# Questions tagged [modular-forms]

Questions about modular forms and related areas

1,253
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### Additional symmetries in Theta-like function

cross-posted from https://math.stackexchange.com/questions/4708694/curious-symmetry-in-a-theta-like-function
Let $\Theta : \mathfrak{h}\times \mathfrak{h} \to \mathbb{R}$ be defined as follows
$$ \...

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### Shimura lift is isomorphic iff twisted Hecke $L$ function does not vanish at central point

Let $S_{2k}(1)$ and $S_{k+1/2}(4)$ denote the set of modular forms of weight $2k$ for $SL_2(\mathbb{Z})$ and weight $k+1/2$ for the congruence subgroup $\Gamma_0(4)$, respectively. Consider the Kohnen ...

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### Using Ramanujan-type "Legendrian" sequences to find new formulas for $\frac1{\pi}$?

I. Recurrences
In Cooper's 2012 paper, "Sporadic sequences, modular forms and new series for 1/π", he did a computer search for the recurrence relation,
$$(n+1)^3 u_{n+1} = (2n+1)(an^2+an+b)...

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### Can you identify this irrational number?

There is a certain number, say $v$. I can prove it is irrational. That would be more interesting if it is expressible in terms of known values ... zeta functions, Catalan's number, L-functions, etc. ...

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### Why can Hecke operators be regarded as finite flat cohomological correspondence?

I'm reading the paper "Higher Hida and Coleman theories on the modular curve" by G.Boxer and V.Pilloni. But I'm confused with the different views towards Hecke operators.
$N$ is an integer ...

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166
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### Experiments with Voronoï summation

In order to test my understanding of the Voronoï summation formula, I tried to apply it to a simple estimation of partial sums of Fourier coefficients of cusp forms. The result I obtained cannot ...

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### Finding a new level-$7$ pi formula using the relation $j_{7A}(\tau) = \left(\sqrt{j_{7B}(\tau)}+\frac{7}{\sqrt{j_{7B}(\tau)}}\right)^2$

(Note: This third method continues from this post.)
There are level-$7$ pi formulas based on the McKay-Thompson series $T_{7A}$ and Cooper's $s_7$ sequence in this paper. This third method, among ...

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### Ramanujan's pi formulas with a twist (nine years later)

(Note: The second method described here continues this post.)
About nine years ago, I made an MO post "Ramanujan's pi formulas with a twist". An answer was informative, but not completely ...

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### Discrete subgroups of $\text{Sp}_{4}(\mathbb{Q})$ parameterizing polarized Abelian surfaces plus torsion data

I want to start by considering a familiar congruence subgroup of the integral symplectic group $\text{Sp}_{4}(\mathbb{Z})$. For a positive integer $N$, let $\Gamma _{0}^{(2)}(N) \subset \text{Sp} _{4}(...

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### Transformations of Ramanujan's 1/pi formulas $\sum_{n=0}^{\infty} s(n)\frac{An+ B}{C^n}$ and Monster moonshine functions

Someone with many papers on Ramanujan's work asked me how I managed to find the closed-forms for the binomial sums of level $10$ in a recent MO post. (A colleague of his wasn't able to find them.) I ...

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332
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### On levels $6$ and $10$ of the McKay-Thompson series of the Monster

After the McKay-Thompson series of levels $1,2,3,4$ of the Monster were mentioned in this MO post, levels $6$ and $10$ have very interesting relations as well, and there are unanswered questions for ...

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155
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### What is the image of the Hecke operator $U_p$?

Let $N \geq 1$ be an integer and let $p$ be a prime not dividing $N$. For $r \geq 1$, let $M_2(\Gamma_0(Np^r))$ denote the space of weight $2$ modular forms of level $\Gamma_0(Np^r)$. Let $$U_p: M_2(\...

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### Definition of modular curve associated to $\Gamma(N)$

For a positive integer $N$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{...

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### On Ramanujan's pi formula $\frac 1\pi=\sum_{k=0}^\infty\frac {(4k)!}{k!^4}\frac {Ak+B}{396^{4k}}$ and the solvable quintic $z^5-5z-396 = 0$?

I. Four quintics?
The general quintic can be transformed in radicals to at least three one-parameter forms. For simplicity, assume this free parameter to be some generic "alpha". Hence,
$$x^...

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456
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### Finding a sextic analogue to the solvable octic $\frac{(x + 1)^6(x^2 + x + 7)}x = -k^3$ where $e^{(\pi/3)\sqrt{d}}\approx k^3+41.999999999999\dots$

I. Degree 8
Assume the $j_i$ to be free parameters. The following octics in $x$ belong to $8T43,$ have group $\text{PGL}(2,7)$, and order $2\times168 = 336.$
\begin{align}
{j_1}\; &=\frac{(x^2 + ...

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252
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### On Elkies' $\text{9T32}$ nonic and a shared property with j-function formulas

I. First Set
Before going to Elkies' nonic, we start with something a bit simpler. There is a list of j-function formulas in this MSE post. For example, for prime levels $p = 5,7,13,$ we have,
$$j=\...

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### An explicit version of the Shimura correspondence for vector-valued modular forms or Jacobi forms

In several papers one can find references to an isomorphism $$S_{k,\rho}^\text{new}\cong S_{2k-1}^{\text{new},\pm}(N)$$ between a space of half-integer weight $k$ vector-valued cusp forms with ...

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### When are modular forms linearly independent modulo $p$?

Let $M_2(\Gamma_0(N))$ be the space of weight $2$ modular forms for $\Gamma_0(N)$ and let $f_1, \dots, f_r$ be a basis of normalized eigenforms for $M_2(\Gamma_0(N))$. Given a rational prime $p$, I'll ...

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466
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### Igusa's $\chi_{10}$ and Borcherds products

Igusa defined a genus 2 Siegel modular form $\chi_{10}$, which vanishes on the Humbert surface $G_{1}$ (the image of a "degenerate" Hilbert modular surface, the product of modular curves, ...

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139
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### Petersson norms of quaternionic modular forms

How is the Petersson norm of a quaternionic modular form defined?
Background: In Tamiozzo, On the Bloch-Kato conjecture for Hilbert modular forms, section 3.3, it is written "We normalize $f_B$ ...

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### what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$?

what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$ ?

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### Nonnegativity of coefficients of a modular form defined in terms of the Jacobi thetanull functions

Question
Let
\begin{align*}
\theta_2(q) & = \sum_{n=-\infty}^{\infty} q^{(n+1/2)^2}
\\
\theta_3(q) & = \sum_{n=-\infty}^{\infty} q^{n^2}
\\
\theta_4(q) & = \sum_{n=-\infty}^{\infty} (-1)^n ...

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### Are topological theta series (taking values in tmf(N)) of lattices good for anything?

I'm going to start with Mike Hopkins' great survey article in the ICM on topological modular forms (https://arxiv.org/abs/math/0212397). In it, he outlines a construction, for even unimodular lattices,...

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### $p$-adic analogue of modular forms, upper half-plane, and $L$-functions

In the classical picture, there is the (complex) modular form, defined on the (complex) upper half plane, which is related to the (complex) $L$-function via the Mellin transform. As I have recently ...

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### References for the construction of Beilinson's motivic Eisenstein classes

According to some authors, it is built in A.A.Beilinson "Higher regulator of modular curves" a class $\mathbf{Eis}_{\phi}$ in the motivic cohomology of the modular curve where $\phi$ is a ...

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80
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### Cohomology of modular curve

(A follow-up on this). Consider the modular curve $X_0(N)$. I'm trying to make the jump from understanding the cohomology $H^1(X_0(N), \mathbb{Z})$ to understanding $H^1(X_0(N), \mathcal{O})_\mathfrak{...

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165
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### Context for Wiles defect criterion and patching

This is not a homework or a project question, just me trying to get acquainted to the subject: I am an MSc student who recently came across the Wiles defect numerical criterion (see, for example, ...

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256
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### Fourier series of Eisenstein series — elegant and very good approximation

I played around with the Fourier series of the Eisenstein series resp. divisor sums and did some calculations, see below. Although the deduction is not rigorous / wrong (as the power series for the ...

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### Eta product of squared tau function

The Ramanujan tau function is the coefficient of the 24th power of the Dedekind eta function.
$$ \eta(x)^{24}= x\prod_{m=1}^\infty (1 - x^m)^{24} =\sum_{n=1}^\infty \tau(n)\,x^n , $$
I want to know ...

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142
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### Evaluation of mock modular forms at elliptic points

The holomorphic function
$$F(\tau)=-\frac{1}{\vartheta_4(\tau)}\sum_{n\in\mathbb Z}\frac{(-1)^nq^{\frac{n^2}{2}-\frac 18}}{1-q^{n-\frac12}}=2q^{\frac38}(1+3q^{\frac12}+7q+14q^{\frac32}+\dots),$$
is a ...

3
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1
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231
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### Siegel modular forms in Mathematica

Is there a convenient way to work with Siegel modular forms in Mathematica? I am interested in doing analytic computations using the $\chi_{10}(\Omega)$ Siegel modular form, where $\Omega$ is the $2\...

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### Langlands-Shahidi method in classical language

The Langlands-Shahidi method says that the $L$-functions of automorphic representations appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and ...

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156
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### Completing some of Ramanujan's results on $p = 5, 7, 9, 13, 25$?

This gathers scattered results together to see if they can be extended. The question is at the end. Given the Ramanujan theta function $f(a,b)$ and define Ramanujan's theta ratio formula,
$$r_k = (-1)^...

2
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2
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334
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### On V. Arnold's trinities regarding PSL(2,5), PSL(2,7), and PSL(2,11)?

Given the Ramanujan theta function,
$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$
Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}.$ Then the following functions for levels $...

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514
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### On the Klein quartic and the similar $a^2b+b^2c+c^2a$?

Given the Ramanujan theta function,
$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$
Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}$.
I. Degree 5
\begin{align}
a &= q^{11/...

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### Hecke operators acting on the Weierstrass $\wp$-function

Roughly speaking, my question is the following: Let $\wp(\tau, z)$ be the Weierstrass $\wp$-function, where $\tau \in \mathbf{H}$ and $z \in \mathbf{C}$. If $p$ is a prime number, can we define $T_p\,\...

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### Using the Rogers-Selberg identities to solve certain septics?

Given the Ramanujan theta function $f(a,b)$ and the Rogers-Selberg identities,
\begin{align}
U_1 &= \frac{f(-q,-q^6)}{f(-q^2)} = \sum_{n=0}^\infty \frac {q^{2n^2+2n}} {(q^2;q^2)_n\,(-q;q)_{2n+1}} =...

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3
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279
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### Solving solvable septics using only cubics?

After the satisfying resolution of my question on the Kondo-Brumer quintic, I decided to revisit my old post on septic equations.
I. Solution by eta quotients
The septic mentioned in that post may not ...

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130
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### What are the modularity conjectures for Artin motives?

Classically, singular cohomology is an important tool for studying topological spaces, in particular, complex varieties. In the mid-twentieth century it was realized that there are many analogues of ...

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123
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### What pre knowledge does Mumford's Tata collections on theta need?

I am a sophomore and have the foundation of complex analysis and abstract algebra. I have learned Legendre elliptic integral and Jacobian elliptic function, and it is through this that I know theta ...

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### What is the relationship between the Leech lattice and Dedekind eta function?

Like this old question, A conceptual proof of Jacobi's product formula for $\Delta$ ?, I am asking again for a conceptual proof of Jacobi's miraculous product formula for $\Delta$ (the unique ...

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0
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103
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### Is there an analog of Weil pairing for modular forms?

Given a newform $f(z)$ (of weight $k$) and a prime $p,$ consider the classical Galois representation
$$\rho_{f,p}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}/p\mathbb{Z}).$...

2
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143
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### Motivation for $p$-stabilization in Hida theory

I'm currently reading Hida's paper "A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms". The setup is the following: let $f$ be a weight $2$ newform ...

5
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730
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### Can Taniyama-Shimura conjecture be generalized to curves of higher genus (within Langlands framework)?

The Shimura-Taniyama-Weil conjecture asserts that if E is an elliptic curve over Q, then there is an integer N ≥ 1 and a weight-two cusp form f of level N, normalized so that a1(f) = 1, such that ap(E)...

5
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75
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### Compute the principal polarization on $J_0(N)$ in terms of modular symbols

If we consider the modular curve $X = X_0(N)$ as a curve over $\mathbb C$ then one can describe the jacobian $J(X)$ as $H^0(X,\Omega^1_X)^\vee/H_1(X,\mathbb Z)$ as one can do for any curve $X$. ...

2
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187
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### Voronoï summation for cusp forms with characters

In an attempt to solve an unrelated problem, I was led to the task of estimating/bounding from above sums of the form
$$\sum_{m=1}^\infty\lambda(m)e\left(-\frac{am}{q}\right)h(m)$$
where $\sum_{m=1}^\...

5
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1
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566
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### Lots of questions about modular forms

For the past year and a half, I have been working my way through Diamond & Shurman's "A First Course in Modular Forms", and I have just finished it. I Have Some Questions.
What is so ...

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1
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323
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### Sum of three squares as class numbers and Waldspurger's formula

It is known that the number of ways to express $n \in \mathbb{Z}_{\geq 0}$ as a sum of three squares (let's denote it as $r_3(n)$) can be expressed as Hurwitz-Kronecker class number (certain weighted ...

2
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2
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194
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### Special values of non-cm $L$-functions

For the sake of simplicity, assume $f$ is a non-cm eigenform of weight $k$ on the group $\mathrm{SL}(2, \mathbb{Z})$. Are there any known results or conjectures regarding any special values of the ...

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0
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108
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### Why does Deligne's construction of the Galois representation attached to the new cuspidal forms require that the Kuga-Sato manifold be regular?

The origin of this question is related to the construction of Galois representations of Deligne attached to $f$ a new cuspidal form (of weight $k\geq 2$). To do this, we consider the fiber product $k$-...