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      Questions tagged [modular-forms]

      Questions about modular forms and related areas

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      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 ...
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      2answers
      360 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 ...
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      1answer
      120 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 (...
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      1answer
      126 views

      complement 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, ...
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      63 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
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      1answer
      95 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 ...
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      139 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 ...
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      56 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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      57 views

      Reference for Selmer-Group coming from Galois representation associated with modular form

      Is there any good reference (lecture notes) for the construction of Selmer-Groups associated with the Galois representation? In particular, i want to understand how they are using Deligne's and Mazur'...
      5
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      1answer
      764 views

      Origin of Hecke operators

      What is the original paper in which Erich Hecke had first introduced the Hecke operators?
      21
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      1answer
      366 views

      Which even lattices have a theta series with this property?

      This is a slight generalization of a question I made in Math StackExchange, which is still unanswered after a month, so I decided to post it here. I am sorry in advance if it is inappropriate for this ...
      2
      votes
      1answer
      375 views

      What does the Langlands philosophy have to say about the weight and the level?

      I have recently attempted to read some number-theoretic texts. Here is an excerpt from a paper by Breuil-Conrad-Diamond-Taylor: Now consider an elliptic curve $E/\mathbb{Q}$. Let $\rho_{E, l}$ (...
      2
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      1answer
      94 views

      Norms of elements in a quadratic order - can you do it elementarily?

      Let $\mathcal O$ be an order in an imaginary quadratic field $K$. Does there exists an element $\lambda\in \mathcal O$ such that the norm $N(\lambda)$ is not a square? Does there exists an element $\...
      5
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      1answer
      270 views

      Cube root of the $j$-invariant

      Let $$\Gamma=\bigg \lbrace \begin{pmatrix} a&b\\c&d\end{pmatrix}\in\Gamma(1):b\equiv c~(\text{mod }3)\text{ or } a\equiv d\equiv 0~(\text{mod }3)\bigg \rbrace.$$ Then $\Gamma$ has exactly ...
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      119 views

      Terminology question for the cohomology of the Hilbert modular group

      Let $\Gamma$ be the Hilbert modular group of determinant one matrices with entries in the ring of integers of a real quadratic field $F$, and let $M$ be a $\Gamma$-module. Is there a standard name for ...

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