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      Questions tagged [modular-forms]

      Questions about modular forms and related areas

      6
      votes
      1answer
      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 $\Gamma\subset \operatorname{SL}_2(\mathbb Z)$. ...
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      votes
      0answers
      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{...
      3
      votes
      1answer
      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 ...
      1
      vote
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      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 $J\geq 0$: \begin{equation} f(q,x)=\sum_{n=-J}^\infty q^{n^2}x^n. \end{equation} My question is whether ...
      2
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      0answers
      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 ...
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      votes
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      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 ...
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      votes
      0answers
      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 ...
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      0answers
      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 ...
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      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 ...
      6
      votes
      2answers
      331 views

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

      Given the Dedekind eta function $\eta(\tau)$, define, $$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$ $$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$...
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      vote
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      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 ...
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      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. ...
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      vote
      0answers
      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....
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      votes
      1answer
      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 ...
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      0answers
      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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