# 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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### Why aren't holomorphic modular forms bounded?

Let $f$ be any non-zero integral weight (holomorphic) modular form with respect to $SL_2(\mathbb{Z})$ and of weight $k, k\geq 4$. Since it is holomorphic at infinity, for given $\epsilon > 0$, it ...
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### About Sturm's bound

The next theorem is known as Sturm's bound. Theorem:Let $\mathfrak{m}$ be a prime ideal in the ring of integers $\mathcal{O}$ of a number field $K$, and let $\Gamma$ be a congruence subgroup of of ...
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### Modular parametrization from equality of $L$-functions

In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways 1) if there exists a nonconstant morphism $X_0(N) \to E$, 2) if there exists a modular form $f$ ...
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### Show that $(f|M)(z) := (cz+d)^{-r/2}f(Mz)$ has a weight $r/2$.
The following text is from Complex Analysis by Freitag : For $r ∈ \mathbb{Z}$ the modified Petersson notation is defined : $$(f|M)(z) := \sqrt{cz+d}^{-r}f(Mz)$$ for $M ∈ SL(2, \mathbb{Z})$. In the ...