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The standard reason why $e^{\pi\sqrt{N}}$ is a near integer for some $N$ is that there is some modular function $f$ with $q$-expansion $q^{-1} + O(q)$, such that substituting $\tau = \frac{1 + i\sqrt …

I can say a little about the work of Brown and Schretz, since Brown gave a talk at BIRS last month. If you take a graph with $N$ edges and some restriction on valence (called a Feynman graph), there …

A bit of mastication of Katz-Mazur Theorem 13.7.6 and the surrounding text seems to yield the following description of the special fiber of $Y(p)$: It is fundamentally $p+1$ copies of $\mathbb{P}^1$ …

In addition to the power-series methods using Eisenstein series and $\Delta$, and the modular equation methods using, e.g., $h_5$ given in the other answers and comments, there are transcendental meth …

We can say something stronger. Theorem: (Helling 1976) Consider the family of subgroups of $SL_2(\mathbb{C})$ that are commensurable with a conjugate of $SL_2(\mathbb{Z})$. The maximal elements o …

The two-variable elliptic genus is a topological invariant of almost-complex manifolds that takes values in power series. These power series turn out to describe weak Jacobi forms when the manifold i …

Following the comments, here is perhaps the simplest counterexample (once you know the Breuil-Conrad-Diamond-Taylor modularity theorem). The curves $y^2 + y = x^3$ and $y^2 + y = x^3 + 2x$ both reduc …

You can think of the space of positively oriented covolume-one bases of $\mathbb{R}^2$ as a torsor under $SL_2(\mathbb{R})$, i.e., it is a manifold with a simply transitive action of the group. If yo …

The thrice-punctured sphere can be represented as a quotient of the upper half-plane by $\Gamma(2)$. One may take as a fundamental domain the region contained in the geodesics between $i\infty, 0, 1, …

Edit: Here's a rather silly method that should work if SAGE is just giving you cusp forms: $\Gamma_0(4)$ has a single normalized cusp form of weight 6, given by $\eta(2\tau)^{12} = q - 12q^3 + 54q^5 …

Supersingular primes are those primes p for which all supersingular elliptic curves over an algebraic closure of Fp have j-invariant in Fp. There is a theorem of Deuring that implies the j-invariant …

Characters of rational vertex operator algebras tend to yield modular functions. This is due to the space of torus partition functions in a chiral conformal field theory being a complex moduli invari …

$Y_0(M,N)$ can be reinterpreted as the moduli space of diagrams $E_1 \to E \leftarrow E_2$ of elliptic curves, where the arrows are cyclic isogenies of degree $M$ and $N$. From this viewpoint, it is …

The coefficients of a modular form of non-positive weight can be given by an explicit formula that depends only on the poles at cusps (and constant terms when the weight is zero). The asymptotics are …

I'm not an expert on black holes, but I can give you a couple pointers. From work of Bekenstein and Hawking in the 1970s, we are pretty sure that macroscopic black holes in our 3+1 dimensional univer …