2,563 research outputs found
Densities of 4-ranks of
Conner and Hurrelbrink established a method of determining the structure of
the 2-Sylow subgroup of the tame kernel for certain
quadratic number fields. Specifically, the 4-rank for these fields was
characterized in terms of positive definite binary quadratic forms. Numerical
calculations led to questions concerning possible density results of the 4-rank
of tame kernels. In this paper, we succeed in giving affirmative answers to
these questions.Comment: 11 page
Vanishing of eigenspaces and cyclotomic fields
We use a result of Thaine to give an alternative proof of the fact that, for
a prime p>3 congruent to 3 modulo 4, the component e_{(p+1)/2} of the p-Sylow
subgroup of the ideal class group of \mathbb Q(\zeta_{p}) is trivial.Comment: 6 pages, minor corrections made, to appear in the International
Mathematics Research Notice
A remark on a conjecture of Borwein and Choi
We prove the remaining case of a conjecture of Borwein and Choi concerning an
estimate on the square of the number of solutions to n=x^2+Ny^2 for a
squarefree integer N.Comment: 7 pages, to appear in Proc. Amer. Math. So
Congruences for traces of singular moduli
We extend a result of Ahlgren and Ono on congruences for traces of singular
moduli of level 1 to traces defined in terms of Hauptmodul associated to
certain groups of genus 0 of higher levels.Comment: 8 pages, to appear in The Ramanujan Journa
A note on 4-rank densities
For certain real quadratic number fields, we prove density results concerning
4-ranks of tame kernels. We also discuss a relationship between 4-ranks of tame
kernels and 4-class ranks of narrow ideal class groups. Additionally, we give a
product formula for a local Hilbert symbol.Comment: 9 page
M_2-rank differences for overpartitions
This is the third and final installment in our series of papers applying the
method of Atkin and Swinnerton-Dyer to deduce formulas for rank differences.
The study of rank differences was initiated by Atkin and Swinnerton-Dyer in
their proof of Dyson's conjectures concerning Ramanujan's congruences for the
partition function. Since then, other types of rank differences for statistics
associated to partitions have been investigated. In this paper, we prove
explicit formulas for M_2-rank differences for overpartitions. Additionally, we
express a third order mock theta function in terms of rank differences.Comment: 19 page
Congruences via modular forms
We prove two congruences for the coefficients of power series expansions in t
of modular forms where t is a modular function. As a result, we settle two
recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table
of congruences for numbers which appear in similar power series expansions and
in the study of integral solutions of Apery-like differential equations.Comment: 8 pages, revised version, to appear in Proceedings of the AM
M_2-rank differences for partitions without repeated odd parts
We prove formulas for the generating functions for M_2-rank differences for
partitions without repeated odd parts. These formulas are in terms of modular
forms and generalized Lambert series.Comment: 18 page
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