For questions related to modular forms

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Modular forms are arithmetic objects

What does arithmetic object exactly means? In an article, I found the following statement: modular forms are arithmetic objects. What this should means? Bests.
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1answer
370 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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3answers
48 views

How to solve a pair of simultaneous linear congruences, using algebraic methods [on hold]

What is the smallest whole number $x$ so that $x$ has remainder $14$ when divided by $400$, and $x$ has remainder $5$ when divided by $7572$? In other words: $$x \equiv 14 \pmod{400}$$ and $$ ...
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0answers
19 views

Relations between Eisenstein series and hypergeometric series

It is known that $$E_4(\tau) = {}_{2}F_{1}\left(\frac{1}{12}, \frac{5}{12}; 1; \frac{1}{J(\tau)}\right)^4$$ and $$E_6(\tau) = {}_{2}F_{1}\left(\frac{1}{12}, \frac{7}{12}; 1; \frac{1}{1 - ...
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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} ...
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1answer
58 views

Eisenstein-type series

Is the series, $$1 - 24\sum_{n = 1}^\infty \frac{q^{2n}}{(1 - q^{2n})^2},$$ somehow related to $$E_2(q) = 1 - 24\sum_{n = 1}^\infty \frac{nq^n}{1 - q^{n}},$$ the Eisenstein series of weight 2?
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1answer
76 views

On the square coeffecients of a modular form

Let $k\in \mathbb{N}$. Let $f\in M_k(\Gamma_0(N),\chi)$ be a modular form of weight $k$ on $\Gamma_0(N)$ with a Dirichlet character $\chi$. If $f$ has a Fourier expansion of the form $$ ...
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0answers
34 views

A problem in the Hecke's trick method

In his 'Introduction to modular forms', Don Zagier deals with the Hecke's trick which I don't really understand : Let $$G_2(\tau)=-\frac{1}{24}+\sum_{n=1}^{+\infty}{\sigma_1(n)q^n}$$ and ...
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0answers
37 views

Twisting modular forms by Dirichlet characters

Let $\chi,\chi_1$ be Dirichlet characters modulo $M$ and $N$. In Koblitz's book "Introduction to Elliptic Curves and Modular Forms", Proposition III.3.17, it is proved that if ...
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Do modular forms of higher weight occur in the proof of FLT?

Fermat's last theorem is a consequence of a statement about weight two modular forms. Going through the long proof, does one ever encounter modular forms of higher weight?
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43 views

Problem about Eisenstein series on $\Gamma_1(N)$

I'm learning about Eisenstein series on $\Gamma_1(N)$ and it seems to me that I have misunderstood something. I imagine the following situation : Let $\nu$ be a function on $(\mathbb{Z}/ N ...
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0answers
13 views

Is there an effective bound known for the coefficients of half integer weight cusp forms?

If $f(z)=\sum a_n q^n$ is a cusp form (of integer weight) normalized so that $a_1=1$, we have the inequality $$\vert a_n \vert \leq d(n) n^{(k-1)/2},$$ known as the Deligne bound (in which $d(n)$ ...
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0answers
10 views

On discrete subgroups of modular group and quadratic forms

I'm trying to work my way through a couple of papers on product formulae associated with certain modular forms. In "Borcherds Products Associated with Certain Thompson Series" by Chang Heon Kim, the ...
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32 views

Arithmetic properties of the partition function

Ramanujan mentioned in his paper in 1920 that "it appears that there are no equally simple properties for any moduli involving primes other than these three" $p(5n+4)\equiv0 \mod 5$ $p(7n+5)\equiv0 ...
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1answer
25 views

A question about a form that is related to modular form

According to wikipedia "A modular form of weight $k$ for the modular group $SL(2, \mathbf Z)$ is a complex-valued function $f$ on the upper half-plane $H = {z ∈ C, Im(z) > 0}$, satisfying the ...
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0answers
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A modular form related to Jacobi theta function

The Riemann $\Xi(z)$ function is defined as $$\Xi(z)=2\int_1^\infty A(x)x^{-1/4}\cos\left((1/2)z\ln x\right)dx$$ where $A(x)$ is given in terms of Jacobi theta function ...
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Modular forms on the theta group

The theta group $\Gamma$ is the subgroup of $SL(2;\mathbb{Z})$ generated by $T=\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$ It ...
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1answer
52 views

Identities for L-series and Euler product

It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly ...
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0answers
29 views

Constructing modular forms for subgroups

If $\Gamma \le SL(2;\mathbb{Z})$ is a subgroup for which we know generators, is there a principle for constructing modular forms for $\Gamma$ out of those for $SL(2;\mathbb{Z})$? An example of what I ...
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4answers
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Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
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0answers
28 views

$\dim \mathcal{S}_k(\Gamma_0(N))$

I'm looking for a formula which gives the dimension of $\mathcal{S}_k(\Gamma_0(N))$ the space of cusp forms of weight $k$ and level $N$. I found the following statement for $k\geq 4$ $$\dim ...
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1answer
63 views

Congruences of weights of modular forms modulo primes

I'm trying to prove that for two modular forms $f$ and $g$ of weight $k$ and $k'$ respectively, that are congruent modulo a prime $\ell\ge 5$, their weights are congruent modulo $\ell-1$. This is what ...
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Koblitz - Are chapters III & IV independent of I & II

I am interested in learning about Modular forms and have heard many great things about Neal Koblitz's Introduction to Elliptic Curves and Modular Forms. However, Koblitz doesn't discuss modular forms ...
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34 views

Modular Forms Over Quadratic Number Fields

I'm trying to do a bit of reading and looking into mislay forms of weight one over quadratic number fields, but am finding it difficult to locate any books or papers. I've got a little material from ...
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1answer
40 views

Transformation property for classical Siegel modular forms of weight 2

Let $\mathbb{H}_g = \{ \tau \in GL_g(\mathbb{C}) | \; {^t\tau} = \tau, Im(\tau) >0\}$ be the Siegel upper half space. There are Eisenstein series $$ E_{2k}(\tau) := \sum_{\gamma\in (P_0\cap ...
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0answers
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Proving that $\sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m)\sigma_3(n-m)$ without using modular forms?

This problem appears as a (starred!) exercise in D. Zagier's notes on modular forms. I have to admit that I have no idea how to do it. Here, $\sigma_k(n) =\sum_{d\mid n} d^k$, as usual. This ...
5
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1answer
155 views

Serre's Modularity Conjecture — Weight

I was reading Serre's paper "Sur les Représentations Modulaires de Degré $2$ de Gal($\bar{\mathbb{Q}}/\mathbb{Q}$)" where he states his modularity conjecture (which is now a theorem). Following his ...
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1answer
46 views

q-expansion principle for modular forms

Let $f(z)$ be a modular form of some integral weight $k \geq 0$ and level $\Gamma_1(N)$ (I insist I want $\Gamma_1(N)$, not $\Gamma_0(N)$ or $\Gamma(N)$). Thus for any $d \in (\mathbb Z/N\mathbb ...
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1answer
42 views

Eisenstein series is a modular form

I want to prove that the Eisenstein series $$G_k(z)= \sum_{c,d}(cz+d)^{-k}, (c,d)\in \mathbb{Z}^2 \setminus \{(0,0)\}$$ satisfies the relationship $$G_k((az+b)/(cz+d))= (cz+d)^k\sum_{(c',d')} ...
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0answers
13 views

Jacobi Form and its Fourier expansion

Let k,m be non negative integers. A Jacobi form of weight k and index m is a holomorphic function f on $\mathbb{H} x \mathbb{C}$ (where $\mathbb{H}$ denotes the upper half plane) satisfying the ...
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1answer
45 views

Decomposition of modular group elements

The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by $$h\cdot z=\frac{az+b}{cz+d},\;z\in H,\;h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in PSL_2(\mathbb{Z})$$ with ...
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1answer
48 views

Operations with $\text{SL}_2(\mathbb{Z})$

We define $\Omega :=\left\lbrace z \in \mathbb{H}\colon -\frac{1}{2} \leq \operatorname{Re}z \leq \frac{1}{2} \wedge |z| \geq 1\right\rbrace$. I want to show that the following holds: $$ \forall \, ...
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0answers
43 views

$(\sigma_0^{-1} SL(2,\mathbb{Z}) \sigma) \cap SL(2, \mathbb{Z})$ is a right coset of $\Gamma_0(m)$ in $SL(2, \mathbb{Z})$

Let $\Gamma_0(m)$ be the subgroup of $SL(2,\mathbb{Z})$ which is defined as follows: $$\Gamma_0(m)=\left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL(2, \mathbb{Z} : c ...
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147 views

Introductive Book on Modular Forms

I'm looking for an introductive book on Modular Forms and their applications to Algebraic Geometry and Algebraic Number Theory. Some Ideas? Explaining you my prerequisites, I've a good knowledge of ...
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1answer
63 views

History of the Coefficients of Elliptic Curves — Why $a_6$? [duplicate]

I would like to know what is the motivation behind the naming convention of the Weierstrass form of elliptic curves given as $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ I can see that $a_1,a_2,a_3,a_4$ ...
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1answer
42 views

Identities of Hecke operators

While studying, I recently came across the following interesting problem. Let's say that the (level one) weight $k$ modular forms $M_k(\Gamma(1))$ have dimension $d$. We know by the ring structure ...
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1answer
67 views

Undergraduate Introduction to Modular Forms

What are the best introductory texts (or lecture notes) on modular forms aimed at an advanced undergraduate audience (for a student with a course in complex analysis and two courses in algebra and ...
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1answer
37 views

Hecke equivariance in Poincaré duality.

Consider the first singular homology and cohomology groups of a modular curve, $H^1(X,\mathbb{Z})$ and $H_1(X,\mathbb{Z})$. The Hecke algebra acts on both of them and they are dual to each other under ...
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2answers
472 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
47 views

Holomorphic Eisenstein Series

$$G_k (z)=\sum_{\displaystyle(m,n) \neq (0,0)} \frac{1}{(mz+n)^k}$$ For $k \geq 3$ how to show that $G_k (z)=0$ for odd $k$ and $G_k (\tau z)=(cz+d)^k G_k (z)$ for $\tau \in SL_2(Z), \tau = \left( ...
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1answer
76 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
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1answer
58 views

Rewriting $\tau(p)\Delta(\tau)$ when $p$ is prime

$p$ is a prime, and $\tau$ is Ramanujan's tau function: $$p^{11}\Delta(p\tau)+\frac{1}{p}\sum_{k=0}^{p-1}\Delta\bigg(\frac{\tau + ...
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0answers
25 views

eisenstein part of theta function

If $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the $\textbf{theta series}$ ...
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1answer
75 views

Modular Form, but Cusp Form in Disguise

Suppose I have a holomorphic modular form $f \in M_k(\Gamma_0(N), \chi)$ with $k \in \mathbb Z^+$ and $f = \sum_{n=1}^\infty a(n)q^n$. Considering the $q$-expansion, one may suspicious that $f$ is ...
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1answer
51 views

Why a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$?

It is said that a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$. Why a modular form is a highest weight vector of a ...
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1answer
47 views

Elliptic points on modular curves and a uniformising parameter

I'm working with the modular curves $X_0(p)$ for $p=2,5$, with $$f_2=q\prod_{n\geq1}(1+q^n)^{24}$$ and $$f_5=q\prod_{n\geq1}(1+q^n+q^{2n}+q^{3n}+q^{4n})^6$$ being inverses of the Hauptmodul and ...
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2answers
214 views

Problems using Euler-Maclaurin for $\sum e^{-2 \pi z a^2}$

I'm trying to evaluate $\sum_{a=-\infty}^{\infty} e^{-2 \pi z a^2}$ using Euler-Maclaurin, but I get $\frac{1}{\sqrt{2z}}$. The only alternative I have is to calculate the remainder term directly for ...
2
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1answer
45 views

Diamond Operator and Fourier Coefficients

Let $d\in(\mathbb{Z}/N\mathbb{Z})^\times$, then one has the Diamond operator $\langle d \rangle$ acting on $M_k(\Gamma_1(N))$ via ...
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1answer
26 views

Diamond operator on a meromorphic function

Is there a standard way in which the diamond operator for modular forms is defined for an arbitrary meromorphic function?. I know the definition for a weight k modular form, but since I usually wont ...
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1answer
23 views

Weber function takes on real values?

Why does the Weber function take on real values? In the middle of the proof, the author argues... "cleary it is real-valued." I don't follow the argument. Info about the Weber function ...