Affiliation:
1. Mathematics Department , Vrije Universiteit Amsterdam , Amsterdam , Netherlands
Abstract
Abstract
Let 𝐺 be a linear algebraic group defined over a finite field
F
q
\mathbb{F}_{q}
.
We present several connections between the isogenies of 𝐺 and the finite groups of rational points
(
G
(
F
q
n
)
)
n
≥
1
(G(\mathbb{F}_{\smash{q^{n}}}))_{n\geq 1}
.
We show that an isogeny
ϕ
:
G
′
→
G
\phi\colon G^{\prime}\to G
over
F
q
\mathbb{F}_{q}
gives rise to a subgroup of fixed index in
G
(
F
q
n
)
G(\mathbb{F}_{\smash{q^{n}}})
for infinitely many 𝑛.
Conversely, we show that if 𝐺 is reductive, the existence of a subgroup
H
n
H_{n}
of fixed index 𝑘 for infinitely many 𝑛 implies the existence of an isogeny of order 𝑘.
In particular, we show that the infinite sequence
H
n
H_{n}
is covered by a finite number of isogenies.
This result applies to classical groups
GL
m
\mathrm{GL}_{m}
,
SL
m
\mathrm{SL}_{m}
,
SO
m
\mathrm{SO}_{m}
,
SU
m
\mathrm{SU}_{m}
,
Sp
2
m
\mathrm{Sp}_{2m}
and can be extended to non-reductive groups if 𝑘 is prime to the characteristic.
As a special case, we see that if 𝐺 is simply connected, the minimal indices of proper subgroups of
G
(
F
q
n
)
G(\mathbb{F}_{\smash{q^{n}}})
diverge to infinity.
Similar results are investigated regarding the sequence
(
G
(
F
p
)
)
p
(G(\mathbb{F}_{p}))_{p}
by varying the characteristic 𝑝.
Subject
Algebra and Number Theory
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. A Study on Isogeny Based Cryptography;2024 International Conference on Electronics, Computing, Communication and Control Technology (ICECCC);2024-05-02