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        Questions tagged [modular-forms]

        Questions about modular forms and related areas

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        3
        votes
        1answer
        214 views

        Questions about modular forms and the role of monodromy

        Let $N \geq 3$ and let $\Gamma=\Gamma_1(N)$, so that the moduli problem for elliptic curves with $\Gamma$-structure is fine. Let $Y=Y(\Gamma)$ be the corresponding moduli space. In this context, one ...
        5
        votes
        0answers
        54 views

        Half-integral weight slash operator

        I am aware that the correct way to look at modular forms of half-integral weight is on the metaplectic cover of $\mathrm{SL}_2(\mathbb R)$. Assume however that we insist on considering them as ...
        3
        votes
        0answers
        116 views

        Kronecker limit formula, modular curves, and the class number problem

        Let $$Q(x,y)=ax^2+bxy+cy^2$$ be a positive definite quadratic form with $a>0$ and $D=b^2-4ac<0$. Let $$\zeta_Q(s)=\sideset{}{'}\sum_{m,n}Q(m,n)^{-s},$$ the accent indicating that $(0,0)$ is ...
        1
        vote
        0answers
        99 views

        Level Lowering Galois representations over Totally real fields

        Let $F$ be a totally real number field and $\mathbb{F}$ a finite field. Let $\bar{\rho}:\text{Gal}(\bar{F}/F)\rightarrow \text{GL}_2(\mathbb{F})$ an irreducible Galois representation arising from a ...
        12
        votes
        1answer
        232 views

        How does one compute the Hecke algebra acting on modular forms?

        I asked this on mathstackexchange, but got no answer. Let $N\geq1$ be an integer, and let $\mathbb{T}$ be the Hecke algebra acting on the cusp forms of weight k and level $\Gamma_0(N)$. Then $\...
        0
        votes
        0answers
        162 views

        What is definition of Cohen–Eisenstein series?

        I only find Cohen–Eisenstein series of weight 3/2 (in the paper A note on the Fourier coefficients of a Cohen-Eisenstein series). I founded some general definition in "Modular Forms with Integral and ...
        6
        votes
        0answers
        158 views

        Diophantine applications of Paramodularity

        I’ve asked this question to quite a few people in person and so far haven’t seen a good answer...but I believe one should exist, so here goes! Ok, we all know how to (roughly) prove Fermat’s Last ...
        2
        votes
        0answers
        41 views

        Algebraic independece of modular forms for Fricke group over $\mathbb C(q)$

        Let $q=e^{\pi i \tau}$ for $\tau \in \mathbb H$ and $$ P_2(q)=\frac{P(q)+2P(q^2)}{3}, \quad Q_2(q)=\frac{Q(q)+4Q(q^2)}{5}, \\ R_2(q)=\frac{R(q)+8R(q^2)}{9}, $$ a Fourier series of quasi-modular form ...
        5
        votes
        2answers
        383 views

        About different cohomology theories used to study Shimura varieties

        The classical theory of Eichler-Shimura realizes the space of cusp forms of certain weights and levels in the parabolic cohomology of modular curves, which is the image of the cohomology with compact ...
        2
        votes
        1answer
        124 views

        Characterizing a modular form via its first Fourier coefficients at infinity

        It is well known that a cusp form $$ f = \sum_{n\ge 1}a_n q^n $$ of weight $k$ and level $1$ is determined by its first $d_k = \text{dim } S_k$ coefficients. This follows from the valence formula (...
        3
        votes
        1answer
        194 views

        Сomplement of the set of numbers of the form $ 4mn - m - n$?

        Numbers of the form $4mn-m-n$ where $m,n\in\mathbb{Z}^+$ are $$ A=\{2, 5, 8, 11, 12, 14, 17, 19, 20, 23, 26, 29, 30, 32, 33, 35, \ldots\} $$ The set complement of the above set is $$ B=\{1, 3, 4, 6, ...
        3
        votes
        0answers
        67 views

        Maass forms associated with Ramanujan's mock theta functions

        If you perform a fulltext search on the L-functions and modular forms database for "Ramanujan", you get entries related to the $\tau$ function and the modular discriminant $\Delta$. If you want the ...
        2
        votes
        1answer
        97 views

        Complex L-functions for Hermitian modular forms?

        Fix an imaginary quadratic field $K$, and let $\mathcal{O}_K$ be its ring of integers. A Hermitian modular form of genus 1 (i.e., an automorphic form on $GU(1,1)$) of weight $(k_1,k_2)$ on a ...
        8
        votes
        0answers
        141 views

        Representation of the space of lattices in $\Bbb R^n$

        The space of 2D lattices in $\Bbb R^2$ can be represented with the two Eisenstein series $G_4$ and $G_6$. Each lattice uniquely maps to a point in $\Bbb C^2$ using these two invariants, and the points ...
        2
        votes
        0answers
        58 views

        Is there an analogue of theta cycles for more general mod p automorphic forms?

        The theory of $\theta$-cycles (due to Tate, I think) and filtrations is to me a very beautiful and powerful tool in proving many statements about mod $p$ modular forms in more explicit and elementary ...

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