modular form for $\Gamma_0(N)$

I'm doing exercise 1.2.8(e) in Diamond & Shurman's A First Course in Modular Forms. The problem is to show that $G_{2,N}(\tau) := G_2(\tau)-NG_2(N\tau)$ is in $M_2(\Gamma_0(N))$. To show this, I ...
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Relation between Rogers Ramanujan continued fraction and $j$-invariant

While going through this answer I found an interesting but slightly complicated relation between Rogers-Ramanujan continued fraction and the j-invariant. I would like to know an elementary proof of ...

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