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Abstract For every integer k there exists a bound $$B=B(k)$$ B = B ( k ) such that if the characteristic polynomial of $$g\in \textrm{SL}_n(q)$$ g ∈ SL n ( q ) is the product of $$\le k$$ ≤ k pairwise distinct monic irreducible polynomials over $$\mathbb {F}_q$$ F q , then every element x of $$\textrm{SL}_n(q)$$ SL n ( q ) of support at least B is the product of two conjugates of g . We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions ( p , q ), in the special case that $$n=p$$ n = p is prime, if g has order $$\frac{q^p-1}{q-1}$$ q p - 1 q - 1 , then every non-scalar element $$x \in \textrm{SL}_p(q)$$ x ∈ SL p ( q ) is the product of two conjugates of g . The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.
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Counting points on smooth plane quartics
Abstract We present efficient algorithms for counting points on a smooth plane quartic curve X modulo a prime p . We address both the case where X is defined over $${\mathbb {F}}_p$$ F p and the case where X is defined over $${\mathbb {Q}}$$ Q and p is a prime of good reduction. We consider two approaches for computing $$\#X({\mathbb {F}}_p)$$ # X ( F p ) , one which runs in $$O(p\log p\log \log p)$$ O ( p log p log log p ) time using $$O(\log p)$$ O ( log p ) space and one which runs in $$O(p^{1/2}\log ^2p)$$ O ( p 1 / 2 log 2 p ) time using $$O(p^{1/2}\log p)$$ O ( p 1 / 2 log p ) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $$X/{\mathbb {Q}}$$ X / Q that compute $$\#X({\mathbb {F}}_p)$$ # X ( F p ) for good primes $$p\leqslant N$$ p ⩽ N in $$O(N\log ^3 N)$$ O ( N log 3 N ) time using O ( N ) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $${\mathbb {P}}^1$$ P 1 , which in combination with previous results addresses all curves of genus $$g\leqslant 3$$ g ⩽ 3 . Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest.
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- Award ID(s):
- 1946311
- PAR ID:
- 10420423
- Date Published:
- Journal Name:
- Research in Number Theory
- Volume:
- 9
- Issue:
- 1
- ISSN:
- 2522-0160
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s40993-022-00397-8
