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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Products of fourier coefficients of distinct eigenforms.

Let $f(z)=\sum\limits_{n=1}^{\infty}a_f(n)q^n$ and $g(z)=\sum\limits_{m=1}^{\infty}b_g(m)q^m$ be two distinct eigenforms in $S(k,N)$, the space of holomorphic cusp forms of weight $k$ and level $N$. Can something be said about the product…
Neha
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Genus of a modular curve mod p

I am reading a paper that states the following without reference: The modular curve $X_0(\mathcal{l})/p$, $l \neq p$ has genus $[\mathcal{l}/12]$. Does anyone know where this comes from? I am finding a ton of references that study the genus of the…
Tom Lewia
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$f$ weakly modular. How to prove that the order of vanishing $v_p(f)$ is well defined for $p\in SL_2(\mathbb{Z}) \setminus H$

I'm studying the proof of Lemma 1.6.6 in this Lecture notes page 16 Lemma If $f$ is weakly modular of weight $k,$ the order of vanishing $ v_p(f)$ is well defined for $p\in SL_2(\mathbb{Z}) \setminus H.$ I have difficulty to understand the…
Med
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Modular forms have values as algebraic numbers?

Can you find examples of modular forms which take on values as e.g. algebraic numbers of degree n ? I'm interested in finding special classes of algebraic numbers particularly when n=3, they don't have to come from modular forms.
hello
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Explicit descriptions of the modular curves $Y(5)$ and $Y_1(5)$

The modular curves $Y(5)$ and $Y_1(5)$ associated to the congruence subgroups $\Gamma(5)$ and $\Gamma_1(5)$ are of genus 0. As such, they are nothing but the Riemann sphere $\mathbb P^1$ minus a finite subset of cusps: $$ Y(5)=\mathbb P^1\setminus…
Lucien
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field of rational functions of a curve

Let $C$ be the algebraic curve defined by the modular polynomial $\phi_N$ of order $N>1$ over the rational numbers, i.e. \begin{equation}C:=\text{specm}(\mathbb{Q}[X,Y]/\phi_N(X,Y)). \end{equation} The singularities of this curve can be removed and…
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Proving that The set of limit points is not empty in an infinite group of linear fractional transformation.

suppose S is an infinite group of linear fractional transformation , show that the set of limit points of S is not empty . I'm studying a modular form course , and I got stuck in this question ,i tried to use the stabilizer ,or to start from SL2(Z)…
Butterfly
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coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$

I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is greater than 1( The additive claim that i hope to reach is clear to the…
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Congruence and first odd primes

I tried to find the solutions to this modular equation: $3^{(5+7+11+13+17+19+\dots +p(m-3)+p(m-2))} \equiv p(m-1) \bmod p(m) $ where $p(m)$ is the m-th odd prime number(note that it's three to the power of 5+7+11+13+...). The only solution that I…
Blue Sky
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Action of automorphisms on Eisenstein series

Let $ f \in \mathcal{M}_{k}(\Gamma) $ and $ \sigma \in \textit{Aut}(\mathbb{C}) $. Suppose $$ f = \sum_{n=0}^{\infty}a_{n}q^{n} .$$ Then we define the action of $ \textit{Aut}(\mathbb{C}) $ on $ \mathcal{M}_{k}(\Gamma) $ by $$ f^{\sigma} =…
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Residue theorem and Angle of modular function

Let $f$ be a meromorphic function on the region $Im(z)>0$, $v_p(f)$ be the order of $p$. (The number $n$ such that $\frac{f(z)}{(z-p)^n}$ is holomorphic and non-zero at $p$.) Moreover, assume $f$ is a $modular$ $function$ for $SL(\mathbb Z)$ of…
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Dirichlet Series of weakly multiplicative characters

Let $$ \sum_{n=1}^{\infty} a(n)n^{-s} = \prod_{p}\left((1-\alpha_{p}p^{-s})(1-\alpha_{p}'p^{-s})\right)^{-1},$$ where $ a(n) $ is weakly multiplicative (i.e $ a(n)a(m) = a(n,m) $ if $ \textit{gcd}(m,n) = 1 $). Then the claim is that $$ a(p^{n}) =…
cfairwea
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How to construct this modular Galois representation?

I want to know how to construct this modular Galois representation: The $\bmod p$ representation is semi-simple. For any lattice $T$ , $\dfrac{T}{p^{3}T}$ does not have a $G$-stable cyclic subgroup of order $p^3$ (i.e., there does not exist the…
Yan Dong
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Why is it so difficult to use geometric methods to construct weight one modular forms?

I am trying to understand why it is that geometric methods of constructing modular forms of weight one for $SL(2,\mathbb Z)$ fail. I have a relatively rudimentary understanding of it, but it would be helpful if someone could give me an overview?
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A question based on proving a result Related to Klein J function

I was unable to solve some questions asked in my mid term of number theory exam and so I am asking it here. If $\tau \in H$ and $x= e^{2\pi i \tau}$, prove that $[{504 \sum_{n=0}^{\infty}\sigma_{5}(n) x^n}]^2 = [j(\tau) -{12}^3 ]…
Avenger
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