Questions about modular forms and related areas

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### How do modular functions of level $N>1$ transform under the full modular group?

Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$
First question
What can we say in general about the factor $j(\...

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55 views

### Is anything known about this class of series involving the divisor function?

I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it!
Let $N\in\mathbb{N}$, let $q$ be a point in the open ...

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vote

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35 views

### Reference request: the dual Coleman family

Recently when I want to understand the construction of triple product p-adic L-function, I am really confused by the notion of dual form. To be precise, assume $f^\circ\in{S_k(N,\chi)}$ is an ...

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votes

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78 views

### Integrals of modular forms

Setup: Write $G = \text{SL}_2(\mathbf{R})$ and $\Gamma = \text{SL}_2(\mathbf{Z})$.
Let $f$ be a modular form on $\mathbf{H}$ of weight $2k$, so that
$$f(gz) = f(z) (cz + d)^{2k} \qquad \text{for} \...

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65 views

### Twisted modular equation

Let $\gamma_2(\tau)=j(\tau)^{1/3}$. The modular equation shows that the functions $$j\left(\frac{a\tau+b}{c\tau+d}\right),\qquad ad-bc=n$$
are integral over $\mathbf Z[j]$. Under what conditions is ...

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90 views

### Expressing modular functions of level 9 and 32 as rational functions

Let
$$\gamma_2(\tau)=j(\tau)^{1/3},\qquad \mathfrak f_1(\tau)=\frac{\eta(\tau/2)}{\eta(\tau)},$$
where $j$ is the modular invariant and $\eta$ is the Dedekind eta function. Cox in his Primes of the ...

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159 views

### Fricke involution on GL(3)

Define $\Gamma_0(N)=\{\begin{pmatrix}
a&b&c\\
d&e&f\\
g&h&i
\end{pmatrix}
\in SL(3,\mathbb{Z})|g\equiv h\equiv 0(\mod N)\}$ be the $N$-level congruence subgroup on GL(3).
...

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votes

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636 views

### Why are the coefficients of the modular equation so large?

The modular equation $\Phi_n(X,Y)$ is a polynomial in $\mathbf Z[X,Y]$ relating the modular invariant $j$ and the functions
$j\left(\frac{a\tau+b}{c\tau+d}\right)$, where $ad-bc=n$.
For example, we ...

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93 views

### A divisibility property of Fourier coefficients of modular forms

Let $f = \sum_{n} a_n q^n$ be a meromorphic modular form with integral Fourier coefficients $a_n$. For various classes of such forms the divisibility property
$$
n|a_n~\forall n\in\mathbb{N}
$$
arises ...

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135 views

### Eisenstein series of Hilbert modular forms

I am reading Shimura's paper "The Special Values of the Zeta Functions Associated With Hilbert Modular Forms" and I do not exactly understand his definition of the Eisenstein series in section 3.
...

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72 views

### Reference request: basics about modular curves

Where can I find a reference (with carefully written proofs) for basic facts about modular curves? Namely:
Congruence subgroups
The open modular curve $Y_\Gamma$ admits the structure of a Riemann ...

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35 views

### Joint Sato-Tate equi-distribution formula for two Hilbert modular forms

Is there any Joint Sato-Tate equi-distribution formula for a pair Hilbert cusp forms ?
Like for classical case we have Joint Sato-Tate equi-distribution formula for a pair cusp forms.

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143 views

### Surjectivity of reduction for Hilbert modular forms

Fix a totally real field $K$, a level $\mathfrak{n}$, a (parallel) weight $k\geq 2$ and a primitive ray class character $\chi$ modulo $\mathfrak{n}$.
Then one can form the space $S_k(\mathfrak{n},\...

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60 views

### Summation formula for twisted L-function

Does any expert here know something about the summation formula of the Voronoi type for the sum $$\sum_{n\le X} a_{f}(n)\chi(n) e\left(\frac{an}{c}\right)?$$
Here $f$ is a newform of level $N$, $\chi$...

**2**

votes

**1**answer

143 views

### An explicit correspondence for reductions of modular curves $Y(N)$

Let $Y(N)$ be the modular curve associated with the principal congruence subgroup $\Gamma(N) \subset \mathrm{SL}(2, \mathbb{Z})$ of level $N \in \mathbb{N}$. It is well known that this curve has a ...

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153 views

### What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal endomorphism ring?

Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added ...

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49 views

### action of formal tori $I^\mathrm{ext}$

this is a question about the action of the formal tori defined in recent papers of Andreatta, Iovita and Pilloni. The notations are heavy, so I will follow the paper Triple product p-adic L-functions ...

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votes

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96 views

### A question about Hasse Invariant and Modular curve

Let $N\geq4$ be a positive integer and p be a prime such that $(p,N)=1$, and $X=X_1(N)$ be the modular curve parameterizing (generalized) elliptic curves with $\Gamma_1(N)$-level structure. Base ...

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65 views

### Modular transformation of affine characters of non-simply connected groups$.$

Consider an (untwisted) affine algebra corresponding to a compact and simply-connected Lie group $G$. Under a modular transformation, its characters transform as (cf. 9612078)
$$
\chi_\mu\to\sum_{\nu\...

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votes

**2**answers

3k views

### How were modular forms discovered?

When modular forms are usually introduced, it is by: "We have the standard action of $SL(2,\mathbb Z)$ on the upper half-plane, so let us study functions which are (almost) invariant under such ...

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votes

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328 views

### Effective bound on the expansion of the $j$-invariant

The $j$-ivariant has the following Fourier expansion
$$j(\tau)=\frac 1q +\sum_{n=0}^{\infty}a_nq^n=\frac{1}{q}+744+196884q+21493760q^2+\cdots.$$
Here is $q=e^{2\pi i \tau}$.
Is there some simple ...

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49 views

### The Geometry of Jacobi Forms and their Asymptotic Expansions

A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying
$$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}...

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149 views

### Generalization of Weil's theorem for L-functions

I have a question reffering to a theorem by Weil, which gives sufficient conditions that a given L-series $$ L(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}$$ which is convergent somewhere comes from a ...

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78 views

### Rankin-Selberg Method

Referencing Don Zagier's notes (Utrecht), and also paper for moderate growth version.
The Rankin-Selberg transform can be defined by $\mathcal{R}(f,s) := \int_0^\infty \tau_2^{s-2} a_0(\tau_2) d\...

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96 views

### Factors of the Jacobian of modular curves

Let $J_1(p)$ be the Jacobian of the modular curve $X_1(p)$ for p an odd prime. We know that $J_1(p)$ is isogenous to a direct sum of abelian varieties $\oplus_{f}A_f$ where the sum runs over Hecke/...

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537 views

### Why are values of Eisenstein $E_2^*$ algebraic integers?

I'm looking for a proof that the following term is an algebraic integer whenever $\tau_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$:
$$A_N:=\sqrt{-N}\cdot\frac{E_2(\...

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241 views

### Zagier's algebraicity of singular moduli

Let $\mathcal M_m\subset M(2,\mathbb Z)$ be the set matrices with determinant $n$. The modular group $\Gamma$ acts on $\mathcal M_m$ from the left and we have the following finite set as a set of ...

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votes

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281 views

### How to compute Coefficients in Chudnovsky's Formula?

My aim is to understand all three coefficients arising in the Chudnovsky-Formula (see also Question 300385). Two of them are easily computed, but I failed with the third:
It is known that for all $\...

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votes

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164 views

### Modular parametrization of a curve of Heegner and Weber

The curve
$$(X-16)^3=XY\tag{1}\label{1}$$
is essential to Heegner's approach to the class number one problem for imaginary quadratic fields. We have the following “modular” parametrization
\begin{...

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votes

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592 views

### Why does this quasi-modular function have integral values?

It is a well-known result that the modular function $1728J(\tau) := \frac{1728E_4(\tau)^3}{E_4(\tau)^3-E_6(\tau)^2}$ has integral values if $\tau$ has class number 1 - for example at $\tau_{163}:=\...

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97 views

### Representatives for the action on an unknown set of matrices by an unknown modular subgroup

Let $\Gamma$ be the modular group and $\mathcal M_n$ the set of all primitive matrices with determinant $n\geq1$. Recall that a primitive matrix has relatively prime entries.
The modular group $\...

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331 views

### About the proof of Weil-Langlands theorem

The statement of the theorem is as follows:
Let $\rho$ be an irreducible two-dimensional representation of $G_\mathbb{Q}=Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ with Artin conductor $N$.
Suppose that $\...

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324 views

### Behavior of a modular form in the lower strip

Let $f$ be an (elliptic) modular form of weight $k>0$, and consider the vertical strip $S_m=\{x+iy\in\mathbb{C}:|x|\le 1/2, y>m$}. For every $m\ll 1$, the fundamental domain for $SL_2(Z)$ is ...

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71 views

### Isomorphism between Group Schemes over $\mathbb{Z}_2$

Consider the Matrix
$$M:= \left(\begin{matrix}
1 & 0 \\
0 & d
\end{matrix}\right)
$$
for an odd $d \in \mathbb{Z}$.
Define the group scheme $Sp(g)$ defined over $\mathbb{Z}$ with ...

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votes

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366 views

### How to compute Dedekind eta function efficiently?

According to wiki: https://en.wikipedia.org/wiki/Dedekind_eta_function, Dedekind eta function is defined in many equivalent forms. But none of them is an explicit description (say in algorithmic ...

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194 views

### The degree of the cube root of the $j$-invariant

I have a question which is fairly elementary, but first I must provide relevant context. Without it, my question would seem rather arbitrary and scarcely interesting. Note also that my question can be ...

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198 views

### The uniqueness of Poincaré metric

The Poincaré metric $ds=\frac{\sqrt{dx^2+dy^2}}{y}$ has the proprety that the action of the group $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I_{2}\}$ on $\mathbb{H}$ preserves the hyperbolic distance.
...

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358 views

### Arithmetic motivations for modularity in higher rank

The classical setting of modularity is that one can associate elliptic modular forms (or automorphic representations of GL(2)/$\mathbb Q$) to elliptic curves over $\mathbb Q$. This has far-reaching ...

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169 views

### Degeneracy maps and cusps

Let $N$ be a positive integer and $p$ a prime not dividing $N$. Let $X_0(N)$ be the modular curve (over $\mathbb{Q}$) associated to the congruence subgroup $\Gamma_0(N)$, which is the subgroup of $\...

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233 views

### Why does the class equation have rational coefficients?

Let $j$ be the Klein modular invariant and let $\Gamma=SL_2(\mathbb Z)$ be the modular group. Consider the set of primitive quadratic forms $ax^2+bxy+cy^2$ of discriminant $d<0$.
The root of a ...

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177 views

### Genus=2 theta functions, Arnold's relation, and KZ connection

Let $C_5:=\{{(z_1 \dots, z_5) \in (\mathbb{C})^5 | z_i \neq z_j \forall i\neq j }\}$ be the configuration space of five distinct ordered points in $\mathbb{C}$. Arnold showed that the holomorphic one ...

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308 views

### How is the Eichler-Shimura congruence related to L-functions?

My understanding is that the Eichler-Shimura relation expresses the Hecke operator $T_p$ in terms of the geometric Frobenius map. Specifically, $T_p = Frob + Ver$ for Frobenius map $Frob$ and it's ...

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80 views

### On the references of Eichler-Shimura isomorphism

Good days, everyone. I am now trying to understand the so-called Eichler-Shimura isomorphism. For the references, I am now reading the book "Introduction to the Arithmetic Theory of Automorphic ...

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114 views

### Weight, Index, and Congruence Subgroup of Classical Jacobi Theta Functions

On the very first page in the Introduction of Eichler and Zagier's text on Jacobi forms, they mention that the theta function
$$\Theta_{x_{0}}(\tau, z) = \sum_{x \in \mathbb{Z}^{N}} q^{Q(x)} y^{B(x, ...

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114 views

### Hecke operators for hermitian modular forms of general level

It has bug me for a while that I don't have a good understanding of the theory of Hecke operators. For elliptic modular forms, it was explained in Koblitz's book that they arose from viewing the ...

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203 views

### Non-vanishing modular forms

Prompted by this MO question, I have the following question about modular forms which do not vanish on the upper-half plane.
Q1. Let $N \geq 1$ be an integer and let $\Gamma(N)$ be the principal ...

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335 views

### Is $\eta(\tau)^2$ a modular form of weight 1 on $\Gamma(12)$?

As we know, the Dedekind eta function $\eta(\tau)$ acquires a phase $\exp(2\pi i/24)$ under the modular transformation: $\tau \rightarrow \tau+1$. Therefore $\eta(\tau)^2$ is invariant under $\tau \...

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223 views

### A question on Deligne-Serre lifting lemma

Good day, everyone. I have been studying the paper "Formes modulaires de poid 1" written by Deligne and Serre. I am confused by lemma 6.11, or the so-called Deligne-Serre lifting lemma in the paper.
...

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92 views

### Explicit description of zeros of some cusp forms of weight 2 on $\Gamma_0(p)$

The question below originates from another post on MathOverflow and a (successful) attempt to prove the Chudnovsky algorithm with modular forms arising from quaternion algebra.
Motivation: As is ...

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116 views

### How does Siegel's Hilbert-Blumenthal fundamental domain differ from Götsky's?

This question has been changed to something related but different from the original question. Thanks to @paulgarrett for chatting with me and helping me hone in on a more interesting part.
The first ...