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Questions about modular forms and related areas

1
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0answers
61 views

Generalized Shimura correspondence

(Sorry for my poor english) Let $f(z)\in S_{2k}(\Gamma_0(N))$ be a newform. Let $\chi$ be a Dirichlet character modulo $N$ and $\chi'$ be an unique even Dirichlet character modulo $4N$ associated to ...
7
votes
4answers
287 views

Fourier expansion at inequivalent cusps

Let $\Gamma\subset SL(2,\mathbb{R})$ be a Fuchsian group of the first kind. Let $c_1, c_2$ be inequivalent cusps of $\Gamma.$ Consider $f\in M_k(\Gamma)$ a weight $k$ holomorphic automorphic form, ...
5
votes
1answer
270 views

Complex Multiplication and algebraic integers

Let $q=e^{2\pi i\tau}$ and $$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty\frac{nq^n}{1-q^n}$$ be the Eisenstein Series of weight $2$ and let $E_2^*(\tau) = E_2(\tau) - \frac{3}{\pi\cdot Im(\tau)}$ be the ...
0
votes
0answers
38 views

Is there a formula for computing the parity of a sequence with discrete alphabet?

Suppose we have a sequence of number, whose alphabet is chosen from a discrete set such as (0,1,2). An example of such sequence is 0210122. Now I would like to determine if it is an odd permutation ...
3
votes
1answer
159 views

Basis for modular forms with Nebentypus character

(Sorry for my poor english..) Let $N$ be a positive integer and $\chi$ be a Dirichlet character modulo 4N. I already know that the $\mathbb{C}$-vector space $S_{k}(\Gamma_1(N))$ has a basis $\{F_1,\...
3
votes
1answer
113 views

Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p)/\operatorname{GL}_2(\mathbb Z_p))$ and Hecke operators

I was reading James Cogdell's notes here on automorphic representations and came to the following claim about the spherical Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p), \operatorname{GL}...
0
votes
0answers
78 views

Fixing quadratic surds

Assume that $n\equiv -1 \text{ }(\text { mod }24 )$ is a positive integer and $(n,48)=1$. Let $$S(n,48)=\bigg\lbrace \begin{pmatrix}r&s\\0& t \end{pmatrix}\colon r>0,rt=n,(r,s,t)=1,48\mid ...
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0answers
208 views

How explain these remarkable empirical observations about mod 3 modular forms of levels 1 and 5?

Define $b(n)$ by $b(0)= 0$, $b(3n)= 9b(n)$, $b(3n+1)= 9b(n) + 1$, $b(3n+2)= 9b(n) + 3$. (This sequence does not show up in the OEIS, but the similar Moser-de Bruijn sequence A000695 appears in many ...
2
votes
0answers
72 views

About newforms of half-integral weight

(Sorry for my poor english..) I have some questions about newforms of half-integral weight. In Mao's paper ("A generalized Shimura correspondence for newforms"), he said: "Ueda defined the set of ...
6
votes
0answers
96 views

$p$ -adic periods of modular curves X_0(71)

I have seen in some papers computation of $p$-adic periods of modular curves $X_0(N)$. Can somebody please explain to me what are the possible applications of such computations? as a concrete ...
1
vote
1answer
177 views

Roots of modular functions

Let $\mathfrak f(\tau)=e^{-\pi i/24}\frac{\eta\left(\frac{\tau+1}{2}\right)}{\eta(\tau)}=q^{-1/48}\prod_{n=1}^{\infty}\left(1+q^{n+1/2}\right)$ be the Weber modular function. The function $\mathfrak f$...
7
votes
1answer
177 views

Existence of newforms which are non-ordinary at a given prime

Let $f$ be a newform of weight $k \geq 2$ and level $N \geq 1$ without complex multiplication. A prime $p$ is said to be ordinary for $f$ if the $p$-th Fourier coefficient $a_p(f)$ is a $p$-adic unit (...
3
votes
1answer
152 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
1answer
107 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 ...
2
votes
1answer
129 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 ...
5
votes
0answers
93 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
1answer
120 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
98 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
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0answers
168 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
644 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
99 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
148 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
76 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
36 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.
8
votes
0answers
154 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
67 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 $N$, $\chi$...
3
votes
1answer
221 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 ...
9
votes
0answers
167 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
50 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
98 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
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0answers
69 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
4k 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
342 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
52 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
155 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
86 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
98 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/...
18
votes
2answers
947 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
312 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
166 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{...
20
votes
1answer
691 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
101 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
345 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
401 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 ...
7
votes
2answers
516 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
196 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
204 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
384 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 ...