For questions related to modular forms

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5
votes
1answer
367 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} ...
8
votes
1answer
210 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 ...
3
votes
1answer
159 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, ...
2
votes
1answer
178 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 ...
0
votes
1answer
150 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 ...
9
votes
1answer
699 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 ...
7
votes
3answers
286 views

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

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} ...
1
vote
0answers
118 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 ...
3
votes
1answer
145 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 ...
1
vote
1answer
71 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 ...
3
votes
3answers
477 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 ...
5
votes
1answer
207 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 ...
3
votes
0answers
105 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 ...
2
votes
2answers
139 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 ...
1
vote
0answers
81 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 ...
0
votes
1answer
89 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 ...
8
votes
1answer
113 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 ...
0
votes
1answer
160 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 ...
3
votes
1answer
193 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 ...
1
vote
1answer
236 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 ...
7
votes
1answer
177 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 ...
22
votes
1answer
854 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 ...
1
vote
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 $ ...
5
votes
2answers
172 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} ...
2
votes
0answers
52 views

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). ...
3
votes
2answers
217 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.
6
votes
1answer
240 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 ...
1
vote
1answer
62 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 $f^{-1}(0)$ ...
1
vote
1answer
187 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 $$ ...
1
vote
1answer
63 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 ...
2
votes
1answer
233 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 ...
0
votes
0answers
249 views

Modular Form Holomorphic at Cusps

Let $f$ be a weight $k$ modular form for $\Gamma$ and $f\vert_{\sigma}(z) = f(\sigma z)(cz + d)^{-k}$ where $\sigma = \begin{pmatrix} a & b\\c & d\end{pmatrix}$. Why is it true that $f$ is ...
6
votes
1answer
254 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 $$ ...
0
votes
1answer
230 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 ...
3
votes
1answer
246 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 ...
1
vote
1answer
229 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 ...
2
votes
0answers
144 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 + ...
7
votes
2answers
464 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 $$ (1-n)\tau(n) = 24\sum_{j=1}^{n-1} \sigma(j)\tau(n-j).$$ ...
1
vote
1answer
383 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 & ...
2
votes
1answer
71 views

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 ...
1
vote
1answer
75 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 ...
0
votes
1answer
235 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 ...
0
votes
0answers
90 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. ...
2
votes
1answer
116 views

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 ...
1
vote
1answer
64 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 ...
9
votes
4answers
319 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
votes
1answer
310 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, ...
1
vote
1answer
207 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 ...
8
votes
1answer
183 views

Converting an infinite product to sum; Ramanujan $\tau$ function

I've gotten what seems most of the way, but I'm quite stuck at this point. Define $\tau(n)$ by \begin{align*} q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty\tau(n)q^n. \end{align*} ...
3
votes
1answer
166 views

Is the derivative of a modular function a modular function

Fix a positive integer $n$. Let $f:\mathbf{H}\longrightarrow \mathbf{C}$ be a modular function with respect to the group $\Gamma(n)$. Is the derivative $$\frac{df}{d\tau}:\mathbf{H}\longrightarrow ...