Figure - uploaded by Semyon Novoselov
Content may be subject to copyright.
Orders of matrices in Sp 4 (F ) as elements of PSp 4 (F ) and their probabilities.
Source publication
The computation of the order of Frobenius action on the $\ell$-torsion is a part of Schoof-Elkies-Atkin algorithm for point counting on an elliptic curve $E$ over a finite field $\mathbb{F}_q$. The idea of Schoof's algorithm is to compute the trace of Frobenius $t$ modulo primes $\ell$ and restore it by the Chinese remainder theorem. Atkin's improv...
Contexts in source publication
Context 1
... the number of matrices in a class is also known, we can calculate the probability of a random matrix M ∈ Sp 4 (F ) to fall in a given class. We give the orders for classes with their respective probabilities in Table 1. ...Context 2
... the data from Table 1, we calculated the distribution of the Frobenius orders for the first 500 primes = 3 . . . 3571. ...Context 3
... for each pair (p, ,) we build a list L of pairs (a 2 1 , (a 2 − 2q) 2 ) corresponding to small orders r = ±1 2 from Table 3. We choose these orders because they appear in many conjugacy classes from Table 1 and the most common orders 2 ±1 2 lead to big lists. For each curve we compute the characteristic polynomial χ p (T ) of the Frobenius endomorphism by built-in methods of SageMath and so we know the exact value of χ p (T ) (mod ). ...Context 4
... the number of matrices in a class is also known, we can calculate the probability of a random matrix M ∈ Sp 4 (F ) to fall in a given class. We give the orders for classes with their respective probabilities in Table 1. ...Context 5
... the data from Table 1, we calculated the distribution of the Frobenius orders for the first 500 primes = 3 . . . 3571. ...Context 6
... for each pair (p, ) we build a list L of pairs (a 2 1 , (a 2 − 2q) 2 ) corresponding to small orders r = ±1 2 from Table 3. We choose these orders because they appear in many conjugacy classes from Table 1 and the most common orders 2 ±1 2 lead to big lists. For each curve we compute the characteristic polynomial χ p (T ) of the Frobenius endomorphism by built-in methods of SageMath and so we know the exact value of χ p (T ) (mod ). ...