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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Eisenstein Series and cusps of $\Gamma_{1}(N)$

Let $q = e^{2\pi i\tau}$, $\operatorname{Im}\tau > 0$ and let $$G(\tau) = 1 - 24\sum_{n = 1}^{\infty}\frac{nq^{n}}{1 - q^{n}}$$ be the weight 2 Eisenstein series for $\Gamma(1)$. Consider the ...
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139 views

Eisenstein series solution

Denote the function $$ \Psi (x,y)= y^{1/2+ik}+ \sum_{g\in SL(2,\Bbb Z)} \frac{y^{1/2+ik}} {|c_{g}z+d_{g}|^{1/2+ik}}\tag{1}$$ My question is if I can write the wave function in terms of the Eisenstein ...
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1answer
162 views

Classical Hecke eigenforms for $\Gamma_0(p)$.

I am in need of classical eigenforms for $\Gamma_0(p)$. In particular I need these to be newforms for each of the even weights 8 to 26 and would settle just for cases $p=2,3$ at this moment in time. ...
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2answers
398 views

Residue of Rankin-Selberg L-function for non-trivial nebentypus

Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as $$ f(z)=\sum_{n\ge 1} ...
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1answer
224 views

Hypergeometric formulas for the Rogers-Ramanujan identities?

Let $q = e^{2\pi i \tau}$. Given the j-function, $$j = j(q) = 1/q + 744 + 196884q + 21493760q^2 + \dots$$ and define, $$k = j-1728$$ Let $\tau =\sqrt{-N}$, where $N > 1$. Anybody knows how ...
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1answer
184 views

Are the eigenvalues of the Hecke operators always real?

Are the eigenvalues of the Hecke operators $T_n$ for $M_k(\text{SL}_2(\mathbb{Z}))$ always real? I think I have an answer but I am not confident with my arguments. If $f$ is a normalized eigenform, ...
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1answer
190 views

Congruence subgroups and modular curves of type (M,N)

I would like to study the "modular curve" $Y(M,N)$, parametrizing an elliptic curve $E$ together with $p \in E[M]$ and $q \in E[N]$ (here and in the following $M$ divides $N$). Let $\Gamma(M,N)$ be ...
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1answer
159 views

Is there an infinite product for $\left(\frac{\eta(13\tau)}{\eta(\tau)}\right)^2$ analogous to the Rogers-Ramanujan identity? [closed]

Given $$ \left(\frac{\eta(5\tau)}{\eta(\tau)}\right)^{6}\;\; =\;\; \frac{r^5}{1-11r^5-r^{10}},\;\;\;\;\;\text{with}\;\;r\; =\; q^{1/5} \prod_{n=1}^\infty ...
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1answer
827 views

How does one graduate from Hecke Operators to Hecke Correspondences?

I've read (skimmed heartily) basic books on the topic of modular forms. (The last being Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.) I strive for an understanding which is as ...
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310 views

An infinite product for $\left(\frac{\eta(13\tau)}{\eta(\tau)}\right)^2$?

Given the Dedekind eta function, $$\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$$ where $q = \exp(2\pi i\tau)$. Consider the following "family", $\begin{align} ...
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124 views

how to prove that a Galois representation is exceptional?

A theorem of Deligne asserts that to cusp forms with Euler product, there is for each prime $\ell$ a Galois representation of $G(K_{\ell}/\mathbb{Q})$, where $K_{\ell}$ is the maximal abelian ...
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1answer
148 views

The filtration of a mod l modular form

I'm reading a paper of Swinnerton-Dyer, "On l-adic representations for coefficients of modular forms." He defines the notion of "filtration" for mod $\ell$ modular forms: If $\tilde{f} \in ...
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1answer
74 views

Algebraic modular form values depending only on coset reps

We have a reductive group $G/\mathbb{Q}$ and a representation space $V$ of this group. Let $K$ be an open subgroup of $G(\mathbb{A}_{\mathbb{Q}})$ (where $\mathbb{A}_{\mathbb{Q}}$ are adeles of ...
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3answers
530 views

Ramanujan 691 congruence

I know how to prove this congruence in two ways (one using basics of modular forms and the other using Hecke operators) and I will be working on proving other such congruences soon. The congruence ...
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1answer
214 views

A question about modular curves and base change

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Suppose that the curve $X\times_{K,\sigma} \mathbf{C}$ is a modular curve for some $\sigma:K\to \mathbf{C}$. Can ...
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107 views

What is a cusp parameter?

I was reading this paper, and on the first page they define a cusp form as $$ f(z) = \sum_{n > -\alpha} a(n) e^{2\pi i (n + \alpha)z}. $$ Is this equivalent to the usual definition of a cusp form ...
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145 views

Epsilon conjecture analog

Recently this question caught my eye. Is there a relation to the modularity problem of elliptic curves over $\mathbb{Q}(\zeta_m)$ and this problem? Namely, if all elliptic curves over ...
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0answers
84 views

Applications and motivation of η-quotient generators and algorithms

I previously asked this question on mathoverflow but got no satisfactory answers so I'm posting it here as well: So initially Dummit, Kisilevsky and McKay found all Dedekind $\eta$-products which are ...
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1answer
90 views

Modular forms have values as algebraic numbers?

Can you find examples of modular forms which take on values as e.g. algebraic numbers of degree n ? I'm interested in finding special classes of algebraic numbers particularly when n=3, they don't ...
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125 views

Flatness of residual representations associated to modular forms

Let $f\in S_k(\Gamma_1(N),\chi)$ be a Hecke eigenform of weight $k\geq 2$, $p$ an odd prime not dividing $N$, and $K_f$ the number field generated by the Hecke eigenvalues of $f$. Fixing a prime ...
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1answer
183 views

Dimension space of modular forms of weight $2k$.

How can I prove that: For a given natural number $k$ the dimension space of modular forms of weight $2k$ is $\lfloor{\frac{k}{6}\rfloor}+1$ if $k \not\equiv 1 \: (\text{mod}\ 6)$ and ...
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1answer
220 views

A question about modular forms in SAGE

I was trying to solve Exercise 1.4.5 in Alvaro Lozano-Robledo's book Elliptic Curves, Modular Forms and Their L-functions, which is about representations of integers as sums of 6 squares and its ...
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1answer
285 views

Is there a fundamental domain for $\Gamma(2)$ contained in the following strip

Let $\Gamma(2)$ be the subgroup of $\mathrm{SL}_2(\mathbf{Z})$ satisfying the usual congruence conditions. It acts on the complex upper half-plane. Does it have a fundamental domain contained in the ...
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186 views

How can I determine in practice whether two elliptic curves over $\mathbb{Q}$ have isomorphic $p$-torsion?

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic ...
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1answer
960 views

Deligne, elliptic curves and modular forms

I'm trying to understand an argument of Deligne (in Courbes elliptiques: Formulaire d'après J. Tate), but I'm not familiar enough with algebraic geometry, so I'm getting quite confused. So even in my ...
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1answer
72 views

modular group differential equation solutions

given the PDE Eigenvalue problem $ y^{2}( \partial _{x}^{2}f(x,y) +\partial _{y}^{2}f(x,y))= E_{n}f(x,y) $ (1) if we are on the poincare disc so i impose the conditions $ x'=x+1$ invariance $ ...
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2answers
223 views

Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?

Given the j-function $j(\tau)$, $j(\tau) = 1728J(\tau)$, where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series, $\begin{align} ...
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What are the branch points of $X(n)\to X(1)$

Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps). ...
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256 views

Is there a precise mathematical connection between hypergeometric functions and modular forms

I've been playing around with Gauss' hypergeometric series $F(a,b,c,z)$ these days and I was wondering if there is some relation with the theory of modular forms.
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257 views

Euler Product of Modular L-series

This question is probably really easy to someone that knows the theory of modular forms well, so I apologize if this is obvious. Suppose $E_1$ and $E_2$ are elliptic curves over $\mathbb{Q}$ and the ...
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1answer
68 views

Does the following define a point of the modular curve $X_1(n)$

Let $X$ be a compact connected Riemann surface of genus $1$ and suppose that there is a finite morphism $X\to \mathbf{P}^1$ of degree $n$ which ramifies totally over $0$ and $\infty$. Let $f^{-1}(0)$ ...
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1answer
204 views

Inferring Jacobi Triple product from $q$-binomial theorem?

Quite a while ago I asked a question about deducing the Jacobi triple product from the $q$-binomial theorem. In fact, from the $q$-binomial theorem $$ ...
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1answer
71 views

Reference for existence of quotients of modular Jacobians?

In the construction of Galois representations attached to modular forms of weight 2, the representation spaces can be found in Tate modules of certain quotients of Jacobians of modular curves by ...
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1answer
262 views

Finiteness of Dimension of $M_{k}(\Gamma)$

Let $M_{k}(\Gamma)$ denote the space of weight $k$ modular forms for the congruence subgroup $\Gamma$. Are there any proofs of the finiteness of the dimension of $M_{k}(\Gamma)$ that don't rely on ...
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256 views

Intermediate step in deducing Jacobi's triple product identity.

An intermediate step deduces Jacobi's triple product identity by taking the $q$-binomial theorem $$ \prod_{i=1}^{m-1}(1+xq^i)=\sum_{j=0}^m\binom{m}{j}_q q^{\binom{j}{2}}x^j $$ and deducing $$ ...
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1answer
234 views

solving this laplacian

given the laplacian $ -y^{2}( \partial _{x}^{2} + \partial _{y}^{2} )f(x,y)=Ef(x,y) $ can we find a solution in the form $ f(x,y)= \sum_{n,m}C_{n,m} \phi (x,E) g(y,E)$ if we impose the extra ...
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1answer
271 views

Preparation for a modular forms course

I'm trying to get a sense of what type of math to brush up on in order to take a course based on the book A First Course in Modular Forms by F. Diamond and J. Shurman. The text claims to be accessible ...
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1answer
245 views

Eisenstein series estimate from Stein-Shakarchi

I need help with this one from Stein & Shakarchi. Let $$E_{4}(\tau)=\sum_{(n,m)\neq (0,0)}{\frac{1}{(n+m\tau)^{4}}}$$ be the Eisenstein series of order $4$. Show that ...
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0answers
158 views

A Dedekind eta function sum of form $y_0^k+y_1^k+y_2^k+y_3^k+… = 0$

Given the Dedekind eta function $\eta(\tau)$. Define, $y_p = e^{\pi i p/6}\,\eta(\tfrac{\tau+2p}{5})$ Prove the multi-grade identity [1], $y_0^k + y_1^k + y_2^k + y_3^k + y_4^k + ...
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525 views

Ramanujan's Tau function, an arithmetic property

The problem: Let $\tau(n)$ denote the Ramanujan $\tau$-function and $\sigma(n)$ be the sum of the positive divisors of $n$. Show that $$ (1-n)\tau(n) = 24\sum_{j=1}^{n-1} \sigma(j)\tau(n-j).$$ ...
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1answer
457 views

On the Fourier Coefficients of a Modular Form

Let $f \in S_2(\Gamma(N))$ be a weight 2 cusp form of full level $N > 5$. It has a Fourier expansion: $$ f(z) = a_1 q^{1/N} + a_2q^{2/N} + \cdots. $$ Now let $S = \left(\begin{array}{cc} 0 & ...
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1answer
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How could $Y(1)$ be the quotient of the upper half plane?

This is a silly question, but I can't resolve this: $Y(1)$ is defined to be $\mathbb{H}/PSL_2(\mathbb{Z})$. So it seems that its universal cover should be $\mathbb{H}$. On the other hand $Y(1)$ is ...
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1answer
79 views

What are these theta functions appearing in Sloane's database

Looking at Sloane's database, I found a neat formula for the lambda-invariant. Let $q:\tau \mapsto \exp(\pi i \tau)$ on the complex upper-half plane. Then $$\lambda(q) = 16q\;\prod_{k>0} \frac{(1 ...
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1answer
267 views

Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a google search didn't give me much. Also, are ...
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95 views

The lambda invariant

Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane. ...
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Real elliptic curves in the fundamental domain of $\Gamma(2)$

An elliptic curve (over $\mathbf{C}$) is real if its j-invariant is real. The set of real elliptic curves in the standard fundamental domain of $\mathrm{SL}_2(\mathbf{Z})$ can be explicitly ...
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67 views

Is the image of this strip equal to the modular curve $Y(2)$

Let $B$ be the image of the strip $$\{x+iy : -1\leq x< 1, y>\frac{1}{2} \} \subset \mathbf{H}$$ in the modular curve $Y(2)$ under the quotient map $\mathbf{H} \to Y(2)$. Is $B=Y(2)$? Is ...
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403 views

Representing a number as a sum of at most $k$ squares

Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to $$ n = x_1^2 + ...
9
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1answer
315 views

Relation between congruence subgroups. $\Gamma(M)\Gamma(N) = \Gamma(\gcd(M,N))$

I'm hoping there's a pleasant way to solve this one. Prove that $\Gamma(M)\Gamma(N) = \Gamma(\gcd(M,N))$. Showing that $\Gamma(M)\Gamma(N) \subset \Gamma(\gcd(M,N))$ is rather straight forward, ...
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242 views

Modular functions of weight zero

The following question was suggested by Sasha's answer to the following question : Is the derivative of a modular function a modular function . Question. What are the modular functions with respect ...