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        Questions tagged [modular-forms]

        Questions about modular forms and related areas

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        35 views

        Constant map from automorphic form is surjective?

        Let $G$ be a connective 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 definition of these ...
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        91 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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        112 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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        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 ...
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        323 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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        “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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        177 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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        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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        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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        133 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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        169 views

        Differential equations satisfied by quasi modular forms?

        It is known that modular forms are solutions of differential equations. More precisely, let me cite the statement from the following question. Differential Equations Satisfied by Modular Forms ...
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        58 views

        generalization of the discreteness of Hecke groups to general reductive groups

        Consider the subgroup $G_{\lambda}$ of $SL_2(\mathbb R)$ generated by $N_{\lambda} = \begin{bmatrix} 1 & \lambda \\ 0 & 1 \end{bmatrix}$ and $S = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{...
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        34 views

        Product of $V_N$ operator(index changing) and its adjoint on Jacobi forms

        In Kohnen & Skoruppa's 1989 inventiones paper, page 549, the operator $V_N: J_{k,1}^\text{cusp} \longrightarrow J_{k,N}^\text{cusp}$ is defined by the action $$ \sum_{\substack{D<0,r \in \...
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        277 views

        Riemann–Hilbert-type problem

        Let $P$ be a fixed pentagon in the hyperbolic plane $\mathbb H^2$ with all the angles equal to $\pi / 3$. Let $w_1, w_2, \dots, w_5$ be the sides of $P$ going in the counterclockwise order. We are ...
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        67 views

        Relation between Fourier coefficients of two (weakly holomorphic) modular forms

        Assume that $\Gamma \leq \mathrm{SL}(2, \mathbb{Z})$ be a certain discrete subgroup (for example, congruence subgroups) and let $\Gamma' = \alpha\Gamma\alpha^{-1}$ for some $\alpha\in \mathrm{SL}(2, \...

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