# 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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### $N$-th root of modular forms

In my research, I have encountered many $q$-series of functions that turn out to be the Fourier expansions of roots of modular forms. Examples are $n$-th roots of Eisenstein series and the $j$-...
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### Is there a dimension formula for modular forms of weight $1$ for any congruence subgroup?

If $M_1(\Gamma )$ denotes the vector space of modular forms of weight $1$ for $\Gamma$, is there any formula for $\text{dim}(M_1(\Gamma ))$?
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### A linear operator on meromorphic functions

I am currently reading Lang's book Introduction to Modular Forms, and perhaps I'm wrong, but I may have spotted something weird he said. Either I don't get it, or there's something abnormal about the ...
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### Limit of $\sum_{d}\sum_{c\neq0}\left(\frac{1}{cz+d}-\frac{1}{cz+d+1}\right)$

To evaluate $$\sum_{d\in\mathbb Z}\sum_{c\in\mathbb Z\smallsetminus\{0\}}\left(\frac{1}{cz+d}-\frac{1}{cz+d+1}\right)$$ where $z$ on the upper half plane of $\mathbb C$. We first consider a finite ...
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### Linear independence of $G_4$ and $G_6$ in entire modular forms

It is known that( it is a theorem) every entire modular form of weight k is a polynomial in $G_4$ and $G_6$ of the type f = $\sum_{a, b} c_{a, b} G_{4}^a G_{6}^b$ where the $c_{a, b}$are complex ...
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### Lower bound for order of pole at $\infty$ of modular functions level $N$

If the modular curve $X_{0}(N)$ has genus $0$, its function field is generated by a single element $j_{N}(z)$ with pole of order $1$ at the cusp $\infty$ and holomorphic at all other cusps (this is ...
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### Modularity in terms of the stack of elliptic curves

Modular forms can be viewed as something like global sections of (some tensor power of) the canonical line bundle on the stack $\mathcal M_{\text{ell}}$ of (generalized) elliptic curves. Is it ...
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### Various variants of modular forms

There are many different generalizations of modular forms. One has Hilbert modular forms, Siegel modular forms, Maass wave forms, Jacobi forms, and then there are various generalities of automorphic ...
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### Proof of Holomorphicity of Eisenstein Series in Serre's 'A Course in Arithmetic'

I was reading the proof of Proposition $4$ in Chapter VII of Serre's 'A Course in Arithmetic'. In order to prove that the Eisenstein Series $G_k(z)$ of index $k>1$ is holomorphic on the upper half ...
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### Identifying Modular Functions of Weight $2k$ with Lattice Functions of Weight $2k$

I have been reading Serre's 'A Course in Arithmetic'. In section $2.2$ he shows that lattice functions $F$ of weight $2k$ are in one-to-one correspondence with modular functions of the same weight, by ...
I was reading the chapter on Modular Forms from Serre's 'A Course in Arithmetic'. In the first line of his proof that the Eisenstein Series $G_k(z)$ is a modular form, he says, "The above arguments ...