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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What is a form?

I have read about differential forms, bilinear forms, quadratic forms and some other r-linear forms but I still have this shred of doubt in my mind on what exactly is a form. I have an assumption that it is to a ring what a vector is to a field. I…
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How does one graduate from Hecke Operators to Hecke Correspondences?

I've read (skimmed heartily) basic books on the topic of modular forms. (The last being Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.) I strive for an understanding which is as mathematically mature as possible. (Read: strive for…
Jenifer
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How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac13,\frac13;\frac56;-\alpha\right)=\beta\,$?

(Note: This is the case $a=\frac13$ of ${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}.\,$ There is also $a=\frac14$ and $a=\frac16$.) In a post,…
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Proving finite dimensionality of modular forms using representation theory?

It is well known how to use algebraic geometry (differentials, divisors, and Riemann-Roch) in order to prove the finite dimensionality of the vector space of modular forms of some fixed weight and level. Is it possible to prove the finite…
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Proving that $\sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m)\sigma_3(n-m)$ without using modular forms?

This problem appears as a (starred!) exercise in D. Zagier's notes on modular forms. I have to admit that I have no idea how to do it. Here, $\sigma_k(n) =\sum_{d\mid n} d^k$, as usual. This identity is traditionally obtained by using the fact that…
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Why is $\sum\limits_{n=1}^{\infty}e^{-(n/10)^2}$ almost equal to $5\sqrt\pi-\frac12$ (agreeing up to $427$ digits)?

The following I saw as an exercise in a text on modular forms. I am lacking the understanding for why the following is true, but it is nevertheless astonishing: Why is the numerical value of $$x:=\sum_{n=1}^{\infty} \exp(-(n/10)^2)$$ so ridiculously…
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How to show that $ \int_0^\infty \frac{1}{\sqrt {t(t^2+21t+112)}} dt=\frac{1}{4\pi\sqrt {7}}\Gamma(\frac{1}{7})\Gamma(\frac{2}{7})\Gamma(\frac{4}{7})$

I've already known that $$ \int \limits_0^\infty \frac{1}{\sqrt {t(t^2+21t+112)}} \mathrm{d}t=\frac{1}{4\pi\sqrt {7}}\Gamma(\frac{1}{7})\Gamma(\frac{2}{7})\Gamma(\frac{4}{7})$$ To get this answer, I let $\mathrm{d}u=\mathrm{d}\sqrt{t}$, then got $$…
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Galois Properties of the Values of Modular Forms

Let $f \in S_0(N)$ be a normalized Hecke eigenform. It is well known that its coefficients are algebraic integers, and $f^\sigma$ lies in $S_0(N)$ for $\sigma \in G_{\mathbb{Q}}$. At CM points $z \in \mathbb{H}$, $f$ takes an algebraic value as…
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Modular interpretation of modular curves

Let $\Gamma$ be a congruence subgroup of level $N$. What is the modular interpretation of $\Gamma\backslash \mathcal{H}^*$, by that I mean what are the elliptic curves + additional structure it parametrizes. I know the answer when $\Gamma$ is one of…
Zorba le Grec
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How did Hecke come up with Hecke-operators?

I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d, \, d > 0} d^{-k } f\left( \frac{az +…
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About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-_{3}F_{2}\left(\frac{1}{4},\frac{1}{4},\frac{1}{2};\frac{5}{4},\frac{5}{4};-1\right).\tag{1}$$ Classical approaches seem to lead…
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Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$?

If $n$ is a natural number, let $\displaystyle \sigma_{11}(n) = \sum_{d \mid n} d^{11}$. The modular form $\Delta$ is defined by $\displaystyle \Delta(q) = q \prod_{n=1}^{\infty}(1 - q^n)^{24}$. Write $\tau(n)$ for the coefficient of $q^{n}$ in…
Richard G
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Converting an infinite product to sum; Ramanujan $\tau$ function

I've gotten what seems most of the way, but I'm quite stuck at this point. Define $\tau(n)$ by \begin{align*} q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty\tau(n)q^n. \end{align*} Prove that $\tau(n)$ is odd if and only if $n =…
Alex
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How should I think about Ihara's Lemma?

I am trying to learn about Ihara's lemma because I see it mentioned in many papers in arithmetic geometry. To be more precise, I would like to know: What is the significance of this result? Why is it so powerful? Intuitively, how should I think…
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How to prove that $\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4)$?

How can we prove that $$\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4) \ ?$$ Here, $\eta(q)$ is the Dedekind Eta Function, which is defined by $$\eta(q)=q^{1/24}\prod_{n=1}^\infty…
Shobhit
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