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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Infinitely $more$ algebraic numbers $\gamma$ and $\delta$ for $_2F_1\left(a,b;\tfrac12;\gamma\right)=\delta$?

Given the complete elliptic integral of the first kind $K(k_\color{blue}m)$, Dedekind eta $\eta(\tau)$, j-function $j(\tau)$, and hypergeometric $_2F_1\left(a,b;c;z\right)$ with $\color{brown}{a+b=c=\tfrac12}$. Conjecture: "The equalities hold for…
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An infinite product for $\left(\frac{\eta(13\tau)}{\eta(\tau)}\right)^2$?

Given the Dedekind eta function, $$\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$$ where $q = \exp(2\pi i\tau)$. Consider the following "family", $\begin{align} \left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{24} &= \frac{u^8}{(-1+16u^8)^2},\;\;\;…
Tito Piezas III
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Whats the difference between modular forms of different levels?

We have a natural surjective group homomorphism: $\phi : SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/(n\mathbb{Z}))$ from which, given any subgroup $H
Eins Null
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What are applications of number theory in physics?

I was reading Goro Shimura's The Map of My Life. He wrote the following quote in the book. It made me come up with the title question. In particular, is there any application of the theory of modular forms in physics? A well known math-physicist…
Makoto Kato
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Show that $\prod\limits_{n=1}^\infty \frac{(1-q^{6n})(1-q^n)^2}{(1-q^{3n})(1-q^{2n})}=\sum\limits_{n=-\infty}^\infty q^{2n^2+n}-3q^{9(2n^2+n)+1}$.

Show that $\displaystyle \prod_{n=1}^\infty \frac{(1-q^{6n})(1-q^n)^2}{(1-q^{3n})(1-q^{2n})}=\sum_{n=-\infty}^\infty q^{2n^2+n}-3q^{9(2n^2+n)+1}$. I can't seem to be able to proceed with this question. I know that $\displaystyle \prod_{n=1}^\infty…
Icycarus
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Hecke operators on modular forms

Would you please explain the importance of Hecke operators on modular forms? I am studying modular forms mostly on my own and I have a pretty good understanding up to Hecke operators. So, I just wonder why we care about them.
GeoffDS
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Why is $12$ the smallest weight for which a cusp form exists?

On wikipedia (here) I have read the following: Twelve is the smallest weight for which a cusp form exists. [...] This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the…
Martin
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Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some links to it. Let $q=e^{2\pi\mathrm{i}\tau}$,…
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Counting points on the Klein quartic

In Moreno's book "Algebraic Curves over Finite Fields", he mentions the following in passing with no further comments ($K$ denotes the Klein quartic defined by $X^3 Y + Y^3 Z + Z^3 X = 0$): The Jacobian of $K$ is a product of three elliptic curves…
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Connection between Hecke operators and Hecke algebras

Hecke operators are things that act on modular forms and give rise to a lot of interesting arithmetical results: http://en.wikipedia.org/wiki/Hecke_operator On the other hand on the wikpedia page for Hecke algebras, which should naively be an…
Denham Forrest
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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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Representing a number as a sum of at most $k$ squares

Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to $$ n = x_1^2 + x_2^2 + \dots + x_k^2$$ with $x_1 \ge x_2 \ge \dots…
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A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain things(infact many) which i couldn't follow and i would like…
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Why $f(z+1)=f(z)$ implies $f$ can be expressed as a function of $e^{2\pi iz}$

I am reading modular forms from J.P.Serre's book, where I came across a complex function which satisfies property $f(z+1)=f(z)$. Then, it is mentioned that we can express $f$ as a function of $e^{2\pi iz}$. I can see that any function expressed as a…
roydiptajit
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Weierstrass elliptic functions and ordinary differential equations

I am studying Elliptic functions for a University project with a particular focus on Weierstrass's theory. For the past few weeks I have been studying various basic properties of the $\wp$ function (the majority of the Elliptic functions section in…
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