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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$f,f',…,f^{(j)}$ is $\mathbb C$-linearly independent if $f$ is a modular form

Conjecture: Let be $f$ a modular form of weight $k$ and $j$ a strictly positive integer, then the set $f,f',...,f^{(j)}$ is $\mathbb C$-linearly independent in $A$. Is that conjecture true or false? ...
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1answer
88 views

Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...
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5answers
2k views

Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of ...
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1answer
386 views

General questions about Eisenstein series and modular forms

I am in a situation where I am trying to get a feel for modular forms type stuff, but don't have anyone to talk to about it (I'm not in academia at the moment). I would like to test my understanding ...
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1answer
131 views

Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: $$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
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1answer
101 views

Condition on infinite component for a cuspidal automorphic representation to be attached to a Hilbert newform

Let $F$ be a totally real field with real primes $\tau_1,\ldots,\tau_n$ and $\pi$ a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbf{A}_F)$. My rough understanding is that in order for ...
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1answer
479 views

how to calculate $q$-expansion coefficients of a modular form?

Can you please explain for me how we can calculate the coefficients of $q$-expansion series of a modular form function $f$? I am really confused.
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1answer
144 views

Are the coefficients of this eigenform totally real?

Let $f \in S_2(\Gamma_0(N))$ (cusp forms) be a normalized Hecke eigenform, and let $K_f$ be the number field obtained by adjoining all its Fourier coefficients to $\mathbb{Q}$. Then is $K_f$ totally ...
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1answer
118 views

Why is the Petersson inner product positive definite?

The Petersson inner product is defined on the space $\mathcal{S}_k(\Gamma)$ of weight $k$ cusp forms of level $\Gamma$, and takes values in $\mathbb{C}$. First of all, I wonder: what does it mean for ...
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1answer
63 views

A certain combination of fourier coefficients of modular forms

Let $f$ be a modular form of weight 2 for the group $\Gamma_0(N)$, with Fourier expansion $$ \sum_{n\geq 0} a(n)\ q^n. $$ Let $d$ be an integer dividing $N$, and consider the Fourier series $\sum ...
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91 views

Number of $SL_2(\mathbb{Z})$-translates lying within a fixed ray in $\mathfrak{h}$

This is probably an easy question, but I can't seem get the right idea so here goes: Let $\tau \in \mathfrak{h}$ (where $\mathfrak{h}$ denotes the upper halfplane) be given. For any two positive real ...
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1answer
59 views

Problem about decomposition of modular space into eigenspace

It is said we can use the operator $$ \pi_\chi=\frac{1}{\phi(N)}\sum_{d\in\mathbb{Z}_N^*}\chi(d)^{-1}\langle d\rangle $$ to project function in $\mathcal{M}_k(\Gamma_1(N))$ into the $\chi-$eigenspace ...
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1answer
92 views

What is the difference between the dimension of $\mathcal{M}_1(\Gamma)$ and $\mathcal{S}_1(\Gamma)$?

There is a formula given in the book saying that $$ \dim{\mathcal{M}_1(\Gamma)}=\dim\mathcal{S}_1(\Gamma)+\frac{\varepsilon^{\text{reg}}_\infty}{2} $$ Where $\varepsilon^{\text{reg}}_\infty$ means ...
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2answers
181 views

Lambert series help

How can one go about proving lambert series identities like, $$\left(1+240\sum_{n=1}^\infty \frac{n^3q^n}{1-q^n} \right)^2=1+480\sum_{n=1}^\infty \frac{n^7q^n}{1-q^n}$$ All the papers I have looked ...
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1answer
70 views

Why $\Delta(N\tau)\in\mathcal{S}_{12}(\Gamma_0(N))$

There is an argument saying that because $\Delta(\tau)\in\mathcal{S}_{12}(\text{SL}_2(\mathbb{Z}))$, then we have $\Delta(N\tau)\in\mathcal{S}_{12}(\Gamma_0(N))$. I don't understand the logic here. ...
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29 views

Problem about the order of a automorphic function on an elliptic curve.

Let $f:X(\Gamma)\rightarrow\mathbb{C}$ be an automorphic function of weight 0, saying that $f$ is $\Gamma$-invariant. Now at each noncusp point $\tau\in\mathbb{C}$, $f$ has an order $\nu_\tau(f)$, ...
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1answer
152 views

In one version of the Modularity Theorem, what does “arise from modular forms” mean?

One version of Modularity Theorem says that The elliptic curves with rational $j$-values arise from modular forms. Where $$j(\tau)=1728\frac{g_2^3(\tau)}{\Delta(\tau)}$$ I know every ...
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1answer
70 views

Question about a property of elliptic function

Let $E=\mathbb{C}/\Lambda$ be a complex elliptic curve where $\Lambda=\omega_1\mathbb{Z}\oplus\omega_2\mathbb{Z}$ Let $f$ be a nonconstant elliptic function with respect to $\Lambda$. Let ...
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1answer
71 views

A question about complex tori.

Let $\Lambda$,$\Lambda'$ be two lattice in $\mathbb{C}$ and $m\neq 0\in\mathbb{C}$ satisfying $$ m\Lambda\subset\Lambda' $$ The, the book I'm reading says that by the theory of finite Abelian groups ...
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1answer
102 views

Why the kernel of isogeny is finite?

It is said that the kernel of a isogeny is finite because it is discrete and complex tori are compact. I have some questions about this. 1. Following is my reason for the kernel is discrete. ...
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1answer
390 views

Question about weight 2 Eisenstein series

I'm new to modular form, reading the book A First Course in Modular Forms We have the weight 2 Eisenstein series $$ G_2(\tau)=\sum_{c\in\mathbb{Z}}\sum_{d\in\mathbb{Z}_c'}\frac{1}{(c\tau+d)^2} $$ ...
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1answer
90 views

Why does a certain equality in Iwaniec's Topics in classical automorphic forms hold?

I have been reading the book Topics in classical automorphic forms for a while, where I encountered a formula in the middle of Sec 8.6 (p143) $$F(z;\beta,\gamma)=\frac ...
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2answers
303 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 ...
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1answer
84 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 ...
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1answer
117 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 ...
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69 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 ...
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1answer
203 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 ...
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104 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, ...
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1answer
190 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: ...
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197 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 ...
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1answer
504 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 ...
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1answer
86 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 ...
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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 ...
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487 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 ...
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1answer
141 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
172 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
412 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
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1answer
228 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
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1answer
192 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
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1answer
196 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
160 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
941 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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3answers
312 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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0answers
130 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
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1answer
155 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
77 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
567 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
224 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
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0answers
111 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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2answers
147 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 ...