# 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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### 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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• 103
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### Dedekind eta and Lie algebras

Reference request and a formal question I am currently self-studying modular forms and read the Wikipedia page for Dedekind's eta function. There it reads that: "The theory of the algebraic ...
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### q-series Expansion at Cusp and L-Functions

We know that the coefficients at $\infty$ of a modular form that is an eigenfunction of all Hecke operators in, say, $\Gamma_0(q)$, give an $L$-function with an Euler product and a functional equation....
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### On proving that an Artin L-function is cusp form

I am reading the article by JEAN-PIERRE SERRE, On a Theorem of Jordan, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 2003. I have problem in the following part: Since $S_{3}$ is a dihedral group, ...
1 vote
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### Sketching Fundamental domains for $SL_2(\mathbb{Z})$ [closed]

I am familiar with the fundamental domain for the full modular group $\Gamma(1)=SL_2(\mathbb{Z})$, given by $$\{z\in\mathbb{H}:-1/2<Re(z)<1/2\textrm{ and }|z|>1\}.$$ ...
1 vote
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### Road map to learning about $\ell$-adic representations of $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I am interested in learning about the congruence relations satisfied by the coefficients of modular forms, in particular I am interested in learning more about $\tau(n)$, the coefficients of the ...
1 vote
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### On the cusps of $\Gamma_0(3) \cap \Gamma(2)$

I've been trying to compute the number of non-equivalent cusps of $\Gamma_0(3) \cap \Gamma(2)$. My approach so far has been the following: I believe that this is finite index subgroup of $\Gamma_0(3)$ ...
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### Questions on sequences and modular forms

Let me apologize ahead of time since I am not at all well versed in the theory of modular forms. I have seen some nice examples where modular forms are used to study certain interesting numbers. For ...
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### Background for Gross-Zagier paper

I have been reading the paper "Heegner points and derivatives of $L$-series" by Gross and Zagier. Link to the paper. In section III of the paper, they use intersection theory to express a ...
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### How to come up with the generating function of an elliptic curve?

Having watched the otherwise splendid Numberphile video with Edward Frenkel explaining the Langlands program, two mysteries remained completely open to me: Given the equation $y^2 + y = x^3 - x^2$ how ...
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### $\tau(n) \equiv n\sigma_9(n)\,(\mathrm{mod}\,1050)$ is known?

I found the congruence in the title today (where $\tau(n)$ is the Ramanujan tau function), and I wonder if this is new or already known (or easy consequence of known results). I read the wikipedia and ...
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### Proof that principal congruence subgroups $\Gamma(N)$ are torsion free.

I have read the fact that the principal congruence subgroups $\Gamma(N)$ of $\mathrm{GL}_n({\mathbb{Z}})$ are torsion free for $N \geq 3$ several times, but only saw proofs for very specific ...
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### Modular symbols, Manin symbols, two and three term relation

Maybe someone could help me out. I consider a $SL_2(\mathbb{Z})$-module $\Omega$. We set \begin{align} S:=\left(\begin{array}{rr} 0 & 1\\ -1 & 0\end{array}\right) \ \mathrm{and} \ U:=\left(\...
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### Way to determine when a modular form divides another?

Given modular forms $f, g$ for $SL_{2}(\mathbb{Z})$ of weights $l, k$ resp. where $l < k$, is there a way to see if $f \mid g$? That is, is there a way to see if $g = fh$ for some modular form $h$ ...
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### Hecke operators on q-expansion of cusp forms

I am trying the following exercise. Let $V=S_2(\Gamma _1 (16))$ and we are given the following basis of $V$, their $q$ expansion at the $\infty$ cusp. \begin{align} f_1 =& q − 2q^3 − 2q^4 + 2q^...
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### Why is the Function Field of the Modular Curve $X(N)$ defined over $\mathbb{Q}(\mu_n)$?

Following Section 7.6 in Diamond & Shurman, the algebraic model of $X(N)$ is constructed (a priori) over $\mathbb{Q}$ by first defining its function field as \begin{equation*} \mathbb{Q}(j,f_{(0,1)...
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### On congruence subgroups of $GL_2^+$

I am reading the book Holomorphic Hilbert Modular Forms by Paul Garrett. The author considers congruence subgroups of general linear groups of positive determinant. More precisely, let $F$ be a ...
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### Weights of Vector Valued Siegel Modular Forms

I am looking for a reference request for a fact I believe to be true (or a confirmation that it is wrong). We may parameterize the algebraic representations of $GL_n(\mathbb{C})$ by non-increasing ...
I am trying to show that there exists a family of matrices $(M_n)$ in $GL(2, \mathbb{R})$ such that $GL(2,\mathbb{Z}) M_n GL(2,\mathbb{Z})$ is the same for every n and $GL(2,\mathbb{Z}) M_n$ is ...
Consider the (geometrically reducible) modular curve $Y(N)$ over $\mathbb{Q}$, representing the functor Y(N)(T)=\{(E/T,\alpha),\; E/T \; \text{elliptic curve}, \;\alpha:(\mathbb{Z}/N\mathbb{Z})_T^2\...