All Questions
Tagged with computational-number-theory modular-forms
24 questions
5
votes
0
answers
141
views
Compute weight of modular form from its Fourier coefficients
It is known that Hecke eigenform $f \in S_{k}(\Gamma_0(N), \chi)$ is uniquely determined by first $C_{k,N}$ many Fourier coefficients, where $C_{k,N}$ is a constant only depends on $k$ and $N$. For ...
- 2,215
5
votes
1
answer
172
views
Isogenous elliptic curves and canonical modular polynomials
Let $\ell$ and $p$ be two primes. We are looking for a method for checking whether two supersingular elliptic curves over the finite field $F_p$, given through their $j$-invariants, are $\ell$-...
- 81
2
votes
0
answers
71
views
Simultaneous computation of the three Weber modular functions
Recall that the three classical Weber modular functions are defined by
$f(\tau)=e^{-\pi i/24}\eta((\tau+1)/2)/\eta(\tau)$,
$f_1(\tau)=\eta(\tau/2)/\eta(\tau)$, and
$f_2(\tau)=\sqrt{2}\eta(2\tau)/\eta(\...
- 13.1k
4
votes
0
answers
143
views
Road map for learning about the computational/general theory of modular curves/isogenies of abelian varieties for cryptography
I am a graduate math/crypto student. So I've had some free time last year and I heard about elliptic curves in cryptography and how a resilient cryptosystem got demolished by a spectacular attack ...
- 41
4
votes
0
answers
206
views
What are the modularity conjectures for Artin motives?
Classically, singular cohomology is an important tool for studying topological spaces, in particular, complex varieties. In the mid-twentieth century it was realized that there are many analogues of ...
- 815
4
votes
0
answers
102
views
Reconstructing coefficients of an elliptic curve L-series from the modular form divisor
Let $E$ be an unknown elliptic curve over $\mathbb{Q}$.
Let $L(E, s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ be the L-function of $E$ and write $f(q) = \sum_{n=1}^{\infty} a_n q^n$.
I'm in a setting ...
- 5,637
0
votes
0
answers
172
views
An application of Koike's Trace Formula
Koike's Trace Formula states that
\begin{equation}
\mbox{Tr}((U_p^{\kappa})^n) = - \sum_{0 \leq u < \sqrt{p^n}\\
(u,p)=1}H(u^2-4p^n)\frac{\gamma(u)^\kappa}{\gamma(u)^2 - p^n}-1,
\end{equation}
...
2
votes
0
answers
118
views
Computing coefficients of theta functions associated to quadratic forms
If we take an integral positive definite quadratic form $Q$ and set $\Theta_Q(z) = \sum_{k\geq 0}R_Q(k)e^{2\pi ikz}$, what are the most efficient algorithms to compute the $R_Q(k)$? I am aware e.g. of ...
- 323
1
vote
0
answers
583
views
Langlands program and complexity theory
Back when I was an undergraduate, I spent some time reading the about the modularity conjecture, but the details are fuzzy now.
One of the motivations I imagined for the Langlands program was for ...
- 59
2
votes
2
answers
312
views
Coefficient field of a newform using Magma
It is well-known that, for a newform $f = \sum c_nq^n \in \Gamma_0(N)$, the coefficient field $K_f := \mathbb{Q}(a_1, a_2, a_3, \cdots )$ is a number field.
I am introducing myself in Magma, and I was ...
- 29
6
votes
0
answers
93
views
Computing all eta quotients of given weight and level
I have written a rather naive program for finding all holomorphic eta quotients of
given weight and level (and varying character). When the level has few divisors it is
very fast, but incredibly slow ...
- 13.1k
5
votes
0
answers
195
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 ...
user130124
8
votes
2
answers
1k
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 ...
- 475
4
votes
1
answer
1k
views
Computing coefficients for the slash operator of a modular form
Suppose $f$ is a classical modular form of weight $r$ for a (congruence) group $\Gamma$. Let $\gamma$ be any matrix in $\operatorname{SL}_2(\mathbb{Z})$. Then the slash operator $|_\gamma$ is usually ...
- 295
10
votes
2
answers
624
views
Computing millions of coefficients of non self-dual modular forms
To test some conjectures made by some colleagues, I need to compute millions of coefficients of non self-dual modular forms, preferably in low weight (say 2 or 3). A form such as this.
For elliptic ...
- 361
7
votes
1
answer
353
views
Numerically double-checking formula with L-values
I'm working with a special case of Ichino's triple product formula, which for classical holomorphic newforms $f$, $g$ ,$h$ of weights $k$, $m-k$, $m$ (and central characters satisfying $\chi_f \chi_g =...
- 131
6
votes
1
answer
412
views
Computing an eigencuspform in $S_2(\Gamma_0(1776))$
Consider
$$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$
the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then $...
- 10.9k
3
votes
0
answers
203
views
Fourier expansions of newforms at width-1 cusps
Let $f_E$ be the newform attached to the Elliptic Curve $E$ with cremona label
$\textbf{100a1}$ and let $\alpha = \left[\begin{matrix} 1&0 \\ 10&1 \end{matrix}\right] \in SL_2(\mathbb{Z})$. ...
- 221
8
votes
3
answers
2k
views
Numerical evaluation of the Petersson product of elliptic modular forms
It is known how to compute the Fourier expansion of elliptic modular forms using modular symbols, and it is known how to get numerical evaluations of $L$-functions of various type ; it's possible to ...
8
votes
3
answers
1k
views
Effective detection of CM modular forms
Say $f$ is a newform of weight $k$ and level $\Gamma_1(N)$. $f$ is called CM if, for example, there is an imaginary quadratic field $K$ such that for all $p\nmid N$ which are inert in $K$, the $p$th ...
- 4,807
27
votes
2
answers
2k
views
How to explicitly compute lifting of points from an elliptic curve to a modular curve?
Say $E$ is an elliptic curve over the rationals, of conductor $N$. There's a covering of $E$ by the modular curve $X_0(N)$, and if you rig it right then you can define this map over $\mathbf{Q}$: ...
- 41.4k
5
votes
2
answers
477
views
Density stability; questions for those who like computer calculation
BACKGROUND: The question, which has its roots in a question asked on MO by O'Bryant, concerns the relative density of certain subsets, $B$, of ${\mathbb N}$ in congruence classes modulo a power of 2. ...
- 5,422
15
votes
3
answers
3k
views
Finding zeroes of classical modular forms
There are several papers which compute zeroes of modular forms for genus 0 congruence subgroups, such as "Zeros of some level 2 Eisenstein series" by Garthwaite et al published in Proc AMS and work of ...
- 419
5
votes
0
answers
228
views
Example of level one cuspidal Hecke Algebra T_k^0 such that p divides its index in its normalization, and p≥k-1?
The question is strongly focused on computations concerning modular forms and Hecke algebras. It is already in the title, but I will repeat it, adding a few details.
Let $S_k$ be the complex vector ...
- 2,406