Let A be an elliptic curve over Q of square free conductor N . Suppose A has a rational torsion point of prime order r such that r does not divide 6N . We prove that then r divides the order of the cuspidal subgroup C of J0(N). If A is optimal, then viewing A as an abelian subvariety of J0(N), our proof shows more precisely that r divides the order of A ∩ C. Also, under the hypotheses above, we...

Given a $p$-group $G$ and subgroup-closed class $\mathfrak{X}$, we associate with each $\mathfrak{X}$-subgroup $H$ certain quantities which count $\mathfrak{X}$-subgroups containing subject to further properties. We show in Theorem I that one of the said is always $\equiv 1 \pmod p$ if only same holds for others. In II supplement above result by focusing on normal III obtain sharpened version c...

1. Introduction. Let p be a prime number and F an algebraic closure of the finite field F p with p elements. Let n and N denote positive integers, N prime to p. We are interested in representations ρ of the Galois group G Q = Gal(¯ Q/Q) into GL(n, F), unramified at all finite primes not dividing pN. (We shall say ρ is unramified outside pN.) In this paper, representation will always mean contin...

For each odd prime p, we prove the existence of infinitely many real quadratic fields which are p-rational. Explicit imaginary and bi-quadratic p-rational also given for p. Using a recent method developed by Greenberg, deduce Galois extensions Q with group isomorphic to an open subgroup GLn(Zp), n=4 n=5 at least all primes p<192.699.943.

In this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. We study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. For strongly z-ideals, we analyze prime ideals using the concept of zero sets. Moreover, it is proven that the intersection of all zero sets of a prime ideal of C(L),...

In the [16] has been proven that the multiplicative group Z/pZ∗ is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group. M...

in a previous paper, the second author established that, given finite fields $f < e$ and certain subgroups $c leq e^times$, there is a galois connection between the intermediate field lattice ${l mid f leq l leq e}$ and $c$'s subgroup lattice. based on the galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product $c rtimes {rm gal} (e/f)$...

Many cryptographic protocols and cryptosystems have been proposed to make use of prime order subgroups of Z n where n is the product of two large distinct primes. In this paper we analyze a number of such schemes. While these schemes were proposed to utilize the diiculty of factoring large integers or that of nding related hidden information (e.g., the order of the group Z n), our analyzes reve...