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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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Why aren't holomorphic modular forms bounded?

Let $f$ be any non-zero integral weight (holomorphic) modular form with respect to $SL_2(\mathbb{Z})$ and of weight $k, k\geq 4$. Since it is holomorphic at infinity, for given $\epsilon > 0$, it ...
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Cusps of congruence subgroup

Let $\Gamma$ be a congruence subgroup and let $F$ be its fundamental domain. My first question is: Do the cusps of $\Gamma$ can be defined as elements of $\partial F\cap\mathbb{P^1(\mathbb{Q})}$? ($\...
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63 views

Test vector for local zeta integral with ramified character

Suppose $\pi$ is an unramified principal series representation of ${\rm GL}_2(F)$, where $F$ is a non-archimedean local field with integers $\mathfrak{o}$. Let $W$ be a function in its Whittaker model....
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51 views

Computation of $\Phi(T,[\gamma])$ in a paper of Poonen and Rodriguez-Villegas

I am reading the paper "Lattice Polygons and the Number 12" by Bjorn Poonen and Fernando Rodriguez-Villegas (a copy of this paper can be found, e.g., on the first author's webpage). In it, the authors ...
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Linear relation between theta functions $\theta(z;u) \in S_8(\Gamma_0(4))$?

Using spherical Harmonics, I can modular forms on $\Gamma_0(4)$ as follows. Let $u(\vec{m}) \in S^2$ be a spherical harmonic: $$ \theta(z; u) = \sum_{m \in \mathbb{Z}^4} u(m) e(|m|z) = \sum_{(a,b,c,...
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For a cusp modular form of weight $k$, what is remarkable about $y^{k/2}f$ staying bounded as $y\rightarrow \infty$?

A holomorphic modular form is said to be a cusp form if its constant Fourier coefficients at the cusps are zero. In the case of a modular form of weight $k$ for the full modular group $\mathrm{SL}_2(\...
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35 views

Cusp form becomes a bounded function on $\operatorname{SL}_2(\mathbb R)$

I'm reading Gelbart's notes on the decomposition of $L^2(\operatorname{SL}_2(\mathbb Z) \backslash \operatorname{SL}_2(\mathbb R))$, and am stuck on a small detail about reinterpreting modular forms ...
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44 views

Theta function squared is a weight $1$ modular form

Let $$\vartheta(\tau) = \sum_{n\in\mathbb{Z}}e^{\pi in^2\tau}.$$ I know that $\vartheta$ satisfies the transfromation properties $$\vartheta(\tau + 2) = \vartheta(\tau), \quad \vartheta\left(-\frac{1}{...
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35 views

If $f$ is a modular form and $y^{k/2}f$ stays bounded as $y\rightarrow \infty$, its constant Fourier coefficients at the cusps are zero

Let $f(\tau)$ be a modular form of weight $k$, where $\tau = x + iy \in \mathbb H$. Then $f$ is a cusp form if its constant Fourier coefficients at the cusps are zero. By modularity, it suffices to ...
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Generating function for sum of two squares is a modular form

Let $$\theta(\tau) = \sum_{n = -\infty}^{\infty}q^{n^2}, \quad q:=e^{2\pi i\tau}$$ be the classical theta series. The Fourier coefficients $r_2(n)$ of $\theta(\tau)^2$ give the number of ways of ...
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16 views

Consider more cusps in modular form?

A holomorphic function on $\mathbb{H}$ is called a modular form of weight $k$ for SL$_2(\mathbb{Z)}$ if it satisfies $f(Mz)= (cz+d)^kf(z) \ \ \ \ \ \ \ \forall M \in$ SL$_2(\mathbb{Z)}$ $f$ is ...
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44 views

Showing existence of modular function

I am studying Mock Modular forms in the book Harmonic Maass Forms and Mock Modular Forms: Theory and Applications and i have problems with a theorem. In Theorem 6.4 it says If $Z_E^+ $ has poles in $...
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69 views

Eisenstein and Weierstrass zeta - series identity

Let $\zeta_\Lambda$ be the Weierstrass $\zeta$-function for lattice $\Lambda$ and $G_2$ the Eisenstein series of weight $2$. The quasiperiod is defined by $\eta_\Lambda(\lambda) := \zeta_\Lambda(z + \...
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Order of modular form $f$ at $p$ is equal to order of $f$ at $g(p)$

Let $f\ $ be a modular form of weight $k$ and $G=\operatorname{SL}_2(\mathbb{Z})/\{\pm 1\}$ be the modular group. In Lattices and Codes from Ebeling, they say that the identity $$f(\tau)=(c\tau +d)^...
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34 views

Bibliography request for differential equation involving Dedekind Eta Function

I would like to find the reference or bibliographic source of the fact that the following differential equation: $$36(y')^2-24y''y+y''' = 0$$ is satisfied by $y(z) = \frac{\eta'(z)}{\eta(z)}$ ($\eta(...
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Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms

Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients. (1) Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know ...
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Factoring out Kloosterman Sum

I'm reading Iwaniec's book and he says Kloosterman sum factors into $S(n,n;c)=S(n\bar{q},n\bar{q};r)T(n\bar{r},n\bar{r};q)$ where $n$ is square free and $c=rq$ such that $(q,n)=(q,r)=1$(i.e. $q$ is ...
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19 views

Another algebraic connection from a Ramanujan theorem.

I think I have interpreted a Ramanujan theorem well, but I ask someone for help to confirm it. In the Notebook II of Ramanujan, we read: Berndt (Vol. IV p. 24 Entry 13), algebraically demonstrates ...
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Hecke operator on half integral weight modular form

(Sorry for my poor english skill..) Let $N$ be an odd integer and $k$ be a positive integer. Let $\chi$ be a Dirichlet character modulo $4N$ and $f=\sum_{n=1}^{\infty} a(n)q^n \in S_{k+\frac{1}{2}}(\...
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1answer
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A clarification of Ramanujan's theorem.

I would like to know if my interpretation is right. Who can give me comfort? In Notebook IV, Berndt reports Entry 11 (p. 328) and the proof that from, it's purely algebraic. However, I am sure ...
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28 views

Do all hauptmoduln have integral coefficients in their Fourier expansion

Define Klein's modular $j$-invariant as $$j(\tau) = 1728 \frac{g_2^3}{\Delta},$$ where $$ g_2 = 60 \sum_{(m,n)}' \Big( \frac{1}{m + n \tau} \Big)^4$$ and $$ g_3 = 60 \sum_{(m,n)}' \Big( \frac{1}{m + ...
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34 views

Hypergeometric functions and modular forms

May I please ask if it is possible to write Hypergeometric functions in terms of Jacobi theta functions? I am trying to bring the following Hypergeometric expression (pg.9, eq 4.3) into a known ...
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1answer
35 views

Generalized Jacobi theta functions - Laurent series expansion of H(w,q,S).

Can someone please assist me with the missing steps in the proof of 'proposition 3' in M. Kaneko and D. Zagier paper (https://people.mpim-bonn.mpg.de/zagier/files/progmath/129/165/fulltext.pdf, pg. 4)....
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1answer
43 views

Subgroup of $\text{PSL}(2,\mathbb{Z})$ generated by $S$ and $T^2$

In the group $PSL(2,\mathbb{Z})$ (which acts on the upper half plane of $\mathbb{C}$), suppose $S$ is the inversion and $T$ is the translation by $1$, i.e. $$S= \left( {\begin{array}{cc} 0 & ...
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24 views

Find weight 2 newforms with given coefficients using websites or softwares, if exists

Given a finite set $P$ that consists of "small" prime numbers $$P=\{p_1, ..., p_N\}.$$ Further if given integers $a_{p_i}$ for each $p_i \in P$, then are there websites or software that could give us ...
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101 views

How to prove the integrality of this Eisenstein Series?

Cohen and Strömberg included in their book Modular Forms: A Classical Approach the chapter "A Brief Introduction to Complex Multiplication" (pp. 199-203). In this chapter (p. 202) we find Proposition ...
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43 views

How to know the j-invariant of the modular elliptic curve from the modular form?

How do people compute the $j$-invariant of an elliptic curve $E$ over $\mathbb Q$ from the associated modular form $f=\sum_{n=1}^{\infty} a_nq^n$? In other words, how to compute (at least giving some ...
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Does Weil's converse theorem holds for weight 1?

I'm reading Iwaneic's "Topics in classical automorphic forms". Now, I'm reading the proof the theorem that for any Hecke character $\xi$ of a quadratic field $K/\mathbb{Q}$, there exists a $\mathrm{GL}...
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35 views

Fundamental domains of modular groups $\Gamma_0(N)$

For the modular group $\Gamma_0(N)$, where $N\in \mathbb{Z}_+$, there exists a fundamental domain $D_N$ which lies in the strip $-\frac{1}{2} < z < \frac{1}{2}$ of the upper half plane, since ...
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49 views

$p$-th coefficient of weight $2$ new form with $p | N$ must be $1$ or $0$ or $-1$?

Let $f= \sum_{n=1}^{\infty}a_nq^n \in S_2^{new}(N)$ be a normalized new form of weight $2$ with respect to $\Gamma_0(N)$ and assume $p|N$ is a prime. Then must $a_p=0$ if $p^2|N$ and belongs to $\{-1,...
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75 views

Do general discrete subgroups of $\operatorname{SL}_2(\mathbb R)$ have fundamental domains in the upper half plane?

Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb Z)$. The quotient $\Gamma \backslash \mathbb H$ has the structure of a one dimensional complex manifold, such that the quotient ...
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25 views

Show $-\arg{j_{AB}(z)}+\arg{j_{A}(Bz)}+\arg{j_{B}(z)}$ does not depend on $z$.

I'm trying to show $-\arg{j_{AB}(z)}+\arg{j_{A}(Bz)}+\arg{j_{B}(z)}$ does not depend on $z\in\mathbb{H}$ where $A,B\in SL_2{(\mathbb{R})}$ and $j_A(z)=cz+d$, $A= \begin{bmatrix} a & b \\ ...
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How to interpret action of $SL_2(\mathcal{O}_d)$

Given a lattice $\wedge = \{\omega_1, \omega_2 \}$ in $\mathbb{C}$, $\omega_1 / \omega_2 \not\in \mathbb{R}$, we know that $\wedge' = \{\omega_1', \omega_2' \}$ defines the same lattice precisely when ...
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58 views

Functional equation for $GL(3)\times GL(2)\times GL(1)$ L-functions

For two Maass forms $$f(z)=\sum_{n\neq 0}a(n)\sqrt{2\pi y}K_{v_1-\frac{1}{2}}(2\pi|n|y)e^{2\pi inx}$$ and$$g(z)=\sum_{\gamma\in U_2(\mathbb{Z})\backslash SL(2,\mathbb{Z})} \,\,\,\,\,\sum_{m=1}^{\...
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37 views

Definition of the weight $k$ hyperbolic Laplacian

I saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $\mathbb ...
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61 views

About Sturm's bound

The next theorem is known as Sturm's bound. Theorem:Let $\mathfrak{m}$ be a prime ideal in the ring of integers $\mathcal{O}$ of a number field $K$, and let $\Gamma$ be a congruence subgroup of of ...
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1answer
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On complex powers of complex numbers

For $z,s \in \mathbb{C}$ and $z\neq 0$, set $z^s = \exp(s\,\log z)$ and $-\pi < \arg z \leq \pi$. In this setting, I am worried about the cases where I have to be careful in assuming $(z_1z_2)^s = (...
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J invariant as a solution to a cubic

In Starks proof of the class number 1 problem, on page 18 he mentions an equation derived by Weber that says: $\exists a,b,c\in Q(j(\frac{-3+\sqrt{d}}{2}))$ s.t. $j(\frac{-3+\sqrt{d}}{2})^3+aj(\frac{-...
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25 views

Coefficient of Maass cusp forms are bounded

Let $$\phi(z)=\sum_{\gamma\in U_2(\mathbb{Z})\backslash SL(2,\mathbb{Z})} \,\,\,\,\,\sum_{m_1=1}^{\infty}\,\,\sum_{m_2\neq 0}a(m,n)W_{\text{Jacquet}}\left(\begin{pmatrix} |m_1m_2| & & \\ &...
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35 views

q-expansion principle and the constant term of modular form

If I have a modular form $f=\sum_{n=0}^\infty a_nq^n$ of weight k, level N and character $\chi$. Assume all $a_n$ except $a_0$ generate a number field, must $a_0$ also lie in this number field? ...
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21 views

How is the bijection obtained?

I am reading in Serge Lang's book " Introduction to Modular Forms " . On page 8 there is written that there is a bijection between functions of lattices , homogenous of degree -k and functions g on H (...
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32 views

The Dual Maass form for $SL(3,\mathbb{Z})$

Let $\phi(z)$ be a Maass form of type $(v_1,v_2)\in \mathbb{C}^2$ for $SL(3,\mathbb{Z})$. Then the dual Maass form $$ \tilde{\phi}(z):= \phi(w.(z^{-1})^{\intercal}.w)\,,\,\,\,\,\,\,\,\,\,\,w=\begin{...
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20 views

Modular parametrization from equality of $L$-functions

In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways 1) if there exists a nonconstant morphism $X_0(N) \to E$, 2) if there exists a modular form $f$ ...
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1answer
30 views

Reformulating Theta Function Symmetry as a Modular Form

If $\theta$ is the Jacobi theta function $\theta(\tau) = \sum e^{\pi i n^2 \tau}$, then $\theta$ satisfies the Modular symmetries $\theta(\tau + 2) = \theta(\tau)$ and $\theta(-1/\tau) = \sqrt{-i \tau}...
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Maps between spaces of modular forms over different congruence subgroups

Basically I would like to know if there exists maps similar to the double coset operator (bijective or preserving generating sets for example) between spaces of modular forms over different congruence ...
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1answer
25 views

Polynomial growth of L-function

Let $f$ be a newform, $L(f,s)$ the related L-function with Ramanujan-Petersson conjunction $|\lambda(n)|\leq \sigma_0(n)$ (divisor counting function). How can I see that it grows only polynomially in ...
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79 views

Value of the J Invariant at $\frac{1+\sqrt{-163}}{2}$

For a while I've wanted to be able to show why $e^{\pi\sqrt{163}}\approx 744+640320^3$, but I have no idea how to show that $j(\frac{1+\sqrt{-163}}{2})=-640320^3$. I considered using the fact that $\...
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84 views

Show that $(f|M)(z) := (cz+d)^{-r/2}f(Mz)$ has a weight $r/2$.

The following text is from Complex Analysis by Freitag : For $r ∈ \mathbb{Z}$ the modified Petersson notation is defined : $$(f|M)(z) := \sqrt{cz+d}^{-r}f(Mz)$$ for $M ∈ SL(2, \mathbb{Z})$. In the ...
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38 views

Show that set of all cusp classes is finite.

A cusp $κ$ of a congruence subgroup $Γ \le SL_2(\mathbb{Z})$ is by definition an element of $\mathbb{Q}∪{\{i∞\}}$. In mapping $κ \to \frac{aκ+b}{cκ+d}$ if we take $κ=a/b \in \mathbb{Q}$ then $A=\begin{...
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1answer
33 views

Show that $I_r(MN, z) = w_r(M,N) I_r(M,Nz) I_r(N, z)$.

Following is from Complex Analysis by Freitag : My questions: 1- The text gives an example for $w_r(M,N)$ but it doesn't explain it or give a clear definition of it. For example where does $w_1(-S,-...