22 research outputs found
Higher -Groups of Smooth Projective Curves Over Finite Fields
Let be a smooth projective curve over a finite field with
elements. For let be the curve over the finite field
, the -th extension of Let be the
-group of the smooth projective curve
In this paper, we study the structure of the groups If is a
prime, we establish an analogue of Iwasawa theorem in algebraic number theory
for the orders of the -primary part of . In
particular, when is an elliptic curve defined over our
method determines the structure of Our results can be applied to
construct an efficient {\bf DL} system in elliptic cryptography.Comment: 20 page, 8 table
The 4-rank of for real quadratic fields F
1. Introduction. Let F be a number field, and let be the ring of its integers. Several formulas for the 4-rank of are known (see [7], [5], etc.). If √{-1) ∉ F, then such formulas are related to S-ideal class groups of F and F(√(-1)), and the numbers of dyadic places in F and F(√(-1)), where S is the set of infinite dyadic places of F. In [11], the author proposes a method which can be applied to determine the 4-rank of for real quadratic fields F with 2 ∉ NF. The author also lists many real quadratic fields with the 2-Sylow subgroups of being isomorphic to ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/4ℤ. In [12], the author gives a 4-rank formula for imaginary quadratic fields F. By the formula, it is enough to compute some Legendre symbols when one wants to know 4-rank for a given imaginary quadratic field F. In the present paper, we give a similar formula for real quadratic fields F. Then we give 4-rank tables for real quadratic fields F = ℚ√d whose discriminants have at most three odd prime divisors
The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields
1. Introduction. Let F be a number field and the ring of its integers. Many results are known about the group , the tame kernel of F. In particular, many authors have investigated the 2-Sylow subgroup of . As compared with real quadratic fields, the 2-Sylow subgroups of for imaginary quadratic fields F are more difficult to deal with. The objective of this paper is to prove a few theorems on the structure of the 2-Sylow subgroups of for imaginary quadratic fields F.
In our Ph.D. thesis (see [11]), we develop a method to determine the structure of the 2-Sylow subgroups of for real quadratic fields F. The present paper is motivated by some ideas in the above thesis
Lehmer's totient problem over
In this paper, we consider the function field analogue of the Lehmer's
totient problem. Let and be the
Euler's totient function of over where
is a finite field with elements. We prove that if and only if (i) is irreducible; or (ii) is the product of any non-associate irreducibes of degree or
(iii) is the product of all irreducibles of degree all
irreducibles of degree and and the product of any irreducibles one
each of degree and .Comment: 12 page