We prove that for a subring R⊆Q and free group F of rank at least 2 the length Bousfield's HR-localization tower is ω+ω. The key ingredient proof theory polynomial functors over Q.

We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ characteristic $p$ global fields with arbitrary abelian group $G$ and Drinfeld module $E$ defined over certain Dedekind subring $F$. For this data, we define $G$-equivariant $L$-function $\Theta_{K/F}^E$ prove an equivariant Tamagawa number formula for Euler-completed versions its special value $\Theta_{K/F}^E(0)$. This generalizes T...

Let k be a subring of the eld of rational functions in x; v; s which contains x 1 ; v 1 ; s 1. If M is an oriented 3-manifold, let S(M) denote the Hommypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the Hommypt skein relations: (1) x ?1 L + ? xL ? = (s ? s ?1)L 0 ; (2) L with a positive twist = (xv ?1)L; (3) L t O ...

in this paper, various elementary properties of vague rings are obtained. furthermore, the concepts of vague subring, vague ideal, vague prime ideal and vague maximal ideal are introduced, and the validity of some relevant classical results in these settings are investigated.

A smooth intersection Y of two quadrics in $$\mathbb {P}^{2g+1}$$ has Hodge level 1. We show that such varieties have a multiplicative Chow–Künneth decomposition, the sense Shen–Vial. As consequence, certain tautological subring Chow ring powers injects into cohomology.

Andrew V. Sutherland A key ingredient to improving the efficiency of elliptic curve primality proving (and many other algorithms) is the ability to directly construct an elliptic curve E/Fq with a specified number of rational points, rather than generating curves at random until a suitable curve is found. To do this we need to develop the theory of complex multiplication. Recall from Lecture 7 ...

Given a natural number n, Waring’s problem asks for the minimum natural number sn such that every natural number can be represented as the sum of sn nth powers of integers. In this paper, we will answer this question for the case n = 2. To do this, we will examine the properties of two rings, one of which is a subring of the complex numbers, the other of which is a subring of the quaternions. T...