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A very nice example in my eyes is Serre's proof of Riemann-Roch: Sometimes, you are just not satisfied with existing proofs, and you look for better ones, which can be applied in different situati …

Given a commutative $\mathbb{Z}[\frac1n]$-algebra $R$, we can consider the ring of modular forms $M_*(\Gamma_1(n), R)$. If $R$ is a subring of $\mathbb{C}$, these can be defined as those (holomorphic) …

Recall that for a subgroup $\Gamma \subset SL_2(\mathbb{Z})$ a modular form $f$ of weight $k$ is a holomorphic function from the upper-half plane into the complex numbers such that for any $\begin{p …

I do not think that your definition of Katz modular form is exactly correct. A (Katz) weight-$k$ modular form for $\Gamma$ is a section of the line bundle on $\mathcal{Y}(\Gamma)_R$ that evaluates on …

For any $n\geq 1$, one can define a functor $\mathcal{M}_1(n)\colon \mathrm{Schemes}/\mathbb{Z}[\frac1n] \to \mathrm{Groupoids}$, sending a scheme to the groupoid of elliptic curves over it with a cho …

Congruences between modular forms are certainly a big topic in number theory, maybe with $$E_{p-1}\equiv 1 \mod p \qquad \text{for a prime }p\geq 5$$ as the easiest example. Sometimes, $p$ might be r …

I will answer Q2: N=2: Denote by $Y^1(2)$ the moduli of elliptic curves with point of order 2 and fixed invariant differential. It is not hard to show that $Y^1(2) = \mathrm{Spec}\, \mathbb{Z}[\frac12 …