For questions related to modular forms

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5
votes
2answers
251 views

Klein's j-invariant and Ford circles

Klein's j-invariant has structure which seems to resemble Ford circles: The latter show up all over number theory (continued fractions, Rademacher's expansion for p(n), etc.) Can someone explain ...
2
votes
1answer
73 views

Divisor of a modular form

I am trying to compute the divisor of $\Delta(z)/\Delta(pz)$ on the modular curve $X_0(p)$ where $p$ is a prime. I know that as a function on the full modular group, the $\Delta$ function has only a ...
1
vote
1answer
99 views

Cusps for higher dimensional hyperbolic spaces

Take your favourite model of the hyperbolic plane $\mathfrak h$. It is well known how to define cusps, and we know those are $\mathbb Q \cup \{ \infty \}$. Now pick a higher dimensional hyperbolic ...
2
votes
0answers
53 views

How to prove that $G_3>0$ in this case?

Let $\Lambda=\{a+be^{2\pi i/3}|a,b\in Z\}$, then $G_{3}(\Lambda)=\sum_{\omega\in\Lambda-\{0\}}\frac{1}{\omega^{6}}$ should be real and nonzero, but how can one prove that it's positive? Moreover, in ...
3
votes
1answer
184 views

Elliptic Points of Modular Group in Upper Half Plane

This is a very small question. Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq ...
4
votes
0answers
91 views

Connection between modular forms and line bundles

I need a good reference about the connection between modular forms and line bundles. I found only Milne's note that treats briefly this argument. I've already checked, but without finding anything, ...
1
vote
1answer
161 views

Dimension of modular forms for a congruence subgroup using contour integration.

There is a well known proof for finding the dimension of modular forms for the full modular group using the residue formula. It is on page 71 of these online notes: ...
9
votes
1answer
182 views

Connection between the $L$-function of a modular form and the $L$-function of its associated $\ell$-adic representation as defined by Bloch-Kato

Let $f\in\mathcal{S}_k(\Gamma_1(N),\epsilon)$ is a normalized eigenform for all the Hecke operators, with character $\epsilon$, $k\geq 2$, and assume the $q$-expansion of $f$ has rational ...
18
votes
1answer
404 views

The importance of modular forms

I'm studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice $\Lambda$ is fixed) called ...
4
votes
1answer
83 views

Local invariants of the discrete Galois module associated to a $p$-ordinary newform

Let $f=\sum_{n=1}^\infty a_nq^n$ be a $p$-ordinary newform of weight $k\geq 2$, level $N$, and character $\chi$, and let $\rho_f:G_\mathbf{Q}\rightarrow\mathrm{GL}_2(K_f)$ be the associated $p$-adic ...
2
votes
0answers
110 views

Evaluating the Hauptmodul of $X_{1}(5)$

The function field of $X_{1}(5)$ is generated by the function $$t(\tau) = q\prod_{n = 1}^{\infty}(1 - q^{n})^{5\left(\frac{n}{5}\right)}$$ where $q = e^{2\pi i\tau}$ and $\left(\frac{n}{5}\right)$ is ...
8
votes
2answers
410 views

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 ...
1
vote
1answer
120 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 ...
2
votes
1answer
155 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. ...
5
votes
1answer
374 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
218 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
168 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
181 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
152 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
729 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
290 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
121 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
146 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
493 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
210 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
140 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
114 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
162 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
201 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 ...
2
votes
1answer
253 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
180 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 ...
23
votes
1answer
879 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
206 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
53 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
234 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
245 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
189 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
65 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
238 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
255 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
255 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
250 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
234 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 ...