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# Questions tagged [modular-forms]

Questions about modular forms and related areas

931 questions
531 views

### Galois theory of modular functions

Let $\mathcal M_m$ be the set of $2$-by-$2$ primitive (relatively prime entries) matrices with determinant $m$. Let $\alpha \in \mathcal M_m$ and let $\Gamma\subset \operatorname{SL}_2(\mathbb Z)$. ...
96 views

### Newman's conjecture of Partition function

(Sorry for my poor english....) Let $p(n)$ be a partition function and $M$ be an integer. Newman conjectured that for each $0\leq r\leq M-1$, there are infinitely many integers $n$ such that \begin{...
184 views

### Why are Poincare series defined as they are?

We know the Poincare series are defined as the following: The $m^{th}$ Poincare series of weight $k$ for $\Gamma$ is: $$P_{m}^{k} (z) = \sum_{(c,d)=1} (cz+d)^{-k} e^{2 \pi in(\tau z)}.$$ The ...
66 views

### theta function with a low bound in the sum

I encounter a sum similar to the Jacobi Theta function except there is a lower bound $-J$ with $J\geq 0$: $$f(q,x)=\sum_{n=-J}^\infty q^{n^2}x^n.$$ My question is whether ...
87 views

### How to express the cuspidal form in terms of Poincare series?

Sorry to disturb. I recently read Blomer's paper. There is a blur which needs the expert's help here. Blomer's paper says "For any cusp form $f$ of integral weight $k$, level $N$, $f$ can be written ...
108 views

### Is constant map from automorphic form surjective?

Let $G$ be a connected reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of $G$ and and $K$ a 'good' maximal compact subgroup of $G$. (For precise definitions of these ...
97 views

### Automorphy Factor from Vector Bundles on Compact Dual

So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...
135 views

### Additive and multiplicative convolution deeply related in modular forms

From the fact spaces of modular forms are finite dimensional, from the decomposition in Hecke eigenforms, and from the duplication formula for $\Gamma(s)$ there are a lot of identities mixing additive ...
80 views

### Deforming Modular Symbols

This is probably a silly question, as I don't really know a whole lot about modular symbols over arbitrary rings. How do modular symbols over a finite field square with Katz modular forms? If they ...
331 views

### Are these 5 the only eta quotients that parameterize $x^2+y^2 = 1$?

Given the Dedekind eta function $\eta(\tau)$, define, $$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$ $$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$...
98 views

### “Modularity” of generalized theta series

The most basic theta series is defined by $\theta(z)=\sum_{n=-\infty}^{\infty}q^{n^2}$, where $q=e^{2\pi iz}$. This is connected to the question of how many representatations does an integer $n$ have ...
189 views

### What role do Hecke operators and ideal classes perform in “Quantum Money from Modular Forms?”

Cross-posted on QCSE An interesting application of the no-cloning theorem of quantum mechanics/quantum computing is embodied in so-called quantum money - qubits in theoretically unforgeable states. ...
82 views

### Igusa curve at infinite level

In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent....
115 views

### Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms

Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients. Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know the ...
145 views

### Geometric interpretation of the rationality of the $j$-invariant

Consider the modular curve $X_0(N)$. Let $\Phi_N(X,Y)$ be the modular equation. Then the curve $\Phi_N(X,Y) = 0$ can be interpreted as a model for $X_0(N)$ because the function field of $X_0(N)$ is \$\...

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