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Questions tagged [modular-forms]

Questions about modular forms and related areas

5
votes
2answers
102 views

What do we know about the ramification of the modularity map $X_0(N)\to E$?

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $N$ be its conductor. By the modularity of elliptic curves over $\mathbb{Q}$, there exists a surjective map $f:X_0(N)\to E$, where $X_0(N)$ is ...
7
votes
1answer
133 views

Invariants in relative cohomology and compact support cohomology of the quotient

Let $\cal H$ be the Poincare upper half-plane and $\overline {\cal H}$ the union of $\cal H$ with the set of cusps $\bf P^1 (\bf Q)$, provided with its usual topology. Let $\Gamma$ a congruence ...
9
votes
1answer
265 views

Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?

Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$? Do there exist ...
1
vote
0answers
39 views

A conjectural formula for the “minimal degree function”, $k\rightarrow d\min(k)$, attached to recursion, $f\rightarrow A(f)$, in char $3$

THE RECURSION: $f\rightarrow A(f)$ $A: t(\mathbb{Z}/3)[t^3]\rightarrow t(\mathbb{Z}/3)[t^3]$ is the $\mathbb{Z}/3$-linear map with $A(t)=0, A(t^4)=t, A(t^7)=t^4$, and $A((t^9)f)=(2t^9)A(f)+(t^3)A((t^...
4
votes
0answers
99 views

Hecke operators that lower level

I am working with weakly holomorphic modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N)...
4
votes
0answers
148 views

Is there some computational evidence of the $pq$ analog of Serre's conjecture?

The $pq$ analog of Serre's conjecture (see "Mod pq Galois representations and Serre's Conjecture"- Khare, Kiming) states that if $\bar{\rho}_1:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F}_p)$ is a ...
2
votes
1answer
98 views

When does the double coset representative for a congruence subgroup contain a $\text{SL}_2(\mathbb{Z})$-conjugacy class?

In the paper p-adic L-functions and p-adic periods of modular forms, Greenberg/Stevens assert that if $\sigma_l:=\begin{pmatrix}l&0\\0&1\end{pmatrix}$ is the usual Hecke operator at $l$ double ...
23
votes
1answer
691 views

Intuitive reason why the $j$-invariant is a cube?

Let $\tau$ be a CM point of discriminant $D$. Assume that $D$ is not divisible by $3$. Then $j(\tau)$ is an algebraic integer of degree equal to the class number $h(D)$. Let $ \gamma_2(\tau)=j(\tau)^{...
1
vote
0answers
38 views
+100

Probability distribution from equidistribution (prime and semiprime case) - III

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ norm of vector $(x',y')$ as $(x',y')$ ranges over all non-zero integral solutions to $(x',y')\...
4
votes
1answer
173 views

Modular forms and Period Polynomials

1.) What is the importance of special values of L functions in connection to weakly holomorphic modular forms? Why is the study of special values a subject of intense study except the fact it is ...
4
votes
0answers
159 views

A website which explains Mazur's torsion point theorem

I'm about to read Mazur's paper "Modular curves and the Eisenstein ideal". It's so long and difficult for me, but I found a website which shows the Mazur's theorem. This is very short and very very ...
1
vote
0answers
77 views

Probability distribution from equidistribution - II

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$ and denote $N(a,b)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
3
votes
0answers
110 views

Probability distribution from equidistribution - I

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
2
votes
1answer
85 views

Probability density from standard domain

Pick $x+iy$ at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$. What does the probability distribution function of $\frac1{\sqrt y}$ look like?
3
votes
1answer
125 views

Conductor of Galois representation attached to newform

(Sorry for poor my english skill..) Let $k$ and $N$ be positive integers and $\chi$ be a Dirichlet character modulo $N$. Let $F$ be a newform with number field $K_{F}$. (All coefficients of $F$ in $...
5
votes
0answers
108 views

List of applications of the Deformation theory of Galois Representations in contexts other that $R=T$

The Deformation theory of Galois representations developed by Mazur has been extensively used in proving that Galois representations with special properties are modular/automorphic through the ...
5
votes
1answer
139 views

The $\mathbb{Q}$-rational cuspidal group of $J_0(N)$

Let $N$ be a positive integer and consider the modular curve $X_0(N)$ over $\mathbb{Q}$. Also, consider the Jacobian variety $J_0(N)$ of $X_0(N)$, which is an abelian variety defined over $\mathbb{Q}$....
2
votes
0answers
94 views

Non-cuspidal Hecke eigenforms and Eisenstein series

It's a direct check that ${\displaystyle E_{2k}(z )=\frac{\zeta(1-2k)}{2}+\sum _{n=1}^{\infty }\sigma _{2k-1}(n)q^{n}}$ is an eigenform for every Hecke operator $T_n$ with eigenvalue $\sigma_{2k-1}(n)$...
5
votes
0answers
85 views

Congruence between modular forms

This might be a very vague question since I am not very familiar with the theory of automorphic forms. Let $G$ be a connected reductive algebraic group defined over $F$ (a number field). Suppose we ...
4
votes
1answer
417 views

Prerequisites for reading papers of arithmetic such as Ribet, Mazur, Faltings, Wiles

I've studied some fundamentals of algebraic geometry and number theory, and now I want to read papers which seem to be the "main stream" of frontier research on arithmetic. I've heard that Mazur's "...
3
votes
1answer
97 views

Newform of Half-integral weight modular forms

(Sorry for my poor english..) Let $k$ and $N$ be integers. Let $f\in S_{k+\frac{1}{2}}(\Gamma_1(4N))$ be a half integral weight modular form. I know that if $g \in S_{k}(\Gamma_1(N))^{new}$ in ...
9
votes
1answer
283 views

Is $j(\tau)^{1/3}$ the hauptmodul for the congruence subgroup generated by $\tau\rightarrow\tau+3, \tau\rightarrow-1/\tau$?

The 3rd root of the modular invariant $j$ is $$ j(\tau)^{1/3}=q^{-1/3}(1+ 248q+ 4124q^2+ 34752q^3+\cdots),$$ where $q=e^{2\pi i \tau}$. I was wondering if $j(\tau)^{1/3}$ the hauptmodul for the ...
1
vote
0answers
69 views

Dimension formulae for Jacobi forms

I'm interested in locating dimension formulae for (more general) Jacobi forms associated with a lattice $L$ (where the Jacobi forms of Eichler-Zagier correspond to $L=A_1$). Unfortunately, the ...
7
votes
1answer
256 views

Algebraic independence of $P,Q,R$ or $E_2,E_4,E_6$ over $\mathbb C(z)$

Let $P,Q,R$ be the Fourier series of the Eisenstein series $E_2,E_4,E_6$, that is, $$ P(q)=1-24\sum_{n=1}^{\infty}\sigma_1(n)q^n, $$ $$ Q(q)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n, $$ $$ R(q)=1-504\...
1
vote
0answers
70 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
346 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, ...
6
votes
1answer
293 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
40 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
169 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
121 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
79 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 ...
9
votes
0answers
214 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
77 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 ...
7
votes
0answers
110 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
184 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
180 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
160 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
116 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
132 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
99 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
121 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
99 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
172 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
649 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
102 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
159 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
84 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
37 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
156 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
68 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 $N$, $\chi$...