Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

Questions about modular forms and related areas

3
votes
1answer
121 views

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(\...
7
votes
0answers
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 ...
1
vote
0answers
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 ...
4
votes
0answers
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} \...
2
votes
0answers
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 ...
2
votes
0answers
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 ...
3
votes
0answers
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). ...
10
votes
1answer
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 ...
7
votes
0answers
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 ...
5
votes
0answers
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. ...
3
votes
0answers
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 ...
3
votes
0answers
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.
6
votes
0answers
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},\...
0
votes
0answers
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
1answer
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 ...
8
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
1
vote
0answers
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\...
50
votes
2answers
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 ...
10
votes
2answers
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 ...
1
vote
0answers
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}...
3
votes
0answers
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 ...
3
votes
0answers
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\...
5
votes
0answers
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/...
9
votes
1answer
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(\...
2
votes
0answers
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 ...
5
votes
1answer
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 $\...
2
votes
1answer
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{...
19
votes
1answer
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}:=\...
3
votes
1answer
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 $\...
5
votes
1answer
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 $\...
6
votes
1answer
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 ...
2
votes
0answers
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 ...
4
votes
1answer
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 ...
3
votes
1answer
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 ...
3
votes
0answers
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. ...
13
votes
1answer
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 ...
3
votes
1answer
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 $\...
7
votes
1answer
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 ...
8
votes
0answers
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 ...
7
votes
1answer
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 ...
2
votes
0answers
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 ...
1
vote
1answer
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, ...
4
votes
1answer
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 ...
9
votes
0answers
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 ...
13
votes
1answer
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 \...
4
votes
1answer
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. ...
5
votes
0answers
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 ...
1
vote
1answer
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 ...