Questions tagged [modular-forms]

Questions about modular forms and related areas

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Computing coefficients of theta functions associated to quadratic forms

If we take an integral positive definite quadratic form $Q$ and set $\Theta_Q(z) = \sum_{k\geq 0}R_Q(k)e^{2\pi ikz}$, what are the most efficient algorithms to compute the $R_Q(k)$? I am aware e.g. of ...
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6 votes
2 answers
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Freeman Dyson's approach to string theory [closed]

Context: In celebrating the centenary of Ramanujan's birth, Freeman Dyson presented the following career advice for talented young physicists [1]: My dream is that I will live to see the day when our ...
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1 vote
1 answer
127 views

Explicit Chebotarev density theorem for Galois representations associated to newforms

Let $f \in S_2(\Gamma_0(N))$ be a newform with associated residual Galois representation $\rho: \operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \operatorname{GL}_2(\mathbf{F})$, $\mathbf{F}$ ...
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-1 votes
1 answer
51 views

Characterization of tori/elliptic curve isogenies

I am reading Chapter 11 of Dale Husemöller's Elliptic Curves Springer book and I got stuck on Theorem (1.4) (c.f., image below). Notation and definitions: Let $L$ and $L'$ be two complex lattices ...
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5 votes
3 answers
391 views

Representation theory of $\text{SL}(2,\mathbb{Z})$

The group $\text{SL}(2,\mathbb{Z})$ is the group of two-by-two matrices with integer entries and determinant one. This is a very simple definition. Yet its representation theory seems quite wild to me ...
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2 votes
0 answers
81 views

Modular discriminant and ordinary/supersingular points

Let $p=2,3,5,7,13$. Denote by $\Delta$ the modular discriminant. How can I prove that if $z\in X_0(1)$ is a point of supersingular reduction, then $v_p(\Delta(z))=0$ ? If $z\in X_0(1)$ is a point of ...
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5 votes
2 answers
158 views

Irreducibility of the $n$th symetric power of the reduction of the Galois representation of a non-CM newform

In "On $\ell$-adic representations attached to modular forms II", Ribet proved that the $\ell$-adic representation $\rho_{f,\ell}$ attached to a non-CM newform form $f$ satisfies $${\rm SL}...
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1 vote
0 answers
85 views

Modular Formulas in $X_0(2)$

Let $f_2(z) = \frac{\Delta(2z)}{\Delta(z)}$, where $\Delta$ is the modular discriminant and $z\in X_0(2)$. How can I prove that $$\frac{(2E_2(2z) - E_2(z))^6}{\Delta(z)} = \frac{(1 + 2^6f_2(z))^3}{f_2(...
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2 votes
0 answers
154 views

Two basic questions on congruence subgroups

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I have two questions related to congruence subgroups. Let $$\Gamma=\Gamma_0(N)=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \...
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1 vote
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Whether or not the Maass form for $\Gamma _0(N)$ on $GL(3)$ covers the classical symmetric lift of a newform on $GL(2)$?

I have a blur which needs some help from the experts here, and may look naive for some experts. Recently I read Zhou's paper "The Voronoi formula on $GL{(3)}$ with ramification" (https://...
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  • 151
6 votes
1 answer
389 views

Ramanujan-Petersson conjecture at various cusps

Suppose that $f \in S_k(\Gamma_0(N)) $ be a Hecke eigenform whose Fourier expansion at $ i\infty $ is given by $$ f(z) = \sum_{n=1}^{\infty} \lambda(n) n^{\frac{k-1}{2}} \exp(2\pi i n z), $$ ...
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1 vote
0 answers
76 views

Determine whether $\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$ is positive rational (given $x$)

Denote the complete elliptic integral of the first kind by $$K(x)=\int_0^{\pi /2}\frac{d\varphi}{\sqrt{1-x^2\sin^2\varphi}}$$ and $$f(x)=\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$$ Question: Given a ...
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  • 111
1 vote
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81 views

Modular cycles?

It is well known that cocycles (differential forms) and cycles share many properties through duality (e.g., de Rham). I've been reading about modular forms recently and I came with a very naive ...
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2 votes
0 answers
456 views

Confusion regarding Proposition 1.1 in Wiles's Fermat paper

This is from p. 459 of Wiles's Fermat paper. Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of ...
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3 votes
0 answers
71 views

Hoffstein–Lockhart for non-congruence subgroups

Let $\Gamma$ be a non-congruence subgroup of $\operatorname{SL}(2,\mathbb{Z})$ of finite index and let $f$ be a holomorphic cuspidal modular form of weight $k$ for the group $\Gamma$. For simplicity, ...
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1 vote
0 answers
63 views

Definition of Hecke operators on Jacobi forms (with level and character)

How are the Hecke operators on the space of Jacobi forms $J_{k,m}(M,\chi)$ with weight $k$, index $m$, level $M$ and Dirichlet character $\chi\pmod M$ is defined? A reference will be good enough. Some ...
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  • 177
2 votes
1 answer
161 views

Existence of congruences between modular forms / elliptic curves

I'd like to ask two questions about congruences: one about modular forms and one about elliptic curves. Suppose we are given a cusp form $f$ of weight $2$ and level $\Gamma_0(N)$. Given a good ...
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  • 695
8 votes
2 answers
390 views

What is the rank of the period lattice of modular forms?

Let $f$ be a weight $2$ cusp form for the group $\Gamma_0(N)$. I was experimenting with integrals of the form $$ \int_r^s f(z) \, dz$$ where $r, s \in \mathbf{P}^1(\mathbf{Q})$ and the integral above ...
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  • 695
0 votes
1 answer
136 views

Lower bound related to derivative of $j$-invariant

Recall the $j$-invariant function, namely, $$ j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k, $$ where $q=e^{2\pi i \tau}$ and the coefficients $(c_k)_k$ are in the OEIS sequence A000521. By using some ...
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  • 355
1 vote
0 answers
47 views

A lower bound for a sum related to the $j$-invariant function

There are some days that I am thinking in the following problem. For any positive integer $x$, let $t(x)$ be a real number which a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \...
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  • 355
3 votes
3 answers
331 views

Growth of the coefficients of the inversion of the $j$-invariant function

We have the $j$-invariant defined as I have that $$ j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k, $$ where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$. The inversion ...
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  • 355
1 vote
0 answers
75 views

Relation between $L$-values of elliptic curves and Manin constants

Given an elliptic curve $E$ over $\mathbf{Q}$, we can attach two numbers two it. the so-called Manin constant $c_E$. (Defined below the fold.) the "algebraic $L$-value" given by $L(E,1)/\...
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  • 695
9 votes
2 answers
353 views

Computing the Petersson norm of newforms of weight 2 from the symmetric square $L$-function

Let $f \in S_2(\Gamma_0(N))$ be a newform with trivial character. I want to compute the Petersson norm $\lVert f\rVert^2$ of $f$, not normalized by $1/[\operatorname{SL}_2(\mathbf{Z}):\Gamma_0(N)]$, ...
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4 votes
0 answers
114 views

What is "the quotient of the universal ordinary Hecke algebra corresponding to an ordinary $\Lambda$-adic form"?

Let $\Lambda := \Bbb Z_p[[T]]$ be the usual Iwasawa algebra. In Jha and Sujatha - On the Hida deformations of fine Selmer groups on page 181, the authors refer to the quotient $\Bbb H^{\text{ord}}_{\...
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6 votes
0 answers
229 views

Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)

For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as ${\displaystyle \eta (q) =q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$ By an $\eta$-quotient ...
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  • 591
2 votes
1 answer
146 views

On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev"

I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying ...
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2 votes
0 answers
193 views

Ambiguity about the exact definition of coefficients of modular forms

You can see the parts after my questions in the boxes. I received the answer to my first question in the comments. I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
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4 votes
0 answers
81 views

Sign error in $\pm$-parts of modular symbols?

I am trying to connect the definition of $\pm$-modular symbols given in [Pollack, pg. 529] and [MTT,pg. 11] to those appearing in [Greenberg-Stevens, pg. 200 in #20 here], but I can't seem to ...
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  • 325
3 votes
0 answers
152 views

Difficulty about Jordan decomposition, (and also an ambiguity about the quadratic forms in indecomposable Jordan components of quadratic modules)

I am trying to understand a concept through solving some exercises, but I can't solve one of them, and I need a hint and guide. I asked my questions in the boxes (See the end of this question). (I ...
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24 votes
1 answer
2k views

Are there mistakes in the proof of FLT?

This semester, I teach a graduate course in epistemology of mathematics and as a case study, I assigned students a discussion on the epistemological status of Fermat's Last Theorem according to ...
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  • 9,480
5 votes
1 answer
233 views

Explicit computation of the effect of the Atkin-Lehner operator/Fricke involution's effect on $q$-expansion

As a part of the research with which I am involved, I would like to understand how to compute the effect of the Atkin-Lehner operator/Fricke involution $W_2 = \begin{pmatrix} 0 & 1 \\ -2 & 0 \...
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3 votes
0 answers
110 views

What do congruences between modular forms tell us about $\mu$-invariants of elliptic curves?

This question is based off these notes by Preston Wake about Iwasawa invariants and Hida Families. In the notes, the author asks "why" the elliptic curve $11A3$ has $\mu$-invariant equal to $...
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  • 695
2 votes
0 answers
216 views

Generalized Siegel Weil formula

I am studying the following Poincare-like series, \begin{equation} F_k(\tau,\bar{\tau})=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma}\sqrt{\text{Im}\gamma\tau}(q_{\gamma}\bar{q}_{\gamma})^k, \end{...
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1 vote
0 answers
51 views

Relationship between modular form at transformed input and original form

As part of a project I'm working on, I need to find some way to relate some $\Gamma_0(2)$ leveled modular form $f(\tau)$ to a transformed version of it at $f(-\dfrac{1}{2\tau})$. So far, I've found ...
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5 votes
1 answer
484 views

How do you compute modular symbols?

In John Cremona's book, he defines the modular symbol of an elliptic curve in the following way. Let $E/\mathbf{Q}$ be an elliptic curve and let $f_E$ be the modular form associated to $E$. The ...
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  • 695
7 votes
1 answer
300 views

Questions on the $j$-invariant

The j-invariant as a modular function is typically defined $$j(\tau) = \frac{E_4(\tau)^3}{\Delta(\tau)}$$ since $E_4$ is a modular form of weight 4 and $\Delta$ has weight 12, it follows that $j$ is a ...
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  • 811
2 votes
1 answer
141 views

Does the modular form associated to cubic twist of a elliptic curve $E$ corresponds to some twist of $f_E$?

Let $E$ be an elliptic curve defined over $\Bbb Q$ and $f_E$ be the modular form associated with the elliptic curve $E$. Suppose the elliptic curve $E^D$ is a quadratic twist of $E$. I understand that ...
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1 vote
0 answers
56 views

Polar growth property for harmonic Maass forms

The definition of a harmonic Maass form consists of three properties; (1) that it is modular, (2) that it is harmonic, and (3) that it has at most polar growth at the cusps (ordered in accordance with ...
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  • 802
4 votes
0 answers
95 views

Values at 1 of symmetric power L-functions of Maass cusp forms

I have a blur that whether one has $L(1,\text{sym}^2f)\ll \log^A q$ for some $A>0$? Here $f$ is assumed to be a Maass cusp form of square-free level $q$. If any experts here know something about ...
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  • 151
6 votes
0 answers
205 views

Why are the $p$-adic $L$-functions for a modular form with $a_p=0$ conjugates?

I have a question about the proof of Theorem 3.5 in Pollack's 2003 paper On the $p$-adic L-function of a Modular Form at a Supersingular Prime. The setup is as follows. Fix an eigenform $f\in S_k(N,\...
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  • 325
5 votes
1 answer
273 views

Coefficients of modular forms and the Sato-Tate distribution

Let $a(n)$ be the $n$th Fourier coefficient of a normalized Hecke eigenform $f(z)=\sum_{n=1}^{\infty}a(n)q^n$ of weight $k$ with respect to the full modular group, where $q=e^{i2\pi z}$. A new paper [...
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  • 51
1 vote
0 answers
117 views

Compute dimension of space of modular forms by counting Galois representations

It is known that we can compute the dimension of the space $S_{k}^{\mathrm{new}}(N, \chi)$ of new forms of weight $k\geq 2$ and level $N$ and Nebentypus $\chi$ via Riemann-Roch theorem or using ...
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  • 1,201
4 votes
0 answers
187 views

$\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the poles of $\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts ...
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  • 151
1 vote
1 answer
230 views

How to relate Rankin triple L-function to its Dirichlet series

I have a very tricky question which may look naive to many experts here. Let $f$ be a newform of level prime $P$, and $g,h$ two newforms of level 1, respectively. These three forms $f,g,h$ are all of ...
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  • 151
6 votes
0 answers
166 views

Does every modular form of higher levels generated by Eisenstein series?

For level 1 modular form, it is known that every modular form can be represented as a polynomial in weight 4 and 6 Eisenstein series. I wonder if this is true for higher levels. Interestingly, I found ...
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  • 1,201
5 votes
2 answers
222 views

A lower-bound for the square-mean of Fourier coefficients of cusp forms at primes argument

There is a basis question which puzzles me for a while. The question is the following: Let $X\ge 2,$ and $\lambda(n)$ be the $n$-th Fourier coefficient of a $GL(2)$ newform of prime level $N>1$, ...
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  • 151
8 votes
1 answer
237 views

Is there an analogue for Ramanujan–Serre derivative for Hilbert modular forms?

If $f$ is a modular form of weight $k$, it is well known that $$ D(f)=f' -\tfrac k{12}E_2f $$ is modular of weight $k+2$. Here $E_2$ is the Eisenstein series. I wanted to ask if there is an extension ...
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  • 121
1 vote
0 answers
445 views

Langlands program and complexity theory

Back when I was an undergraduate, I spent some time reading the about the modularity conjecture, but the details are fuzzy now. One of the motivations I imagined for the Langlands program was for ...
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4 votes
0 answers
129 views

How often is the rank of J_0(p)^- zero

As mentioned in this answer there is a conjecture by Kimball Martin that, formulated slightly informally, has the following special case. Conjecture: On average $J_0(p)$ has 2 simple components when ...
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  • 1,775
6 votes
0 answers
200 views

Eichler-Shimura isomorphism as simple $(k-1)$-fold integration?

I am currently trying to learn about the Eichler-Shimura isomorphism, and all the definitions seems to be somewhat cumbersome and hard to commit to memory as intuitive. On the other hand, the ...
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