# Questions tagged [modular-forms]

Questions about modular forms and related areas

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### Computing coefficients of theta functions associated to quadratic forms

If we take an integral positive definite quadratic form $Q$ and set $\Theta_Q(z) = \sum_{k\geq 0}R_Q(k)e^{2\pi ikz}$, what are the most efficient algorithms to compute the $R_Q(k)$? I am aware e.g. of ...
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### Freeman Dyson's approach to string theory [closed]

Context: In celebrating the centenary of Ramanujan's birth, Freeman Dyson presented the following career advice for talented young physicists : My dream is that I will live to see the day when our ...
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### Explicit Chebotarev density theorem for Galois representations associated to newforms

Let $f \in S_2(\Gamma_0(N))$ be a newform with associated residual Galois representation $\rho: \operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \operatorname{GL}_2(\mathbf{F})$, $\mathbf{F}$ ...
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### Characterization of tori/elliptic curve isogenies

I am reading Chapter 11 of Dale Husemöller's Elliptic Curves Springer book and I got stuck on Theorem (1.4) (c.f., image below). Notation and definitions: Let $L$ and $L'$ be two complex lattices ...
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### Representation theory of $\text{SL}(2,\mathbb{Z})$

The group $\text{SL}(2,\mathbb{Z})$ is the group of two-by-two matrices with integer entries and determinant one. This is a very simple definition. Yet its representation theory seems quite wild to me ...
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### Modular discriminant and ordinary/supersingular points

Let $p=2,3,5,7,13$. Denote by $\Delta$ the modular discriminant. How can I prove that if $z\in X_0(1)$ is a point of supersingular reduction, then $v_p(\Delta(z))=0$ ? If $z\in X_0(1)$ is a point of ...
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### Langlands program and complexity theory

Back when I was an undergraduate, I spent some time reading the about the modularity conjecture, but the details are fuzzy now. One of the motivations I imagined for the Langlands program was for ...
As mentioned in this answer there is a conjecture by Kimball Martin that, formulated slightly informally, has the following special case. Conjecture: On average $J_0(p)$ has 2 simple components when ...