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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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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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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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44 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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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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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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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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35 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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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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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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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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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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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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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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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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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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22 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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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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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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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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31 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 $w_1(-S,-...
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$f$ is a normalized eigenform $\Rightarrow$ so is $f^\sigma$, for $\sigma \in \operatorname{Aut}\mathbb{C}$

Let $f \in S_2(\Gamma_1(N))$ be a normalized eigenform, and $\sigma \in \operatorname{Aut}\mathbb{C}$. Then is $f^\sigma$ a normalized eigenform? This is stated in theorem6.5.4. of Diamond, Shurman's ...
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Construction of eigenfunction for Hecke operators $T_p$

I am reading Theorem 2 from Atkin-Lehner's Hecke operators on $\Gamma_0(m)$. Let $u=(u_1(\tau),u_2(\tau),..,u_n(\tau))$ be an orthonormal basis for the space of cusp forms on $\Gamma_0(m)$ with weight ...
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Modulus and argument of a quadratic expression

Please how can i find the modulus and the argument of the complex number $s^2 + s + 10?$ For modulus can i write that : $|s^2 + s + 10|=|10-\omega^2+j\omega|= \frac{\omega}{\sqrt {(10-\omega^2)^2}}$ ...
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Uniqueness of $L$-series of cusp forms

For a cusp form $f$, one gets an $L$-series by taking the Mellin transform as we have $$ \tilde{f}(s) = (2\pi)^{-s} \Gamma(s) L(s,f). $$ My question is: is this operation injective? It seems to me ...
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Location of the zeros of Dedekind Eta Function

Just a fast question, since I have not been able to find any answer for it online. Where are the zeros of Dedekind eta function $\eta(s)$ located? Apart from the trivial one as $s \to i \infty$, ...
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Show that $\Delta \ne 0$ and LI of $G_4^3$ and $G_6^2$ are equivalent.

In Complex Analysis by Freitag it is claimed that $\Delta \ne 0$ and LI of $G_4^3$ and $G_6^2$ are equivalent; that is, if $\Delta = (60G_4)^3 - 27(140G_6)^2 \ne 0$ then $G_4^3$ and $G_6^2$ are ...
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G(k,X) is a modular form of weight k and character X

I'm trying to proof the transformation property of the Eisenstein series G(k,X) defined on page 17: https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf I already ...
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72 views

Lower bound for sum of Hecke eigenvalues

Let $\lambda$ be weakly multiplicative, $\lambda(n)\geq0$, $p$ prime and $S(x)=\sum_{n\leq x}\lambda(n)\log(\frac{x}{n})$ for real $x$. How can I show $S(x)\gg \left(\sum_{p\leq \sqrt{x/3}}\lambda(p)\...
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53 views

Convexity Bound of Rankin-Selberg L-Function

Let $f,g$ be primitive modularforms of arbitrary levels $N_1,N_2$, trivial nebentypus and same weight $k$. Let $L(f\otimes g,s)=\zeta(2s)\sum_{n\geq1}\frac{\lambda_f(n)\lambda_g(n)}{n^s}$ be the ...
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Bijection between subgroups of $\text{Mat}_2(\mathbb{Z})$ and $\mathbb{Z^2}$?

The following is from the book Modular Forms by W Stein: I don't understand the whole sentence from "Note that the set ..." : For example, 1- How is the proving bijection? 2- What does $L=\mathbb{...
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3answers
36 views

How to prove that $M_{k-12} \to S_k$ is an isomorphism?

The following is from the book Modular Forms by W Stein: My questions: 1- Why multiplication by a nonzero holomorphic $\Delta$ defines an injective map? 2- How showing that "if $f \in S_k$ then $\...
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What is a form?

I have read about differential forms, bilinear forms, quadratic forms and some other r-linear forms but I still have this shred of doubt in my mind on what exactly is a form. I have an assumption that ...
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1answer
37 views

How $\Gamma(N)$ is contained in $\Gamma_1(N)$?

The following is from the book Modular Forms by W Stein: By the very same book "a congruence subgroup is a subgroup of $SL_2(\mathbb{Z})$ that contains $\Gamma(N)$ for some $N$". So $\Gamma(N)$ must ...
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Frobenius for modular curves.

Actually, I am a bit confused about the notation and what is called Frobenius for modular curves. Let $N\geq 4$ be an integer, let $R$ be an $\mathbb{F}_p$-algebra, where $p$ is a prime not dividing $...
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75 views

On the set of cusps $\mathbb{P^1(Q)}$

The following is from the book Modular Forms by W Stein: My questions: $1-$ The very same book defines a cusp form as a modular form when $f(\infty)=a_0=0$. Is the set of cusps a different ...
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1answer
49 views

Theory of finite Abelian groups on Modular Forms

The following is from Diamond and Shurman's A First Course in Modular Forms book: I have two questions (as underlined above): 1- Which and how the theory of finite Abelian groups are related to the ...
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1answer
59 views

Related to Proposition 2.1.1 in Diamond 's Modular Forms

The following is from Diamond and Shurman's A First Course in Modular Forms book: I can't understand not even a fraction of all of underlined statements: 1- How homeomorphism comes out? 2- How Prop....
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Riemann hypothesis for $L(s,\chi)$ and $L(s,\chi^\sigma)$

If $\sigma \in \text{Gal}(\mathbb{Q}(\zeta_{\infty})/\mathbb{Q})$ do we know or expect that two Dirichlet L-functions $L(s,\chi)$ and $L(s,\chi^\sigma)$ have more in common, especially in term of ...
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1answer
47 views

Elliptic Curves in Cryptography [closed]

Elliptic curve cryptography is based on finding intersections of lines and elliptic curves: $$y^2 = x^3 + ax + b ~~\text{and}~~ y = ax + b$$ It make sense when you see it on the graph, but the ...
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1answer
37 views

A holomorphic map between complex tori (proposition 1.3.2 in Diamond–Shurman)

The following is from Diamond and Shurman's A First Course in Modular Forms book: I had studied Munkres Topology a few years ago but for lifting I had to review the materials again, but I still don't ...
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1answer
36 views

Lemma 1.2.2. Diamond's Modular Forms

The following is from Diamond's Modular Forms book: I don't understand the first three lines after the lemma, at all. Also, from which part of the lemma it is a consequence? I don't see a ...
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1answer
47 views

Transformation of a theta function

Given $\Theta(\tau)=\sum_{n \in \mathbb Z}exp(2\pi in² \tau)$ and $\tau \in \mathbb H$ I am trying to prove the following identity: $\Theta(-\frac{1}{2\pi})=\sqrt{\frac{\tau}{i}}\Theta(\frac{\tau}{2}...
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A simple modular transformation of an integral

I want to perform modular $S$ and $T$ transformations for a period integral. The $S$ transformation takes $\sigma \to -1/\sigma$ and the $T$ transformation takes $\sigma \to \sigma + 1$. Here, $\sigma ...
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1answer
38 views

A Gauss's third order modular equation.

I would want a match for a typographical error (I think!). In a formula in the work of “CARL FRIEDRICH GAUSS, WERKE BAND III. GÖTTINGEN 1866” (www.archive.org). In chapter “ZUR THEORIE DER ...
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50 views

The $\heartsuit$ operator on $\mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})$

In Goldfeld's text Automorphic forms and L-functions for GL(n,R), for a fixed prime $p$ the operator $\heartsuit \colon \mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})\to\mathcal{L}^2_{cusp}(SL_2(...
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Condition of Shimura correspondence

(Sorry for my poor english.) Let $N$ be an integer and $t$ be a square-free integer. Let $\chi$ be a Dirichlet character modulo $4N$. In Shimura's paper, Shimura defined 'Shimura correspondence', \...
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25 views

Given the first 6 terms of a Fourier series expansion, is there a cusp form of weight 50 whose Fourier expansion begins that way

Is there a cusp form of weight 50 whose Fourier expansion begins with.. the first 6 terms being $q + 13q^2 + 611q^3 − 6546q^4 + 17727398783055q^5 + 5003897687242243q^6$. My guess is that I am supposed ...