Questions tagged [modular-forms]

Questions about modular forms and related areas

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Why is this a modular function

Suppose we have a function $\phi\colon \mathfrak H \longrightarrow \mathbb C$ such that $\phi^{24}$ is a modular function of level $5$. $\phi(\tau)=\sum_{n=-1}^{\infty}a_{n}q^{n/5}$, $a_{-1}\neq 0,q=...
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2answers
141 views

Coefficient field of a newform using Magma

It is well-known that, for a newform $f = \sum c_nq^n \in \Gamma_0(N)$, the coefficient field $K_f := \mathbb{Q}(a_1, a_2, a_3, \cdots )$ is a number field. I am introducing myself in Magma, and I was ...
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2answers
226 views

About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

NOTE: I post this question on math.stackexchange but nobody answered, so I try here. For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-...
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Zagier's “From 3-manifold invariants to number theory”?

Zagier lectures on "From 3-manifold invariants to number theory" - do you know about texts of that or on the discussed web of ideas? ([https://www.mpim-bonn.mpg.de/de/node/10791])
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The mystery of the jumps of functions with the prescribed jumps: Eisenstein series and hidden symmetries(?)

Say that a function $f(t)$ “changes only by jumps” if $f(t) + \text{const} = C ∑_k j_k θ(t-t_k)$ for a certain constant $C$. Here $θ(t)$ is the Heaviside step function which has a jump 1 at $t=0$ (it ...
4
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1answer
211 views

An explicit equation of the canonical morphism $X_1(N) \to X_0(N)$

I know there are some research about explicit equations for affine models in $\mathbb{A}^2$ of many modular curves over $\mathbb{Q}$, for example of $X_i(N), X(N)$ (where $i = 0, 1, 2$) for small $N$. ...
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2answers
658 views

Modular forms with finitely many or very few non-zero Fourier coefficients

I have an elementary question on modular forms, but which I don't know how to solve. a) Is there a congruence subgroup $\Gamma \leq \mathrm{SL}_2(\Bbb Z)$, an integer $k \in \Bbb Z$ and a non-...
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130 views

Reference Request - New proof of Ribet's level lowering by Khare and Wintenberger

I'm currently following the note of Sug Woo Shin's course at Berkeley with notes taken by Rong Zhou. In Section 24.3 (Page 86), Ribet's level lowering theorem is stated: [Theorem 24.7] $E = E_{a^{\...
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162 views

Diophantine consequences of the Buzzard–Diamond–Jarvis conjecture

Serre's modularity conjecture famously implies Fermat Last Theorem. More generally, Serre's conjecture implies that certain generalized Fermat equations have no non-trivial solutions (see Section 4.1 ...
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60 views

Ordinary primes for a weak form corresponding to a CM newform

Setup: Let $f$ be a harmonic Maass form of weight $2-k$ ($k \in \mathbb{N}$), level $N$, and character $\chi$. Letting $q := e^{2\pi i z}$ and considering the Fourier expansion of any harmonic Maass ...
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169 views

“Moonshine” basics?

Having browsed recently a bit about "moonshine", which looks to me like some weird surrealist landscape, I wonder if: 1. sporadic groups could be seen as seemingly isolated special points of ...
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229 views

Why is Dedekind sum?

The Dedekind function is defined as follows $$\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),\qquad q=e^{2\pi i\tau}.$$ We have $$\eta(\tau+1)=\zeta_{24}\eta(\tau),\qquad \eta\left(-\frac{1}{\tau}\right)...
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83 views

A computation of the rank of the Jacobian of a hyperelliptic curve over a number field using MAGMA

In this paper, the authors says that, in order to show the rank of a Jacobian over $\mathbb{Q}$ is 0, they use the L function. In the section 3.3, the authors compute the rank of the Jacobian of $X_1(...
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On Fourier coefficients of Bianchi modular forms, l-ordinary

Let $f\in S_2(\Gamma_1(N))$ be a Hecke eigenform and $\ell$ a prime number does not divide $N$. Let $a_f(\ell)$ be the $\ell$-th Fourier coefficient of $f$. Then $a_f(\ell)$ is is called $\ell$-...
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169 views

Maximum value of newform from Galois representation

One can attach $\ell$-adic Galois representations to holomorphic cuspidal newforms of weight $2$ on the upper half-plane. If a newform is $L^2$-normalized, can one extract its maximum value from the ...
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1answer
187 views

Meaning of Atkin-Lehner eigenvalues

Suppose I have $f\in S_2(\Gamma_0(N))$ a classical modular newform of level $N$. I want to understand what information (if any) is carried by its Atkin-Lehner eigenvalues for primes $p\mid N$, as ...
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166 views

Modular forms on $\Gamma(N)$

I'm wondering where I can find a good reference about what is known about modular forms (especially cuspidal eigenforms) of full principal level $\Gamma(N)$, in terms of their Hecke theory, old/...
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1answer
427 views

Hurwitz numbers and $t$-cores

For integers $k \geq 0$ and $d \geq 1$ let $H(k,d)$ be the Hurwitz number which, for the purposes of this posting, will be defined by: \begin{equation} H(k,d) \, := \ d! \, \sum_{\lambda \, \vdash d}...
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1answer
225 views

Modular form not meromorphic at $\infty$

Is there a function $f$ with the following properties $f$ meromorphic at the upper half plane $\mathfrak h$, $f$ is of weight $k$ under a congruence subgroup of $\operatorname{SL}_2(\mathbb Z)$, $f$ ...
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Is there some precise way in which modular cocycle $c/(c\tau +d)$ “is” the generator of $H^1(\mathcal{M}_{ell}, \omega^2) \cong \mathbb{Z}/12$?

It is semiclassical that the extension class $\text{Ext}^1(\omega^{-1},\omega)\cong H^1(\mathcal{M}_{ell},\omega^2)$ over the modular stack $\mathcal{M}_{ell}/\mathbb{Z}$ is nonvanishing, being cyclic ...
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1answer
114 views

Index and weight of Weierstrass $\sigma$-function as a Jacobi form, versus a statement in a note by Zagier

Let $$\sigma_L(w; \tau):=\frac{w}{\exp\left(\sum_{k\ge 2} 2G_k(q)\frac{w^k}{k!}\right)}$$ be the version of the Weierstrass $\sigma$-function which is used to orient $\text{tmf}$; here $w=2\pi i z$, $...
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1answer
357 views

Modular forms over $\mathbb{Z}$ vs modular forms with integral Fourier coefficients

It is well known that the ring of modular forms over $\mathbb{C}$ is $$ \mathbb{C}[c_4,c_6] $$ where $$ c_4 = 1+240 q + \cdots,\qquad c_6 = 1-504 q - \cdots $$ are the standard Eisenstein series, and ...
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260 views

Work of Atkin on the 26th power of eta

The 26th power of the Dedekind $\eta$ function has been mentioned several times here on MO: A 14th and 26th-power Dedekind eta function identity? What's the status of the following relationship ...
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123 views

Holomorphic automorphic/cusp forms on real Lie groups

An automorphic form on a real Lie group $G$ for a discrete subgroup $\Gamma$ is a function $f:G\to\mathbb{C}$ with some properties (see Borel’s definition in Proceedings of Symposia in PURE ...
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1answer
255 views

An explicit equation for $X_1(13)$ and a computation using MAGMA

By a general theory $X_1(13)$ is smooth over $\mathbb{Z}[1/13]$, and so is its Jacobian $J$. And the hyperelliptic curve given by an affine model $y^2 = x^6 - 2x^5 + x^4 -2x^3 + 6x^2 -4x + 1$ is $X_1(...
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99 views

Moduli interpretation and Ogg's notation for the cusps on modular curves

In Ogg's paper "rational points on certain elliptic modular curves", the author says, using Ogg's notation for cusps, that for fiexed $d$, if $(y, N) = d$, then for any $x$ satisfying $(x, y,...
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1answer
169 views

Statement of classical Ramanujan-Petersson conjecture

I'm preparing for an expository talk on some topics in the representation theory of reductive p-adic groups, including tempered representations and Whittaker models, and as motivation I wanted to ...
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56 views

Stabilizers of points in the upper half-plane

Suppose that $\Gamma$ is a group acting discontinuously on $\mathcal{H} = \{ z \in \mathbb{C} : \operatorname{Im}(z) > 1\}$. In order to keep things simple, suppose that $\Gamma \subseteq \...
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1answer
130 views

Reference request for some fragments of Gauss with dubious origin

Gauss's results on the interconnection between the different values of the arithmetic-geometric mean of two complex numbers as recorded in his private notebooks led him to introduce foundational ...
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143 views

Factorizing classical Eisenstein series

In the course of my research, I found some surprising (for me) factorizations of Eisenstein series in levels $1$, $2$, $3$, and $4$. For instance, in level $1$ set with standard modular form notation $...
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1answer
136 views

Modular forms and number of representations by binary quadratic forms

Let $Q(x,y)$ be a positive definite quadratic form of discriminant $d$. Let $r_Q(n)$ be the number of solutions of $Q(x,y)=n$. It is known that the function $f_Q(\tau)=\sum_{n=0}^{\infty}r_Q(n)q^n$ is ...
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108 views

A formula in Ramanujan's lost notebook and its connection with Chudnovsky series for $1/\pi$

While studying Berndt's Ramanujan's Lost Notebook Vol. 2, page 369 (chapter on Springerlink), I found that Ramanujan gave values of a certain expression $$\frac{1}{\sqrt{Q_n}}\left(\sqrt {n} P_n-\frac{...
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2answers
217 views

Do odd-weight cusp forms have analytic rank 0?

Let $f(z)=\sum_{n\ge 1}a_nq^n$ be a cusp form, where $q=e^{2\pi i z}$. Let $ L(s) = \sum_{n\ge 1} a_nn^{-s} $ be its corresponding L-function. The completed L-function of $L(s)$, $\Lambda(s)$, should ...
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78 views

Logarithmic vector-valued modular functions and quasimodular forms with misleading modular weights

I have a somewhat imprecise question about functions with reasonably nice modular transformations that don't seem to fit nicely into what I understand of the plain vanilla theory of modular and ...
5
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65 views

Totally rational hypergeometric evaluations

This is a followup to the question at which rational points does the Hypergeometric function take rational values asked 10 years ago by Eugene Starling, and is more a challenge than a question. Let $F(...
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59 views

Continuation of $\sum \sigma_\nu(n) a(n) n^{-s}$ for $a(\cdot)$ coming from a half-integral weight form

In some of my work, I've run into a wall trying to understand whether a Dirichlet series has a meromorphic continuation or not. Let $f(z) = \sum_{n \geq 1} a(n) e(nz)$ be a half-integral weight ...
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74 views

How can I compute the derivative of a modular form, for example an Eisenstein series?

Suppose that $f(z)$ is a modular form of weight $k$, therefore for any $z\in \mathbb{H}$, and any matrix in $SL_2(\mathbb{Z})$ we have: $$f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z).$$ I do not ...
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1answer
236 views

Two non constant meromorphic functions over a connected compact Riemann surface, could not be algebraically independent

Let $M$ be a connected compact Riemann surface. Let $f, g$ be two nonconstant meromorphic functions. Why is there a two-variable complex polynomial $F(x,y)$ that vanishes for $(x, y)=(f, g)$, (in ...
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1answer
192 views

Chiral homology for the Virasoro algebra and/or affine Lie algebra

I want to understand what concrete analytical objects are found in chiral homology of higher degree of a vertex algera (-module) $M$. More precisely: I can obtain conformal blocks on a surface $\Sigma$...
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1answer
123 views

What are the known number-theoretic functions, that are related to “the number of ideals of norm $n$, that belong to the class $[c]$”?

Let $L$ be a number field, $\mathcal{O}_L$ its ring of integers, and $\mathcal{Cl(O}_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}_L)$. By $r(n)=r([c], n)$, I mean ...
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1answer
364 views

Modularity of higher genus curves

The modularity conjecture for elliptic curves over number fields is well known, and indeed, is a theorem for all elliptic curves over $\mathbb{Q}$, and at least potentially, over any CM field. What ...
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1answer
423 views

How can I transform $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^k\sin(\pi rn)}$ into a modular form?

Let $$f_k(z)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^k\sin(\pi zn)}$$ be a family of holomorphic functions on the upper-half plane $\mathbb{H}=\{a+bi|b>0\}$ for each odd natural number $k$. These ...
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0answers
80 views

Siegel's formula for generalized theta series with characteristics?

Siegel's formula(Siegel-Weil) directly relates the weighted sum of theta functions to Eisenstein series. (Or equivalently, the weighted sum of the cusp form is zero). I wonder if there is a ...
14
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1answer
279 views

Bounding the fourier coefficient field

Let $f = \sum_n a_n q^n \in S_2(\Gamma_0(N))$ be a normalized, non-CM, newform of weight $N \geq 1$ and level $2$. Let $K_f := {\mathbb Q}(\{a_n\}) \subset {\mathbb C}$ be the number field generated ...
8
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1answer
369 views

Spectral decomposition of product of modular functions

The eigenfunctions of the Laplacian on $SL(2,\mathbb Z)\backslash \mathbb H$ are known to be given by three types: the constant function, the real analytic Eisenstein series (which come in a ...
2
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0answers
49 views

Hecke convergence factor

I was reading a paper here. There the author define an infinite series $$\sum_{ad-cb=1}(cz+d)^{-(k-j)}(az+b)^{-j}$$ where $k$ is an even integer bigger than 2 and $2\leqslant j\leqslant k-2$. Then ...
5
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2answers
561 views

What are the applications of modular forms in number theory?

I am new to the topic, so I'm trying to get an overview. I am aware of the relation between modular forms and $L$-series (but don't know what that does) and FLT. Are there other applications of ...
3
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0answers
136 views

Finiteness of points over the cyclotomic extension for modular forms

Let $\rho(f):G_\mathbb{Q} \rightarrow GL_2(K_f)$ be the Galois representation attached to some cuspidal modular form $f$ where $K_f$ is a finite extension of $\mathbb{Q}_p$. Let $V_f$ be the vector ...
25
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4answers
853 views

Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$

Let $X=\mathbb C\setminus\{0,1\}$, equipped with the hyperbolic structure it inherits from Klein's modular $\lambda$ function $\lambda:\mathbb H \to X$. In each (non-peripheral and nontrivial) free-...
2
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1answer
138 views

Global section of vertical differential 1 forms on universal elliptic curve

Let $B$ be a modular curve (of some level) over a number field $K$ (here, we implicitly assume that $K$ is large enough to make sense the phrase "$B$ is a $K$-variety"). Let $E\to B$ the ...

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