In this talk, I will present a connection between designing low
correlation zone (LCZ) sequences and the results of correlation
of sequences with subfield decompositions. This results
in low correlation zone signal sets with huge sizes over three
different alphabetic sets: finite field of size $q$, integer
residue ring modulo $q$, and the subset in the complex field which
consists of powers of a primitive $q$-th root of unity. A connection between these
constructions and ``completely
non-cyclic'' Hadamard matrices will be shown. I will also provide some open problems
along this direction.
Joint work with Solomon W. Golomb and Hongyeop Song.
Brumer's conjecture states that Stickelberger elements combining values of L-functions at s=0 for an abelian extension of number fields E/F should annihilate the ideal class group of E when it is considered as module over the appropriate group ring. In some cases, an ideal obtained from these Stickelberger elements has been shown to equal a Fitting ideal connected with the ideal class group. We consider the analog of this at s=-1, in which the class group is replaced by the tame kernel, which we will define. For a field extension of degree 2, we show that there is an exact equality etween the Fitting ideal of the tame kernel and the most natural higher Stickelberger ideal; the 2-part of this equality is conditional on the Birch-Tate conjecture.
Suppose E is an elliptic curve defined over a number field k, K/k is a quadratic extension, p is an odd prime, and L is a p-extension of K that is Galois over K. Let c be an element of order 2 in Gal(L/k), and H the subgroup of all elements of G := Gal(L/K) that commute with c. Under very mild hypotheses the Parity Conjecture (combined with a little representation theory) predicts that if the rank of E(K) is odd, then the rank of E(L) is at least [G:H]. For example, if L/k is dihedral and the rank of E(K) is odd, then the rank of E(L) should be at least [L:K].
In this talk I will discuss recent joint work with Barry Mazur, where we prove an analogue of this result with "rank" repaced by "p-Selmer rank".
This will be a series of two introductory lectures on the
distribution of closed points on a scheme of finite type
over the integers. Both general properties and important
examples will be discussed, with an emphasis on p-adic
variation for zeta functions over finite fields.