Questions tagged [modular-forms]

Questions about modular forms and related areas

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How does the trace map affect modular forms modulo $p$?

Suppose $M,N \geq 1$ are positive integers and consider the trace map $$\text{Tr}: M_2(\Gamma_1(MN)) \to M_2(\Gamma_1(N)).$$ Say we have a modular form $f = \sum_{n=0}^{\infty} a_nq^n \in M_2(\Gamma_1(...
Adithya Chakravarthy's user avatar
7 votes
2 answers
693 views

How to get the dimension of Atkin-Lehner eigenspace or do you have any data already obtained?

I need the dimensions of the Atkin-Lehner eigenspace for the paper I'm writing. As is well known, the cuspidal space $S_{k}(\Gamma_{0}(N))$ can be decomposed by Atkin Lehner involution. For example, ...
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Understanding Shimura correspondence in context of Langlands functoriality

Recently, I started to read about automorphic forms and representations on covering groups, e.g. metaplectic groups. I set my first goal as understanding Shimura's correspondence in representation ...
Seewoo Lee's user avatar
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Optimal sphere packings in dimensions different fom 8 and 24

After the groundbreaking work of Viazovska, now we have a proof for the optimal density of sphere packings in dimensions 8 and 24. Both packings emerge from very particular algebraic lattice ...
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Hecke operators for modular form with respect to $\Gamma_{\theta}(2)$ subgroup

The congruence subgroup $\Gamma_{\theta}(2)$ is defined as: $$\Gamma_{\theta}(2)=\left\{\gamma\in SL(2,\mathbb{Z})|\gamma\equiv\left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right) \...
liouville's user avatar
3 votes
1 answer
109 views

Non-holomorphicity of Hecke regularization of weight-2 level-1 Eisenstein series

Originally asked on stackexchange since I figured it was a very elementary and silly error, but since I haven't had any response or interest in a couple days, I thought I'd post it here. I have a &...
xir's user avatar
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3 votes
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Decomposition of real quasimodular forms of depth 1

Let $\widetilde{M}_k^{\leq \ell}$ be the space of weight $k$ depth $\leq \ell$ quasimodular forms, and $\widetilde{M}_{k,\mathbb R}^{\leq \ell}$ be a subspace of $\widetilde{M}_k^{\leq \ell}$ whose ...
LWW's user avatar
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2 votes
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Extrema of real analytic Eisenstein series and more general modular functions

The real analytic Eisenstein series defined by the Poincare sum $$E(s,z)=\sum_{(m,n)\neq (0,0)} {y^s\over |mz+n|^{2s}}$$ for $z\in{\mathbb H}$ and ${\rm Re}(s)>1$ is a manifestly $SL(2,{\mathbb Z})$...
Yifan's user avatar
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Why are compact arithmetic surfaces defined through quaternion algebras (usually) only over $\mathbb{Q}$?

As in Chapter 6.2 of "Introduction to arithmetic groups" (by D.W. Morris), compact arithmetic surfaces could be defined through quaternion algebras $\mathbb{H}^{a,b}_F=\big(\frac{a,b}{F}\big)...
Local's user avatar
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Entropy of a sequence

I am reading the paper Sign Changes in Hecke Eigenvalues by Matomaki and Radziwill, and in one place they mention the following, It would be interesting to rule out the possibility of $\lambda_f(n)$ ...
Krishnarjun's user avatar
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Ash–Stevens for Hilbert modular forms

In the theory of mod-$p$ modular forms, I learned a while ago about an interesting result that I think is technically due to Serre and Tate, though the proof was first published by Jochnowitz in ...
babu_babu's user avatar
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Symmetric Integral Matrices

Let $SM_n(R)$ be the set of $n\times n$ symmetric matrices with entries in a ring $R$ and let $A\sim B$ for such matrices if $A=C^T\cdot B\cdot C$ for some $C\in SL(n,R).$ It is an equivalence ...
Adam's user avatar
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Upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$

Does there exist an upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$? (Or more generally of $J_\Gamma$ for a congruence subgroup $\Gamma_0 \subseteq \Gamma \subseteq \...
k.j.'s user avatar
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modularity of elliptic curves over function fields in positive characteristic

Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $...
Anwesh Ray's user avatar
4 votes
1 answer
519 views

Are Frey elliptic curves semi-stable?

Are all Frey elliptic curves semi-stable? If so, where exactly is this needed in the modularity approach, now that we know modularity for all rational elliptic curves? Thank you!
did's user avatar
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Modular forms and Petersson inner product via De Rham cohomology, Hodge filtration and cup products

I'm looking for an explanation on how and why you can define modular forms through De Rham cohomology via the Hodge filtration and especially how the Petersson inner product is related to the cup ...
Lukas Heger's user avatar
4 votes
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462 views

Ramanujan's conjecture on modular forms and Riemann hypothesis

I just watched Kannan Soundararajan's talk on the distributions of valus of zeta and $L$-functions at virtual ICM 2022. In his talk, he introduced a theorem on Ramanujan's ternary form $\phi_{1}: x^{2}...
Seewoo Lee's user avatar
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Global irreducible admissible representations analogue

Let ${A}_{\mathbb{Q}}$ denote the adele over the rational numbers. Then it is known that cuspidal modular forms of level $N$ correspond to some unitary automorphic representation of $\operatorname{GL}...
Akash Yadav's user avatar
7 votes
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173 views

Rank 1 curves with prime conductor have trivial torsion. Why?

In the LMFDB database, there are 337912 elliptic curves over $\mathbb{Q}$ for which the rank is 1 and the conductor is a prime number. All of these curves have trivial torsion group. Is there a known ...
Andreas Holmstrom's user avatar
4 votes
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99 views

Reconstructing coefficients of an elliptic curve L-series from the modular form divisor

Let $E$ be an unknown elliptic curve over $\mathbb{Q}$. Let $L(E, s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ be the L-function of $E$ and write $f(q) = \sum_{n=1}^{\infty} a_n q^n$. I'm in a setting ...
Andreas Holmstrom's user avatar
1 vote
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89 views

Iterated integrals on higher dimensional Calabi-Yau manifolds?

I recently read about the construction of closed quasi-periodic differential forms on elliptic curves (1-dim Calabi-Yaus) via the Kronecker-Eisenstein series. I now wonder if similar constructions are ...
DaveWasHere's user avatar
4 votes
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255 views

Geometric interpretation of Theta functions and the Jacobi inversion problem

A great part of complex geometry and, algebraic geometry has been developed to address the theory of abelian integrals/functions. A very special problem that kept many great mathematicians busy was ...
Victor Felipe's user avatar
1 vote
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148 views

Action of $T_p$ on automorphic forms, and error in Gelbart's "Automorphic forms on adele groups"?

Let $f\in\mathcal{S}_k(\Gamma_0(N),\chi)$ be a cuspidal modular form, and $\phi_f\in\mathcal{A}_0(\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A}_\mathbb{Q}),\omega)$ be its corresponding ...
klein4's user avatar
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100 views

Kac-Peterson modular forms and shifted theta functions

Let $\Lambda$ be the root lattice corresponding to an ADE root system $R$ of rank $n$. With the ADE assumption, the weight lattice is simply the dual lattice $\Lambda^{\vee}$. Given any weight vector $...
Benighted's user avatar
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170 views

An application of Koike's Trace Formula

Koike's Trace Formula states that \begin{equation} \mbox{Tr}((U_p^{\kappa})^n) = - \sum_{0 \leq u < \sqrt{p^n}\\ (u,p)=1}H(u^2-4p^n)\frac{\gamma(u)^\kappa}{\gamma(u)^2 - p^n}-1, \end{equation} ...
Cláudio da Silva Velasque's user avatar
4 votes
0 answers
135 views

How to obtain the harmonic theta series via the global theta correspondence explicitly?

I am interested in the following kind of modular forms: let $(L,q)$ be an even unimodular lattice inside $V:=L\otimes \mathbb{Q}$, and $P$ is a harmonic homogeneous polynomial of degree $d$ on $\...
Erica's user avatar
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5 votes
2 answers
744 views

Reference for universal elliptic curves

I've seen the following sentence come up in a few papers: Consider the modular curve $Y_1(N)$ and let $E$ be the universal elliptic curve over $Y_1(N)$. This comes up in Deligne's construction of ...
Adithya Chakravarthy's user avatar
2 votes
0 answers
483 views

On quasi-modular forms with integer Fourier coefficients

It is well-known that the ring $M$ of modular forms has the structure $M=\mathbb{C}[E_4,E_6]$, where $E_k$ are the Eisenstein series. It is also known that one can define the concept of quasi-modular ...
Yuji Tachikawa's user avatar
7 votes
1 answer
373 views

Fricke involution and Atkin operator

Let $f\in S_k(\Gamma_0(N))$ be a cusp form for $N>1$. Consider the following operators acting on $f$ via the natural action of $GL_2^{+}(\mathbb{R})$ : $$ W_N=\begin{pmatrix} 0 & -1\\ N & 0 ...
Akash Yadav's user avatar
3 votes
1 answer
325 views

Overconvergent modular forms and the level at $p$

I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot. The ...
babu_babu's user avatar
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3 votes
0 answers
169 views

A question on the twisted symmetric square L-functions

Sorry to disturb. I have a puzzle which might be naive for many experts here. Let $f$ be a Hecke newform of prime level $N$ on $\mathrm{GL}_2$, and $ \chi$ a primitive character of square-free ...
Fei's user avatar
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3 votes
1 answer
277 views

Non-modular elliptic curves

Do we have any examples of non-modular elliptic curves over number fields $K \neq \mathbb{Q}$? In particular, I came across a paper by Freitas, Le Hung, and Siksek, "Elliptic curves over real ...
did's user avatar
  • 595
3 votes
2 answers
163 views

Explicit expressions for "weakly holomorphic" modular forms of weight 1

I am mainly thinking about the group $\Gamma(N)$. A weakly modular form of weight one is a holomorphic function $f: \mathfrak{H} \to \mathbb{C}$ satisfying $$ f(\gamma \tau) = (c\tau+d)f(\tau), \qquad ...
Monsieur Periné's user avatar
5 votes
0 answers
135 views

How to interpret half integral modular forms? Why not considering more general things such as third integral modular forms? [duplicate]

My knowledge of automorphic forms basically comes from Lang's $SL_2$ and Borel's automorphic forms on $SL_2$. I try to understand them from representation theory. For example, an integral weight $k$ ...
ronggang's user avatar
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2 votes
0 answers
103 views

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 ...
a196884's user avatar
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6 votes
2 answers
2k views

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 ...
Aidan Rocke's user avatar
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1 vote
1 answer
263 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 $\rho: \operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \operatorname{GL}_2(\mathbf{F})$, $\mathbf{F}$ ...
user471019's user avatar
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1 answer
76 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 ...
DaveWasHere's user avatar
6 votes
3 answers
1k 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 ...
DaveWasHere's user avatar
2 votes
0 answers
103 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 ...
Cláudio da Silva Velasque's user avatar
5 votes
2 answers
225 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}...
Fouad Fahmi's user avatar
1 vote
1 answer
235 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(...
Cláudio da Silva Velasque's user avatar
2 votes
0 answers
197 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} \...
Andrew's user avatar
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1 vote
0 answers
105 views

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://...
hofnumber's user avatar
  • 553
7 votes
1 answer
526 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 $$ f(z) = \sum_{n=1}^{\infty} \lambda(n) n^{\frac{k-1}{2}} \exp(2\pi i n z), $$ ...
Krishnarjun's user avatar
2 votes
0 answers
99 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 ...
Nomas's user avatar
  • 121
1 vote
0 answers
104 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 ...
DaveWasHere's user avatar
2 votes
0 answers
467 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 ...
The Thin Whistler's user avatar
3 votes
0 answers
83 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, ...
Krishnarjun's user avatar
1 vote
0 answers
87 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 ...
1.414212's user avatar
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