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.
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Connection between lattice basis vector and algebraic coefficients of a complex elliptic curve
Background
An elliptic curve over $\mathbb C$ is the zero locus of $y^2 = 4x^3-g_2x-g_3$ in $\mathbb C^2$ or $\mathbb CP^2$.
A complex number $\tau \in \mathbb H$, uniquely determines an elliptic ...
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Another proof for $S_2(\Gamma_0(4))=\{0\}$
How to see there are no nontrivial cusp forms for $\Gamma_0(4)$ of weight 2
I am searching for a proof of this that directly uses the provided set of generators for $M_2(\Gamma_0(4))$.
As mentioned in ...
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Fricke involution’s effect on character
I’m using the definition $W_Nf(\tau)=i^kN^{-k/2}\tau^{-k}f(-1/N\tau)$. Now suppose $f\in M_k(\Gamma_1(N),\chi)$, show $W_Nf\in M_k(\Gamma_1(N),\chi^{-1})$.
I know this is pure calculation but I’m ...
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Modular forms of $M_1(\Gamma_1(4), \chi)$
I want to find a basis of the space of modular forms of weight one $M_1(\Gamma_0(4), \chi)$, where $\chi$ is the character
$\chi(d)=\Big(\frac{-1}{d} \Big)$, and
$$\Gamma_0(4)=\Big\lbrace
\begin{...
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How to see there are no nontrivial cusp forms for $\Gamma_0(4)$ of weight 2
I know that $M_2(\Gamma_0(4))$ is generated by
$E_{2,2}(z)=E_2(z)-2E_2(2z)$ and
$E_{2,4}(z)=E_2(z)-4E_2(4z)$,
where $E_2$ is the Eisenstein's series.
This space is supposed to only have $0$ as cusp ...
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What is the history behind the discovery of the almost-integer values of $ e^{\pi \sqrt{d}} $?
It's well-known that $ e^{\pi \sqrt{d}} $ is almost an integer when $ d $ is taken to be one of the large Heegner numbers $ d = 43, 67, 163 $.
I'm interested to know what the history of this discovery ...
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Eisenstein sum.
I have a proof of:
$$S=\sum_{n=1}^{\infty}\frac{n(-1)^{n+1}}{(-1)^n+e^{\pi n}}=\frac{1}{24}.$$
That is related to Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}...
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Additional symmetries in a theta-like function
Let $\Theta : \mathfrak{h}\times \mathfrak{h} \to \mathbb{R}$ be defined as follows
$$ \Theta(z,\tau) = \sum_{\omega_1, \omega_2 \ \in \ \mathbb{Z}\tau + \mathbb{Z}} \exp\Big(-2\pi\frac{| \...
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Definition of modular form
There is a well known definition of modular form
Let $f:\mathbb{H}\rightarrow\mathbb{C}$, $\Gamma=\Gamma_0(N)$ and $k\in\mathbb{Z}$
(i)$f$ is holomorphic on $\mathbb{H}$
(ii)$f|_k\gamma=f$ for any $\...
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What is the motivation to study $\Gamma \backslash \mathbb H $, where $\Gamma \subset Sl_2(\mathbb Z)$ is a congruence subgroup?
Let $\mathbb H \subset \mathbb C$ be the upper half plane. On $\mathbb H$ the group $Sl_2(\mathbb Z)$ acts by Möbius transformations
$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} . z = \frac{...
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How to compute dimensions of modular forms?
I am reading Iwaniec's book on modular forms. After having proven the valence formula, he deduces constraints on the possible zeros. For instance, for the modular group and $k=4$, he deduces that a ...
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Are modular form theory applicable to representation of sum of higher powers
just a disclaimer before my question, I don't really know too much about modular forms, as I just started learning it. However, any answers whether technical or not are very welcomed!
I know that ...
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$q$-expansion of Eisenstein series twisted by a Dirichlet character
Consider the Eisenstein series given by
$$
G_{k}(\chi,z) = \sum_{m,n\in \mathbb{Z}, (n,N) = 1}\frac{\chi(n)}{(mz + n)^{k}}
$$
where $\chi:(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow \mathbb{C}^{\...
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Lambert series and higher order derivatives of the Euler Function
It is well known that:
$$\sum_{n=1}^\infty \frac{n q^n}{1-q^n} = -q \frac{d}{dq} \Phi(q)$$
Where $\Phi$ is defined as follows for $|q|<1$:
$$\Phi(q) = \prod_{n=1}^\infty (1-q^n)$$
Is there any ...
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Proof of Lemma 3.49 in the Diamond-Darmon-Taylor Fermat's Last Theorem Notes
We start with a semistable elliptic curve $E/\mathbb Q$ such that such that its mod 3 representation $\overline{\rho}_{E,3}$ is reducible. We wish to show that $\overline{\rho}_{E,5}$ is modular. This ...
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Why are Jacobian varieties important?
I've recently met Jacobian varieties, in the context of elliptic curves and modular curves. I'm still very new to them and not super comfortable with the machinery, so please forgive me any mistakes ...
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Behavior of weakly holomorphic modular forms at cusps
Let $M_0^{!}(\Gamma)$ be a space of weakly holomorphic modular functions.
Now we consider a subspace $M_0^{!,\infty}(\Gamma)$ which allows pole only at infinity.
Assume that $\Gamma=\Gamma_0(N)$ and $...
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Is there a known description for the composition of two double coset operators?
Let $\Gamma$ and $\Gamma'$ be congruence subgroups of $SL_2(\mathbb{Z})$, and $\alpha\in GL_2^+(\mathbb{Q})$. Then one defines a "double coset operator" or "change of automorphy ...
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$disc^{2}$ divides level of CM newform?
Given a newform $g$ of weight $2$ and level $\Gamma_{0}(N)$ with CM by $K = \mathbb{Q}(\sqrt{-D})$, we have $N = MD$, where $M$ is the norm of a chosen ideal $\mathfrak{m}$ in $K$, and the Hecke ...
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Conjectured identity for the ratio of Ramanujan theta functions
Following Ramanujan, we define theta functions as follows $$\chi(q):=\prod_{n = 1}^{\infty}\left(1+q^{2n-1}\right),\\\phi(q)=\sum_{n=-\infty}^{\infty}q^{n^2},\\\displaystyle \psi(q)=\sum_{n = 0}^{\...
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Why is the Eisenstein series $G_2$ a quasimodular form?
For even $k \geq 4$, the Eisenstein series
\begin{align*}
G_k(\tau) &= \sum_{(n, m)\in \mathbb{Z}^2} \frac{1}{(m + n\tau)^k}
\end{align*}
(omitting the term $(n, m) = (0, 0)$) is a modular form ...
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Why is the j-invariant a modular function but not a modular form?
I read that the j-invariant is a modular function but not a modular form. This is confusing, because the j-invariant doesn't have poles in the upper half plane. What is the difference between modular ...
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Rankin-Cohen bracket in Sagemath
For two modular forms $f, g$ of weights $k_{1}, k_{2}$ respectively, the $n$-th Rankin-Cohen bracket of $f, g$ is given by
$$[f, g]_{n} := \sum_{j=0}^{n} (-1)^{j}\binom{n+k_{1}-11}{j}\binom{n+k_{2}-1}{...
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Why exactly is $e^{\pi\sqrt{163}}$ a near-integer? [duplicate]
Ramanujan's constant, $e^{\pi\sqrt{163}}$, is almost an integer. I know that this comes from the Laurent series for the $j$ function in terms of $e^{2\pi\tau}$:
$$j(\tau)=q^{-1}+744+196884q+\dots$$
...
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Corollary 2.43 in the Diamond-Darmon-Taylor Notes on FLT
On page 79 of the document, at the beginning of the proof of Corollary 2.43, the authors claim that if $\overline{\rho}: G_{\mathbb Q} \to \operatorname{GL}_2(k)$ is a continuous absolutely ...
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Derivative of weight-2 Eisenstein series
The weight-2 Eisenstein series $$E_{2}(z) := 1 - 24\sum_{n\geq 1} \sigma_{1}(n)q^{n}$$ is not a modular form, but is "quasi-modular" in the sense that $E_{2}(-1/z)$ is equal to $z^{2}E(z) +$(...
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Classification of automorphy factors for $\mathrm{SL}_2(\mathbb{R})$ on the upper half-plane
By an automorphy factor (or a factor of automorphy) for $\mathrm{SL}_2(\mathbb{R})$ on the upper half-plane $\mathbb{H}$, I mean a continuous map
$$j \colon \mathrm{SL}_2(\mathbb{R}) \times \mathbb{H} ...
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Proof of Eichler-Shimura isomorphism
For a congruence subgroup $\Gamma \subseteq \mathrm{SL}_2(\mathbb{Z})$ we have the Eichler-Shimura isomorphism
$$
M_k(\Gamma) \oplus \overline{S_k(\Gamma)} \cong H^1(\Gamma,V_k)
$$
with $V_k$ a ...
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Are the cuspidal and Eisenstein parts of an integral modular form also integral?
From the general theory of modular forms, we have the orthogonal decomposition:
$$M_2(\Gamma_1(N))=S_2(\Gamma_1(N))\oplus\mathrm{Eis}_2(\Gamma_1(N)).$$
So for $f\in M_2(\Gamma_1(N))$, we write $f=f_\...
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How to prove this identity involving Eisenstein series
Consider the twisted Eisenstein series
$$
\begin{align}
E_{k \ge 1}\left[\begin{matrix}
\phi \\ \theta
\end{matrix}\right](q) := & \ - \frac{B_k(\lambda)}{k!} \\
& \ + \frac{1}{(k-1)!}\...
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Echelon basis for modular forms $M_{2}(\Gamma_{0}(23))$
This is referring to Example 9.15 in William Stein's book 'Modular forms: a computational approach'.
In this example, we are to calculate the newform of weight 2 level 23 in $S_{2}(\Gamma_{0}(23))$. ...
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Gauss sum involved in the Hecke action on classical Hilbert modular forms
Let $F=\mathbb{Q}(\sqrt{D})$ be a real quadratic field and consider the classical Hilbert modular forms over $F$. Let $\varepsilon_0>1$ be the fundamental unit of $F$ and write $d=\varepsilon_0\...
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Wikipedia wrong on modular forms & line bundles?
The current (25/03/2023 12:30 UTC) version of the Wikipedia article on modular forms has a section "As sections of a line bundle" where it claims the following:
I think there are two ...
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Correspondence for $U_p$ operator
I've seen in many places that one can define the usual Hecke operator $T_p$ via correspondences by $(\pi_1)_*(\pi_2^*(\bullet))$ in the diagram
$$Y \xleftarrow{\pi_1} Y_0(p) \xrightarrow{\pi_2} Y$$
...
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Theorem 1.27 a) in Diamond, Darmon, Taylor, "Fermat's Last Theorem"
I am looking for a reference for part a) of Theorem 1.27 here regarding the proof of the growth of coefficients of cusp forms. Theorem 1.24 gives a very sketchy argument of a less specific fact, but I ...
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Subspace of newforms one-dimensional with CM $\implies$ unique newform a Poincare series.
Let $k \geq 2$. Say $S_{k}^{\text{new}}(\Gamma_{0}(N), \chi)$ is one-dimensional and spanned by a newform with CM, and $S_{k}^{\text{old}}(\Gamma_{0}(N), \chi)$ has positive dimension. Must it be true ...
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Reduced row echelon basis of weakly holomorphic modular forms have algebraic coefficients?
Let $S^{\infty}_{k}(\Gamma_{0}(N))$ be the space of weakly holomorphic modular functions for $\Gamma_{0}(N)$ whose only possible poles lie at the cusp $\infty$ and vanish at all other cusps. There is ...
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Origin of singular moduli
Books on modular forms don't usually attribute the main results to anyone. For instance, who first obtained that values of the $j$ invariant at quadratic imaginary points (singular moduli) are ...
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Connections between Modular forms and Hyperbolic geometry
I am taking a course each on Modular forms and Hyperbolic geometry currently and I have begun to like the nice connections that exists between them. I am still a beginner in both these subjects and ...
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A little proposition for Galois representations on a local field of Diamond-Shurman's book
$\def\gl{\mathrm{GL}}
\def\Q{\mathbb{Q}}
\DeclareMathOperator{\gal}{Gal}
$
I am reading the Galois representation chapter of Diamond-Shurman "A first course in modular forms".
Let $L$ be a ...
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Some questions about modular group [duplicate]
Recently I read 'A Course in Arithmetic (Serre)', and I want to know how to proof the remark in §1, chapter 7: $\left\langle S,T; S^2,(ST)^3\right\rangle$ is a presentation of $G$, or equivalently, ...
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Lie algebra sheaf of abelian varieties and the derived pushforward of its structure sheaf
Let $A$ be an abelian variety over a base scheme $S$, write $\pi: A \rightarrow S$, equipped with the zero section $e: S \rightarrow A$. Let $A^{\vee}$ be the dual variety of $A$. I am hoping to ...
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Why is the quotient of a manifold by a group action singular at a point with non-trivial stabilizer?
My question is essentially the same as the question here, and I'm not sure if I fully understand that answer. I am reading Don Zagier's chapter in The 1-2-3 of Modular Forms, and the author says
The ...
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Normalisation of $L$-function for classical modular forms and automorphic representations
I found that the normalisation of $L$-functions of classical modular forms and corresponding automorphic representations is somewhat confusing for me. Recall that if $f\in S_k(\Gamma_0(N))$ is a ...
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How to define a measure on a quotient space
I have been trying to understand the following fact, consider $\mathbb{H}$ the upper half plane of the complex numbers. And let $\Gamma_0 = SL_2(\mathbb{Z})$ act on $\mathbb{H}$ we know that there is ...
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Action of the complex conjugation on modular Galois representations
Let $p$ be an odd prime, and let $T$ be the Tate module of an elliptic curve defined over $\mathbb{Q}$, or the representation attached to a modular form or to a Hida family of modular forms. Why is it ...
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Unsolved problems for partition function
In number theory, the partition function $p(n)$ represents the number of possible partitions of a non-negative integer $n$. For instance, $p(4) = 5$ because the integer $4$ has the five partitions $1 +...
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A property of simultaneous eigenforms
While reading the following theorem for Apostol's modular functions and dirichlet series in number theory, I have a question:
(Theorem 6.14, page 130)
Assume that k is even and $k\geq 4$. If the space ...
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Generalization of Ramanujan's formulas for derivatives of Eisenstein series
Let $E_2, E_4,$ and $E_6$ be the first three Eisenstein series. There are well-known formulas due to Ramanujan for the derivatives of these quantities:
\begin{align*}
DE_2 &= \frac{1}{12}(E_2^2-...
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What is the space of weakly holomorphic modular forms of weight $\frac12$?
What is the space of weakly holomorphic modular forms of weight $\frac12$ for $\text{SL}(2,\mathbb Z)$?
My thoughts as of yet: The derivative $j_l:=(j^l)'$ of the $j$-invariant for $l\in\mathbb N$ is ...
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