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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.

In mathematics, 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, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology and string theory (Wikipedia).

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Is there a general way to prove series and products are modular?

The following$$\eta(q)=q^{1/24}(q)_\infty$$ $$E_{n}(z)=\sum_{z \in \Lambda\setminus \lbrace0\rbrace}z^{-n}$$ $$F(q)=q^{-1/60} \sum_{n \ge0} \frac{q^{n^2}}{(q;q)_\infty}$$ $$F(q)=q^{11/60} \sum_{n \ge0} \frac{q^{n^2+n}}{(q;q)_\infty}$$ and many other…
Meow
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What are modular forms used for?

I have seen the definition of a modular form, but it seems obscure to me. I get the impression that if I were to read a lot about them, eventually I would see how they can be used. I am curious about the ways in which modular forms are applied. How…
JessicaB
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j-invariant Fourier expansion

I'm reading about Fourier expansion of modular functions, but I have trouble understanding the following equation: Is it some inherent property of the denominator, as it is?
Pavel
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Eigenvalues of Hecke operators are algebraic integers

I want to show that if a normalised modular form of level $1$, say $f$, is a simultaneous eigenform of the Hecke operators $T(n)$ for all $n$, then the corresponding eigenvalues are all algebraic integers. My approach goes like this: Let…
MathManiac
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General questions about Eisenstein series and modular forms

I am in a situation where I am trying to get a feel for modular forms type stuff, but don't have anyone to talk to about it (I'm not in academia at the moment). I would like to test my understanding of Eisenstein series and modular forms by asking a…
user21725
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Fourier expansion of Hilbert Eisenstein series

Suppose $F=\mathbb{Q}(\sqrt{D})$ is a real quadratic field with class number 1. In J. Bruiner's article "Hilbert modular forms and applications", the Fourier series of Eisenstein series $$ G_{k, \mathcal{O}_{F}}(z_{1}, z_{2})=\sum_{(c,d)\in…
Seewoo Lee
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eisenstein part of theta function

If $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the $\textbf{theta series}$ $\theta_Q(z):=\sum_{m\in\mathbb{Z}^{2k}}q^{Q(m)}\ (q=e^{2\pi i z})$ is a modular…
Abdullah.Y
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Approximation of $e^{\pi\sqrt{n}}$ using Ramanujan's Class Invariants

From Wikipedia article we get $\displaystyle \begin{aligned}e^{\pi\sqrt{43}} &\approx 884736743.999777466 \\ e^{\pi\sqrt{67}} &\approx 147197952743.999998662454\\ e^{\pi\sqrt{163}}&\approx 262537412640768743.99999999999925007 \end{aligned}$ In his…
Paramanand Singh
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How are modular forms fundamental operations?

The German mathematician Martin Eichler once stated that there were five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms. This was also mentioned in Simon Singh's book on Fermat's last…
Michael Ulm
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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 form (i.e. $S_2(\Gamma_0(4))=\{0\}$), but I am…
Siegmeyer of Catarina
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Number of cusps of an modular curve $X_0(N)$

Let $X_0(N) = \Gamma_0(N) / (\mathbb{H} \cup \mathbb{P}^1(\mathbb{Q}))$. A lecture note (p. 2) lists the following (very easy!) formula for the corresponding cusps: $$\nu_\infty = \sum_{d\mid N} \varphi(\gcd(d,N/d)).$$ Unfortunately, there is no…
mikemike
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An identity about the Dedekind $\eta$ function

Let $\eta$ be the Dedekind eta function. Show that $\dfrac{\eta(q^9)^3}{\eta(q^3)}=\displaystyle\sum_{a,b\in \mathbb{Z}^2}q^{3(a^2+b^2+ab+a+b)+1}$. I'm pretty sure the RHS is equal to $\theta_2(q^3)\psi_6(q^9)+\theta_3(q^3)\psi_3(q^9)$, but I'm not…
whetham
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Need of cusps (with respect to Modular Forms)

In every introduction about Modular Forms (on $SL_2(\mathbb{Z})$ and congruence subgroups) one reads the term 'cusps'. A Modular Form should be holomorphic in the cusps. Can anybody explain to me, what is so special about cusps? Why is it necessary…
user337073
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Are the eigenvalues of the Hecke operators always real?

Are the eigenvalues of the Hecke operators $T_n$ for $M_k(\text{SL}_2(\mathbb{Z}))$ always real? I think I have an answer but I am not confident with my arguments. If $f$ is a normalized eigenform, then $f$ have real Fourier coefficients. And if $f$…
Vicman Aricheta
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Finiteness of Dimension of $M_{k}(\Gamma)$

Let $M_{k}(\Gamma)$ denote the space of weight $k$ modular forms for the congruence subgroup $\Gamma$. Are there any proofs of the finiteness of the dimension of $M_{k}(\Gamma)$ that don't rely on Riemann-Roch?
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