Questions tagged [modular-forms]
A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group.
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How to prove a property of the product of Eisenstein Series
I want to prove that, if $$ E_i(\tau) = \frac{1}{2\zeta(2i)}\sum_{(m,n)\in\mathbb{Z}^2-\{(0,0)\}}(m\tau+n)^{-k}$$
is the normalized Eisenstein Series of weight $i$, then $E_i(\tau)E_j(\tau)\neq E_{i+j}...
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Modular parametrization of elliptic curves expressed as rational function of $j(\tau)$
I have learned that one can parametrize an elliptic curve using meromorphic modular functions on group $\Gamma_0(N)$ for certain level $N$. Using the example on wikipedia, the parametrization of $y^2-...
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Relation between the field and $\mathbb{Z}$-algebra generated by eigenvalues of modular form
Cross posted to MO: https://mathoverflow.net/questions/475273/relation-between-the-field-and-mathbbz-algebra-generated-by-eigenvalues-of
Let $f$ be a cusp form of weight $k\in\mathbb{Z}$ for the group ...
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If the analytic rank is one then the sign in the functional equation is -1?
Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $L(E, s)$ denote its $L$-function over $\mathbf{Q}$. Also $f$ denotes the weight two cusp form associated to $E$, but this shouldn't be ...
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The $j$-invariant on the 'critical line' and its zeros
It is well-known that for the $j$-invariant, we have $j\left(\frac{1+\sqrt{-n}}{2}\right)\in\mathbb R$ whenever $n>0$. Moreover, $j\left(\frac{1+\sqrt{-n}}{2}\right)=0$ for $n=3$ at the cusp. In ...
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How should the conjecture associated with Taniyama, Shimura, etc. be referred to?
I'm writing an article in which I need to refer to the Modularity theorem before it was a theorem. I always knew it as the first on the list below, but I'm aware there are alternatives, so I need to ...
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Number of copies of a fundamental domain for $\Gamma_0(q)/\{\pm1\} $ needed to cover a region $P(Y) = \{z \mid \Im z > Y, \Re(z) \in ]0,1] \}$ ($Y>0$)
I've been struggling on a claim in W.Duke's paper on the dimension of the space of cusp forms of weight one (see : https://arxiv.org/pdf/math/9411212 Lemma 2 p.5).
Take a cuspidal form $f \in \mathcal{...
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Degree of extension of the field of coefficients of modular forms
I am beginning to study modular forms and I came across an inequality defining the bound for the degree of extension of the field of coefficients of a modular form $f\in S_k(\Gamma_1(N))$. This goes ...
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Sources of non-congruence representations of the modular group
I'd be interested in any references that provide a source of non-congruence representations of the modular group or that discuss such representations. In my own field, vertex algebras, they have ...
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Excercise on j-invariant
I have this problem.
Given $j:\cal{H}\rightarrow\mathbb{C}$ the j-invariant function defined on the upper-half complex plane as $j=1728\frac{g_2(\tau)^3}{g_2(\tau)^3-27g_3(\tau)^2}$, where $g_2,g_3$ ...
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Counting the number of double cosets for subgroup
Let $H,K \le G$ and consider the set of double cosets $ H \backslash G / K$. If we have $H' \le H$, is there a formula expressing $| H' \backslash G / K|$ in terms of $|H \backslash G / K|$, $|H:H'|$...
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Find cusps of $\Gamma_1(p)$, show $\frac{j(\tau)}{j(2\tau)}$ is a modular function of weight $0$ level $\Gamma_1(2)$, which cusps is it holomorphic?
Firstly for notation, let
$$\Gamma_1(p) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid a \equiv d \equiv 1 \pmod p, c \equiv 0 \pmod p \right\},$$
and $j(\tau) = \frac{E_4(\tau)^...
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The Continuity of the Galois Representation Attached to a Weight 2 Newform
A representation $\rho: G_{\mathbb Q} \to \operatorname{GL}(V)$, where $V$ is a finite-dimensional $K$-vector space for some field $K$, must be continuous in order to be considered a Galois ...
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Is the Galois conjugate of a Siegel eigenform another eigenform?
Let $F$ be a Siegel cusp form of weight $k\geq2$ (integer), degree two for the group $\text{Sp}_4(\mathbb{Z})$. Suppose it is an eigenform for all the Hecke operators $T(n)$. Let $\sigma$ be an ...
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Computing torsion subgroup of elliptic curve
Compute the torsion subgroup of the elliptic curve $y^2=x^3+5x^2+3x+7$.
I am only used to computing torsion groups when our equation is in 'short Weirstrass form'; i.e. $y^2=x^3+Ax+B$ for integer $A,...
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