Questions tagged [modular-forms]
Questions about modular forms and related areas
1,423 questions
2
votes
0
answers
123
views
Special $x$-values in $k^2 ≡ 64(x^3-n^2x) \pmod{p}$: connection to elliptic curves?"
I am investigating a system of congruences related to the elliptic curve $y^2 = x^3 - n^2x $ (congruent number curve) with the following conditions:
System:
$$k^2 ≡ 64(x^3 - n^2x) \pmod{p}$$ where 'p' ...
- 337
2
votes
0
answers
66
views
Normalization of Hecke operators on overconvergent Siegel modular forms
In their paper p-adic families of Siegel modular cuspforms, Andreatta, Iovita and Pilloni defined operators $U_{p,i},i=1,2,...g$ on the spaces of overconvergent modular forms of arbitary $p-$adic ...
- 151
0
votes
0
answers
129
views
How to construct elliptic functions with predescribled zeros and poles by means of Weierstrass ℘-function and its derivatives?
Given a lattice $\Lambda=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ in $\mathbb{C}$. It's well known that given $n_i\in\mathbb{Z}, z_i\in \mathbb{C}$ satisfying $\sum n_i z\in\Lambda$ and $\sum n_i=0$, ...
2
votes
0
answers
148
views
Can a holomorphic cusp form become a CM form？
Given a holomorphic cusp form $f$ with weight $k \geq 2$, level $N$ and trivial nebentypus. I am wondering if $f$ can be a CM (dihedral) cusp form, i.e. $f$ is isomorphic to its quadratic twist. ...
- 479
6
votes
1
answer
213
views
Another question on large sieve inequality
Recently, I was interested in the large sieve inequalities. A few days ago, I came up with a question on the large sieve inequality involving 𝐺𝐿(2); see On the large sieve inequality involving $GL(2)...
- 191
3
votes
1
answer
180
views
On the large sieve inequality involving $GL(2)$ harmonics
I have a question on the large sieve inequality involving $GL(2)$ harmonics. Recall that one has the analog for $GL(1)$ harmonics that, for any complex numbers $\alpha_m,\beta_n$, one has
$$\sum_{q\le ...
- 191
5
votes
0
answers
181
views
Where can one find the page of corrections to Lehner–Newman's Weierstrass point paper?
In Ogg, Andrew P., Hyperelliptic modular curves, Bull. Soc. Math. Fr. 102, 449-462 (1974). ZBL0314.10018. p. 450, there is the sentence:
“As LEHNER and NEWMAN noted in a page of corrections attached ...
- 2,284
4
votes
0
answers
114
views
A question on the spectral large sieve
Let $q\in \mathbb{N}$. Denote by ${B}(q,\chi)$ an orthogonal basis of $GL(2)$-Maass cusp forms of level $q$ with nebentypus $\chi\bmod q$. Does there exist a corresponding version for the following ...
- 191
4
votes
0
answers
96
views
Sturm bound for Katz modular forms
Let $\mathfrak{P}$ be a place of $\bar{\mathbb{Q}}$ above a prime number $p$, $k \geq 1$, $N \geq 1$ prime to $p$ and $\varepsilon$ a Dirichlet character mod $N$.
It is well known that a modular form ...
9
votes
0
answers
319
views
Is there an operator-theoretic foundation for Quantum Modular Forms?
As far as I know it seems that quantum modular forms lack a clean operator-theoretic foundation parallel to that of classical and Maass forms. The spectral theory of automorphic Laplacians and the ...
- 91
1
vote
0
answers
93
views
Question about formula for Siegel-Eisenstein series with character
What's the definition for a Siegel-Eisenstein series with 2 characters?
I know in the $\mathrm{SL}_2$ case it's something like this: let $\psi, \chi$ be two characters with modulus $u,v$ respectively, ...
- 115
2
votes
0
answers
119
views
Linear relations between critical $L$-values of cusp forms
Let $k\geq 3$ be an integer, fix a level $N$, consider the critical $L$-values of all cusp forms in this level and weight with algebraic Fourier coefficients:
$$\mathbb{L}_{N,k}:= \{(2\pi)^{k-1-s} L(f,...
- 1,073
0
votes
0
answers
109
views
Determining if a set intersects the orbit of another in Siegel upper half space
Consider Siegel upper half space, consisting of symmetric matrices $X+iY$ such that $Y$ is positive definite. This has an action of $\operatorname{Sp}_{2n}(\mathbb{Z})$ on it by generalized Möbius ...
- 29
5
votes
1
answer
686
views
The Bring quintic and the Baby Monster?
I. Quintic
The general quintic was reduced to the Bring form $x^5+ax+b=0$ in the 1790s, while the Baby Monster was found in the 1970s.
We combine the two together using the McKay-Thompson series (...
- 13.1k
2
votes
1
answer
298
views
Show $\alpha(n)=0$ when $n \equiv 3\mod4$
Consider the modular forms
$$\theta(q):=\sum_{n\in\mathbb Z}q^{n^2},\qquad E_4(q):=1+240\sum_{n\ge1}\sigma_3(n)q^n,$$
$$\sum_{n\ge1}\alpha(n)q^n:=\frac1{16}(\theta^{10}(q)-\theta^2(q)E_4(q^2)).\tag{$\...
- 341