Skip to main content

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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
18 views

Number of copies of a fundamental domain for $\Gamma_0(q)/\{\pm1\} $ needed to cover a region $P(Y) = \{z \mid \Im z > Y, \Re(z) \in ]0,1] \}$ ($Y>0$)

I've been struggling on a claim in W.Duke's paper on the dimension of the space of cusp forms of weight one (see : https://arxiv.org/pdf/math/9411212 Lemma 2 p.5). Take a cuspidal form $f \in \mathcal{...
supermartruc's user avatar
2 votes
1 answer
78 views
+50

Degree of extension of the field of coefficients of modular forms

I am beginning to study modular forms and I came across an inequality defining the bound for the degree of extension of the field of coefficients of a modular form $f\in S_k(\Gamma_1(N))$. This goes ...
roydiptajit's user avatar
0 votes
1 answer
52 views

Sources of non-congruence representations of the modular group

I'd be interested in any references that provide a source of non-congruence representations of the modular group or that discuss such representations. In my own field, vertex algebras, they have ...
JPhy's user avatar
  • 1,766
0 votes
0 answers
33 views

Excercise on j-invariant

I have this problem. Given $j:\cal{H}\rightarrow\mathbb{C}$ the j-invariant function defined on the upper-half complex plane as $j=1728\frac{g_2(\tau)^3}{g_2(\tau)^3-27g_3(\tau)^2}$, where $g_2,g_3$ ...
cespun's user avatar
  • 94
0 votes
0 answers
30 views

Counting the number of double cosets for subgroup

Let $H,K \le G$ and consider the set of double cosets $ H \backslash G / K$. If we have $H' \le H$, is there a formula expressing $| H' \backslash G / K|$ in terms of $|H \backslash G / K|$, $|H:H'|$...
J. S.'s user avatar
  • 155
3 votes
0 answers
113 views

Find cusps of $\Gamma_1(p)$, show $\frac{j(\tau)}{j(2\tau)}$ is a modular function of weight $0$ level $\Gamma_1(2)$, which cusps is it holomorphic?

Firstly for notation, let $$\Gamma_1(p) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid a \equiv d \equiv 1 \pmod p, c \equiv 0 \pmod p \right\},$$ and $j(\tau) = \frac{E_4(\tau)^...
Daniel New's user avatar
0 votes
0 answers
33 views

The Continuity of the Galois Representation Attached to a Weight 2 Newform

A representation $\rho: G_{\mathbb Q} \to \operatorname{GL}(V)$, where $V$ is a finite-dimensional $K$-vector space for some field $K$, must be continuous in order to be considered a Galois ...
Johnny Apple's user avatar
  • 4,325
2 votes
0 answers
54 views

Is the Galois conjugate of a Siegel eigenform another eigenform?

Let $F$ be a Siegel cusp form of weight $k\geq2$ (integer), degree two for the group $\text{Sp}_4(\mathbb{Z})$. Suppose it is an eigenform for all the Hecke operators $T(n)$. Let $\sigma$ be an ...
1.414212's user avatar
  • 285
3 votes
1 answer
65 views

Computing torsion subgroup of elliptic curve

Compute the torsion subgroup of the elliptic curve $y^2=x^3+5x^2+3x+7$. I am only used to computing torsion groups when our equation is in 'short Weirstrass form'; i.e. $y^2=x^3+Ax+B$ for integer $A,...
alidixon222's user avatar
6 votes
0 answers
62 views

Anything interesting known about this generalization of even and odd functions?

Let $n \in \mathbb N$. Let's say a complex function $f: U \rightarrow \mathbb C$ is "of type $k \pmod n$" if for one (and hence every) primitive $n$-th root of unity $\omega$, $$f(\omega z) =...
Torsten Schoeneberg's user avatar
0 votes
1 answer
52 views

Number of cusps of $X_0(p^2)$ for prime $p$

Find the number of cusps on $X_0(p^2)$ (for prime $p$) by finding a complete set of coset representatives for $\Gamma_0(p^2)$ as a subgroup of $\Gamma_0(p)$. Hint: $X_0(p)$ has two cusps, the ...
turkey131's user avatar
  • 135
1 vote
0 answers
34 views

Fundamental group of Y(2)

Let $\mathcal{H}$ be the upper half plane and $Y(2)=\mathcal{H}/\Gamma(2)$ as usual. The modular lambda function yields a bijection $Y(2)\xrightarrow{\sim}\mathbb{C}\setminus\{0,1\}$. Let $a$ and $b$ ...
User0829's user avatar
  • 1,369
0 votes
0 answers
30 views

Finding the right conjugation of $\Gamma_0(nl) \cap \Gamma(l) $

Let $l>2, n \in \mathbb{N} $. In my attempt to compute the genus of $\mathbb{H} / \Gamma_0(nl) \cap \Gamma(l) $, I've stumbled upon the problem of computing the cusps of this modular group or just ...
Meliodas's user avatar
  • 133
0 votes
0 answers
61 views

How would you proceed with this integration question?

Q1: I am tasked to integrate the complete elliptic integral of the first kind $K(k)$ $$\int_{0}^{1}\frac{dk}{k'K(k)^2}$$ where k is the elliptic modulus. From this answer by Setness Ramesory you can ...
ProtoZone's user avatar
0 votes
0 answers
31 views

$ \lambda^{*}(n) $ minimal polynomials

I already asked a closely related question on MSE, but didn't received any answer. Considering the modular lambda function, the values of $ \lambda^{*}(n) $ for some integers are given on here. Is ...
user967210's user avatar
6 votes
3 answers
236 views

How did Jacobi find his connection between theta functions and $q$?

I was 'reading' (I can't actually read german, but I can read math!) Jacobi's derivation of the ODE for $y(q) = \sum_{n=-\infty}^{\infty} q^{n^2} $. On page 2 of the paper Jacobi states the following ...
Sidharth Ghoshal's user avatar
3 votes
1 answer
68 views

How to Construct the Theory of Hecke Operators for Maass Forms

I'm trying to find a construction for the theory of Hecke operators for Maass forms that is analgous to the double coset operator construction for modular forms. For modular forms of weight $k$, this ...
Laan Morse's user avatar
0 votes
0 answers
19 views

how to evaluate the explicit formula for the quotient powers of the Dedekind eta function

I'm working on a thesis in number theory, specifically focusing on modular forms, particularly the Dedekind eta functions. I want to know if there is a way to obtain the explicit expression for the ...
Sofiane Abdelhamid's user avatar
1 vote
0 answers
25 views

How to prove this peculiar relationship between minimal polynomials of Ramanujan class invariants?

The Ramanujan class invariants (a.k.a. "Ramanujan-Weber class invariants") are defined for $n>0$ by $$G_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1+e^{-(2k+1)\pi\sqrt{n}}\right)...
Wolfgang's user avatar
  • 962
0 votes
0 answers
25 views

Reciprocal Discriminant function is a modular form

I came across a problem that any cusp form of order k, (say f) can be written as product of Discriminant function and a modular function of order k-12 say, h. So i took $h=\frac{f}{\Delta}$ since $\...
Singh Naveen's user avatar
2 votes
0 answers
70 views

Dedekind eta and Lie algebras

Reference request and a formal question I am currently self-studying modular forms and read the Wikipedia page for Dedekind's eta function. There it reads that: "The theory of the algebraic ...
ProtoZone's user avatar
2 votes
0 answers
39 views

q-series Expansion at Cusp and L-Functions

We know that the coefficients at $\infty$ of a modular form that is an eigenfunction of all Hecke operators in, say, $\Gamma_0(q)$, give an $L$-function with an Euler product and a functional equation....
Riobaldo's user avatar
1 vote
0 answers
41 views

On proving that an Artin L-function is cusp form

I am reading the article by JEAN-PIERRE SERRE, On a Theorem of Jordan, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 2003. I have problem in the following part: Since $S_{3}$ is a dihedral group, ...
Aster Phoenix's user avatar
1 vote
0 answers
55 views

Sketching Fundamental domains for $SL_2(\mathbb{Z})$ [closed]

I am familiar with the fundamental domain for the full modular group $\Gamma(1)=SL_2(\mathbb{Z})$, given by \begin{equation}\{z\in\mathbb{H}:-1/2<Re(z)<1/2\textrm{ and }|z|>1\}.\end{equation} ...
user avatar
1 vote
0 answers
47 views

Road map to learning about $\ell$-adic representations of $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I am interested in learning about the congruence relations satisfied by the coefficients of modular forms, in particular I am interested in learning more about $\tau(n)$, the coefficients of the ...
Takamoto Yuji's user avatar
1 vote
0 answers
111 views

Residue theorem and theta function identities

Let's use the classical definition $$ \vartheta _1\left( z,q \right) =-i\sum_{n\in \mathbb{Z}}{\left( -1 \right) ^nq^{\left( n+\frac{1}{2} \right) ^2}e^{i\left( 2n+1 \right) z}}\,\,\,\,\,\ q=e^{i\pi \...
Loyar's user avatar
  • 69
6 votes
0 answers
63 views

Number Fields Generated by Fourier Coefficients of Modular Forms

We know that for every normalized Hecke eigenform $f$, its Fourier coefficients generate a number field. I wonder if we have an "inverse Galois" type conjecture regarding which number fields ...
Jason Lee's user avatar
  • 182
0 votes
0 answers
46 views

Finite index subgroup of congruence subgroup $\Gamma_0(4)$ and $\Gamma$

It is standard notation but I define them anyway to avoid ambiguity: $$\Gamma_0(N) := \left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma : c \equiv 0 \text{(mod $...
Chiara M's user avatar
0 votes
0 answers
50 views

The $SL_{2}(\mathbb{Z})$ double coset of diagonal matrix

I have a trouble proving that: For $k\in \mathbb{N}$, the double coset \begin{align*} SL_{2}(\mathbb{Z})\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} SL_{2}(\mathbb{Z})\end{align*} is ...
ImBear's user avatar
  • 21
2 votes
0 answers
35 views

Torsion group over $\mathbb{Q}$ of $Y^2=X^3+DX$ for $D\equiv 2\pmod{3}$.

Let $D\in\mathbb{N}$ with $D\equiv 2\pmod{3}$. Describe the torsion group over $\mathbb{Q}$ of the elliptic curve \begin{equation}Y^2=X^3+DX.\end{equation} Idea: Firstly we recall that for an ...
user avatar
1 vote
0 answers
29 views

For which of the Weierstrass elliptic function periods do this equation of the modular discriminant and the Dedekind eta function apply?

It is often claimed that the following equation holds for the modular discriminant $\Delta=g_2^3-27g_3^2$ of the Weierstrass elliptic funtion and the Dedekind eta ($\eta$) function for period ratio $\...
Arvid Samuelsson's user avatar
3 votes
1 answer
78 views

On the cusps of $\Gamma_0(3) \cap \Gamma(2)$

I've been trying to compute the number of non-equivalent cusps of $\Gamma_0(3) \cap \Gamma(2)$. My approach so far has been the following: I believe that this is finite index subgroup of $\Gamma_0(3)$ ...
Meliodas's user avatar
  • 133
0 votes
1 answer
39 views

Questions on sequences and modular forms

Let me apologize ahead of time since I am not at all well versed in the theory of modular forms. I have seen some nice examples where modular forms are used to study certain interesting numbers. For ...
matt stokes's user avatar
4 votes
0 answers
130 views

Background for Gross-Zagier paper

I have been reading the paper "Heegner points and derivatives of $L$-series" by Gross and Zagier. Link to the paper. In section III of the paper, they use intersection theory to express a ...
Joseph Harrison's user avatar
2 votes
1 answer
147 views

How to come up with the generating function of an elliptic curve?

Having watched the otherwise splendid Numberphile video with Edward Frenkel explaining the Langlands program, two mysteries remained completely open to me: Given the equation $y^2 + y = x^3 - x^2$ how ...
Hans-Peter Stricker's user avatar
0 votes
0 answers
32 views

Almost holomorphic functions

I am trying to compute an integral which goes as: $$I=\int_{\mathcal{F}}\left(\sqrt{Im(z)}\eta(z)\overline{\eta(z)}\right)^{-k}\frac{dzd\bar{z}}{Im(z)^2}$$ Where $\eta$ is Dedekind's $\eta$, $\mathcal{...
Isolated Pole's user avatar
3 votes
1 answer
63 views

$\Gamma(N)$ -inequivalent cusps clarification

We know that $\Gamma(N)$ has at most $[\Gamma(1):\Gamma(N)]$ inequivalent cusps given possibly by $g_i \infty$ where the $g_i$ are coset representatives of the subgroup $\Gamma(N)$. Then I don't ...
Meliodas's user avatar
  • 133
0 votes
1 answer
55 views

Coefficients of a rational function that depend meromorphically on a parameter

Let $D \equiv 1, 2 \, (\textrm{mod }4)$ be a positive, squarefree integer. Let \begin{align} r_D(n) = \{(x, y) \in \mathbf{Z}^2 \mid x^2 + Dy^2 = n\} \end{align} for any positive integer $n$. Consider ...
Joseph Harrison's user avatar
0 votes
0 answers
51 views

Why the field generated by the coefficients of a modular form over $\mathbb{Q}$ is finite over $\mathbb{Q}$

I am going through modular forms and I came to know about that, the coefficients of a modular forms f over $\mathbb{Q}$ define a finite extension $K_f/Q$. Can anyone point me to the proof of this fact?...
roydiptajit's user avatar
0 votes
1 answer
50 views

basis for $\mathcal M_{24}$

When I run the command ModularForms(1, 24).basis() in Sage, I get back the following $q$-expansions: $q + 195660q^3 + 12080128q^4 + 44656110q^5 + O(q^6)$ $q^2 - 48q^...
node196884's user avatar
1 vote
1 answer
60 views

$\tau(n) \equiv n\sigma_9(n)\,(\mathrm{mod}\,1050)$ is known?

I found the congruence in the title today (where $\tau(n)$ is the Ramanujan tau function), and I wonder if this is new or already known (or easy consequence of known results). I read the wikipedia and ...
Seewoo Lee's user avatar
  • 15.2k
2 votes
1 answer
76 views

Proof that principal congruence subgroups $\Gamma(N)$ are torsion free.

I have read the fact that the principal congruence subgroups $\Gamma(N)$ of $\mathrm{GL}_n({\mathbb{Z}})$ are torsion free for $N \geq 3$ several times, but only saw proofs for very specific ...
Staub und Dreck's user avatar
2 votes
0 answers
34 views

Modular symbols, Manin symbols, two and three term relation

Maybe someone could help me out. I consider a $SL_2(\mathbb{Z})$-module $\Omega$. We set \begin{align} S:=\left(\begin{array}{rr} 0 & 1\\ -1 & 0\end{array}\right) \ \mathrm{and} \ U:=\left(\...
Running_mathematics's user avatar
2 votes
0 answers
62 views

Way to determine when a modular form divides another?

Given modular forms $f, g$ for $SL_{2}(\mathbb{Z})$ of weights $l, k$ resp. where $l < k$, is there a way to see if $f \mid g$? That is, is there a way to see if $g = fh$ for some modular form $h$ ...
Freddie's user avatar
  • 1,769
0 votes
0 answers
46 views

Hecke operators on q-expansion of cusp forms

I am trying the following exercise. Let $V=S_2(\Gamma _1 (16))$ and we are given the following basis of $V$, their $q$ expansion at the $\infty $ cusp. \begin{align} f_1 =& q − 2q^3 − 2q^4 + 2q^...
mathemather's user avatar
  • 2,997
1 vote
1 answer
69 views

Why is the Function Field of the Modular Curve $X(N)$ defined over $\mathbb{Q}(\mu_n)$?

Following Section 7.6 in Diamond & Shurman, the algebraic model of $X(N)$ is constructed (a priori) over $\mathbb{Q}$ by first defining its function field as \begin{equation*} \mathbb{Q}(j,f_{(0,1)...
Josu P. Z.'s user avatar
0 votes
2 answers
72 views

On congruence subgroups of $GL_2^+$

I am reading the book Holomorphic Hilbert Modular Forms by Paul Garrett. The author considers congruence subgroups of general linear groups of positive determinant. More precisely, let $F$ be a ...
zc l's user avatar
  • 97
0 votes
0 answers
18 views

Weights of Vector Valued Siegel Modular Forms

I am looking for a reference request for a fact I believe to be true (or a confirmation that it is wrong). We may parameterize the algebraic representations of $GL_n(\mathbb{C})$ by non-increasing ...
Maximilian Klambauer's user avatar
0 votes
0 answers
33 views

Some property of GL(2,R) and GL(2,Z)

I am trying to show that there exists a family of matrices $(M_n)$ in $GL(2, \mathbb{R}) $ such that $GL(2,\mathbb{Z}) M_n GL(2,\mathbb{Z})$ is the same for every n and $GL(2,\mathbb{Z}) M_n $ is ...
RadonMeasure's user avatar
0 votes
1 answer
77 views

Galois action on modular curve

Consider the (geometrically reducible) modular curve $Y(N)$ over $\mathbb{Q}$, representing the functor $$Y(N)(T)=\{(E/T,\alpha),\; E/T \; \text{elliptic curve}, \;\alpha:(\mathbb{Z}/N\mathbb{Z})_T^2\...
Ben's user avatar
  • 1,210

1
2 3 4 5
29