Questions tagged [modular-forms]

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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21 views

Fourier Coefficients of Eisenstein Series $G_{2k}(\tau)=\sum\limits_{(m,n)\in\Bbb Z^2\setminus \{(0,0)\}}\frac{1}{(m+n\tau)^{2k}}$.

Suppose $\tau\in\Bbb C$ and $\Im(\tau)>0$. Also, let $k\in\Bbb Z_{>2}$, and $A=\Bbb Z^2\setminus\{(0,0)\}$. Then the Eisenstein series $G_{2k}(\tau)$ is given by $$G_{2k}(\tau)=\sum_{(m,n)\in A}\...
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Order of vanishing of $F(z)=\Delta(z)\Delta(5z)$ as a cusp form of weight 24 for $\Gamma_1(5)$ at zero

Let $F(z)=\Delta(z)\Delta(5z)$. We can prove that F is a cusp form of weight 24 for $\Gamma_1(5)$. Then, what is the order of vanishing of F at 0?
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$\infty$ is a regular cusp for the group $\Gamma_1(5)$

How can we show that $\infty$ is a regular cusp for the group $\Gamma_1(5)$?
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18 views

Cusp forms with different orders at infinity

Let’s assume we have cusp forms $f_1,...,f_n\in S_k(\Gamma_1(N))$ which they are zero at infinity with different orders. Then, how can we show that they are linearly independent?
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Why do we care whether Hecke algebras are complete intersections?

The title really says it all. I get the impression that proving certain Hecke algebras are complete intersections is a crucial step in the proof of Fermat's Last Theorem. But how do you use a result ...
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Question on paper of Mazur, Tate, Teitelbaum and $p$-adic L functions of modular forms

I'm trying to fill in the details in proposition 14 of this paper by Mazur, Tate, and Teitelbaum. In particular, I'd like to understand the following. Let $f$ be a cuspidal eigenform of weight $k$ ...
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53 views

Dimension of space of Eisenstein series $\mathcal{E}_k(\Gamma)$

Let $\Gamma$ be a congruence subgroup of $\text{SL}_2(\mathbb{Z})$, and define the space of Eisenstein series for $\Gamma$ to be $\mathcal{E}_k(\Gamma):=\{f\in M_k(\Gamma)\colon \langle f,S_k(\Gamma)\...
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Why do the Galois and Hecke action commute on the Picard group of the modular curve $X_1(N)$?

In chapter 9 of Diamond-Shurman's book A First Course in Modular Forms, when they construct the Tate module associated to the modular curve $X_1(N)$, they state that the Galois action and Hecke action ...
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Where (or How) can I find Fourier expansion of antiholomorphic (antianalytic) version of modular J-invariant, i.e. Fourier[ J( ̅t ) ].

It is known that Fourier expansion of the modular J-invariant is J(q)= 1/q + 196884q + ... It would be greatly appreciated if someone advise me where (or how) can I find the Fourier expansion of ...
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Dedekind zeta functions as Mellin transforms

This is surely well known and I might even have seen it before: The Riemann zeta function can be expressed as a Mellin transform: $$\Gamma(s)\zeta(s) = \int_0^\infty x^{s-1}\frac{1}{e^x - 1}dx.$$ ...
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Why are the $p$-oldforms $f(z)$ and $f(pz)$ linearly independent at level $\Gamma_0(pN)$?

Let $f$ be a newform (normalized eigenform) of weight $k$ and level $\Gamma_0(N)$. Fix $p$ not dividing $N$ and set $f_p(z)=f(pz)$. Viewing $f$ and $f_p$ at level $\Gamma_0(pN)$, why are they ...
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61 views

Two congruence subgroups of $SL_2(\mathbb{Z})$

In many references about modular forms, I see that they just only focus on two congruence subgroups of $SL_2(\mathbb{Z})$, namely $$\Gamma_1(N)=\bigg\{ \gamma \in SL_2(\mathbb{Z}) \bigg\vert \gamma \...
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46 views

$N$-th root of modular forms

In my research, I have encountered many $q$-series of functions that turn out to be the Fourier expansions of roots of modular forms. Examples are $n$-th roots of Eisenstein series and the $j$-...
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44 views

Is there a dimension formula for modular forms of weight $1$ for any congruence subgroup?

If $M_1(\Gamma )$ denotes the vector space of modular forms of weight $1$ for $\Gamma$, is there any formula for $\text{dim}(M_1(\Gamma ))$?
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Equalities of the Petersson inner product for two related modular forms

Let $\Gamma_1(N)$ be the usual congruence subgroup of $\text{SL}_2(\mathbb{Z})$ and let $f\in S_k(\Gamma_1(N))_{\text{new}}$ be a normalized primitive form, and write $f=\sum_{n\geq 1}a_nq^n$. Let $...
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Rational independence of $j$-invariants

Suppose $K_1$ and $K_2$ are imaginary quadratic fields of relatively prime discriminant $d_1$ and $d_2$, and let $j_i = j(\mathcal{O}_{K_i})$. Then $[\mathbb{Q}(j_i):\mathbb{Q}]=|C(\mathcal{O}_{K_i})|$...
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Why are the coefficients zero if $n$ contains a prime with prime power $1$ that is not split in $K$?

Let $K=\mathbb{Q}(\sqrt{D})$ with discriminant $D<0$ and $\mathfrak{a}$ be an ideal in $K$. For $Q \in \mathbb{N},\rho \in \mathfrak{a},Norm(\mathfrak{a})=A,$ we define the theta series $$\vartheta(...
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If $\sum_{i=n}^{p-1}\frac{1}{2^i}\binom{i-1}{n-1}\equiv X \pmod p$, can we find $𝑋$ s. t. $𝑋$ is independent of $n$?

Let $p$ be a prime number and $n\in\Bbb N\setminus\{1\}$. We have $$\sum_{i=n}^{p-1}\frac{1}{2^i}\binom{i-1}{n-1}\equiv X \pmod p$$ How can we find $X$ s. t. $X$ is independent of $n$ ( or in a much ...
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Details in Deligne-Serre “Formes modulaires de poids 1”

There are specific details which I'm a little stuck on in Deligne and Serre's paper on attaching Galois representations to modular forms of weight 1. In the proof of Lemma 8.3, they use the ...
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modular symbols and the calculation of homology

We have fundamental domain $F$ but we have constructed a larger triangle $F+?$ Why do we need $F+?$
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Isogenies between $\mathbb{C}^*/q^\mathbb{Z}$ and $\mathbb{C}^*/q'^\mathbb{Z}$

Let $\Delta = \{ q \in \mathbb{C}^* | |q| \lt 1\}$, $q \in \Delta$, and $f : \mathbb{C}^*/q^\mathbb{Z} \to E'$ be an isogeny of degree $n$. Then there exists the unique pair $(a, q')$, where $a$ is ...
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1answer
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Why is the oldform map injective?

Consider the space of cusp forms $S_k(\Gamma_0(N))$; it has two different maps to $S_k(\Gamma_0(Np))$ where $(p, N) = 1$. We can combine them into a map $$S_k(\Gamma_0(N)) \oplus S_k (\Gamma_0(N)) \to ...
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59 views

How do doubly-periodic functions correspond to algebraic curves?

I've been working through some problems in complex analysis (Hille's Analytic Function Theory) and I came across the Weierstrass $\mathscr{P}$ function defined as follows: $\mathscr{P}_{\Lambda}(z)=\...
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Poincare series with character

The classical Poincare series $P(z; m, k, N)$ for $m > 0$ are defined by $$P(z; m, k, N) := \sum_{\gamma \in \Gamma_{0}(N)_{\infty}\setminus \Gamma_{0}(N)} q^{m} \mid_{k} \gamma,$$ where $\...
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Functoriality in Modular Forms

So I'm still a little new to Modular Forms. I have started reading Lang, I have watched Keith Conrad's lectures and I have read his notes. I started to ask myself some questions. Let $\Gamma$ and $\...
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A linear operator on meromorphic functions

I am currently reading Lang's book Introduction to Modular Forms, and perhaps I'm wrong, but I may have spotted something weird he said. Either I don't get it, or there's something abnormal about the ...
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1answer
43 views

Limit of $\sum_{d}\sum_{c\neq0}\left(\frac{1}{cz+d}-\frac{1}{cz+d+1}\right)$

To evaluate $$\sum_{d\in\mathbb Z}\sum_{c\in\mathbb Z\smallsetminus\{0\}}\left(\frac{1}{cz+d}-\frac{1}{cz+d+1}\right)$$ where $z$ on the upper half plane of $\mathbb C$. We first consider a finite ...
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Set of cusps of the congruence subgroup $\Gamma_{1}(p)$, where $p$ is an odd prime number.

Let $p$ be an odd prime and consider the congruence subgroup of $\mathrm{SL}_{2}(\mathbb{Z})=\left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \ \Big | \ a,b,c,d\in\mathbb{Z}, \ ad-bc=1 \...
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Characterization of the subgroups of finite index of that stabilizer of $\infty$ in the modular group $\mathrm{SL}_{2}(\mathbb{Z})$

This text is only to provide some context but it is not needed. Let $\mathrm{SL}_{2}(\mathbb{Z})=\bigg\{\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\mathbb{Z}^{2\times 2} \ \Big| \ ad-bc=1 ...
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1answer
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Linear independence of $G_4$ and $G_6$ in entire modular forms

It is known that( it is a theorem) every entire modular form of weight k is a polynomial in $G_4 $ and $ G_6$ of the type f = $\sum_{a, b} c_{a, b} G_{4}^a G_{6}^b $ where the $c_{a, b} $are complex ...
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26 views

Lower bound for order of pole at $\infty$ of modular functions level $N$

If the modular curve $X_{0}(N)$ has genus $0$, its function field is generated by a single element $j_{N}(z)$ with pole of order $1$ at the cusp $\infty$ and holomorphic at all other cusps (this is ...
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1answer
78 views

Modularity in terms of the stack of elliptic curves

Modular forms can be viewed as something like global sections of (some tensor power of) the canonical line bundle on the stack $\mathcal M_{\text{ell}}$ of (generalized) elliptic curves. Is it ...
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Various variants of modular forms

There are many different generalizations of modular forms. One has Hilbert modular forms, Siegel modular forms, Maass wave forms, Jacobi forms, and then there are various generalities of automorphic ...
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Fundamental domain for congruence subgroup.

Let $\Gamma$ be a congruence subgroup of the modular group $\mathrm{SL}_{2}(\mathbb{Z})$. Let $R$ be coset representatives of the quotient $\Gamma\setminus\mathrm{SL}_{2}(\mathbb{Z})$ and let $\...
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Order of a Modular Function at a Point is Invariant under $SL_2(\mathbb{Z})$ Action: Interpretation and Proof

In chapter VII of Serre's 'A Course in Arithmetic', he defines the order of a meromorphic function $f$ (on the upper half plane $\mathbb{H}$) at $p \in \mathbb{H}$ to be the integer $n$ for which $\...
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Proof of Holomorphicity of Eisenstein Series in Serre's 'A Course in Arithmetic'

I was reading the proof of Proposition $4$ in Chapter VII of Serre's 'A Course in Arithmetic'. In order to prove that the Eisenstein Series $G_k(z)$ of index $k>1$ is holomorphic on the upper half ...
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Identifying Modular Functions of Weight $2k$ with Lattice Functions of Weight $2k$

I have been reading Serre's 'A Course in Arithmetic'. In section $2.2$ he shows that lattice functions $F$ of weight $2k$ are in one-to-one correspondence with modular functions of the same weight, by ...
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1answer
43 views

Proof of Meromorphicity of Eisenstein Series in Serre's 'A Course in Arithmetic'

I was reading the chapter on Modular Forms from Serre's 'A Course in Arithmetic'. In the first line of his proof that the Eisenstein Series $G_k(z)$ is a modular form, he says, "The above arguments ...
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2answers
54 views

$SL(2, \Bbb Z)$ has only one cusp

Let $\Gamma$ be a congruence subgroup of $SL(2, \Bbb Z)$. A cusp is an equivalence of $\Bbb Q\cup\{\infty\}$ under $\Gamma$-action. What's the meaning of "equivalence $\Bbb Q\cup\{\infty\}$ under $\...
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Do all cuspidal automorphic representations of $\operatorname{GL}_2(\mathbb A_{\mathbb Q})$ come from Maass or holomorphic cusp forms?

A normalized cuspidal newform $f$ (either holomorphic or Maass) can be identified with a function on $\phi: \operatorname{GL}_2(\mathbb Q) \backslash \operatorname{GL}_2(\mathbb A_{\mathbb Q})$, and ...
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Computing the width of a cusp of a congruence subgroup of level $N$ “in characterstic $N$”.

Let $\Gamma$ be a congruence subgroup of $\text{SL}_2(\mathbb{Z})$ of level $N$ and let $\pi:\text{SL}_2(\mathbb{Z})\to\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ be the reduction map, which is surjective. ...
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2answers
63 views

Why are the Fourier coefficients of a modular form constant?

Let $f$ be a holomorphic modular form (of given weight and level one). Since it $1$-périodic and of moderate growth, it has a Fourier expansion, but this one is a Fourier expansion in $x$, that is $$f(...
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1answer
45 views

Definition of congruence subgroup $\Gamma_0(N)$ when $N$ is negative

This is probably a confusion or a matter of notation. The congruence subgroup $\Gamma_0(N)$ is defined as $$\Gamma_0(N)= \left\lbrace \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in \text{SL}...
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Allowed ramification of deformations of Galois representations

I am trying to learn about deformations of degree 2 Galois representations mod $p$ and get a grasp of simple intuitions on examples. In basic references the explicit examples of universal deformation ...
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1answer
84 views

Let $\frac{r}{2t}\in\mathbb{Q}$ with $r,t$ odd, then $\frac{r}{2t}\in\Gamma_1(4)*\frac{1}{2}$ with $\Gamma_1(4)$ a principal subgroup.

Let $\Gamma_1(4)$ be the principal subgroup $\bigg\{\gamma\in\text{SL}_2(\mathbb{Z})\colon\gamma\equiv\begin{pmatrix}1&b\\0&1\end{pmatrix}\mod(4)\bigg\}$. I am trying to show that the cusps ...
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23 views

Composition of a modular function with the inverse of a modular function

Let $f$, $g$ be two modular functions. Then, how can we show that $fog^{-1}$ is a meromorphic function on the extended complex numbers? I am aware that it is well-defined.
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37 views

Meromorphic functions on upper half plane and infinity

Is the set of all meromorphic functions on the upper half plane which they are also meromorphic at infinity, a field? I am trying to show that the set of all modular functions for $Sl_2(\mathbb{Z})$ ...
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1answer
52 views

Principal congruence subgroup $\Gamma(DN)$ and $\Gamma(N)$

Let $\Gamma(N)$ denotes the principal congruence subgroup of level $N$ and $\beta$ be a $2 \times 2$ matrix with integral entries and deteminant $D$. Prove that $\beta \Gamma(DN) \beta^{-1}$ is ...
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1answer
40 views

how to find the dual basis

Let $\mathfrak{a}\subset \mathcal{O_K},K=\mathbb{Q}(\sqrt{D}),D<0$ with square free discriminant $D$. I want to calculate the dual lattice $\mathfrak{a}^*$ of $\mathfrak{a} $ relative to the ...
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1answer
32 views

Definition of Diamond Operator

I'm studying modular forms, but I can't understand the definition of diamond operator. Why can I define for all $\alpha$ with $\delta \equiv d$? I can't understand the reason why two different matrix ...

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