Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.3087

Stark units and main conjectures for totally real fields

Kazim Buyukboduk
View original source ↗Read source document ↗Export metadata ↓

Abstract

Main theorem of [Buyukboduk, arXiv:0706.0377v1] suggests that it should be possible to lift the Kolyvagin systems of Stark units constructed in [Buyukboduk, arXiv:math/0703426v1] to a Kolyvagin system over the cyclotomic Iwasawa algebra. This is what we prove in this paper. This construction gives the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement of local Iwasawa theory. This statement, however, turns out to be interesting in its own right as it suggests a relation between solutions to $p$-adic and complex Stark conjectures.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kazim Buyukboduk. 2007-06-21. Stark units and main conjectures for totally real fields. https://doi.org/10.1112/s0010437x09004163

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums

Inspired by Weil's classical result on the zeta function of a projective Fermat curve defined over a finite field, in this paper, we investigate some arithmetic properties of the cyclotomic matrix $$\left[J_p(χ^{ki},χ^{kj})\right]_{1\le i,j\le n-1},$$ where $p\ge3$ is a prime, $1\le k<p-1$ is a divisor of $p-1$ with $p-1=kn$, $χ$ is a generator of the group of all multiplicative characters of $\mathbb{F}_p$ and $J_p(χ^{ki},χ^{kj})$ is the Jacobi sum. For example, let $ζ_p\in\mathbb{C}$ be a primitive $p$-th root of unity and $P_k(T)$ be the minimal polynomial of the algebraic integer $$θ_k=\sum_{x\in\mathbb{F}_p,x^k=1}ζ_p^x$$ over $\mathbb{Q}$. Then we prove that $$\det \left[J_p(χ^{ki},χ^{kj})\right]_{1\le i,j\le n-1}=(-1)^{\frac{(k+1)(n^2-n)}{2}}\cdot n^{n-2}\cdot x_p(k),$$ where $x_p(k)$ is the coefficient of $T$ in $P_k(T)$.

math.NT↗
arXiv2026

Weighted uniform distribution of subpolynomial functions along primes and applications

Let $u(x)$ be a subpolynomial function in a Hardy field. We establish necessary and sufficient conditions for the weighted uniform distribution of the sequences $(u(n))_{n\in\mathbb{N}}$ and $(u(p_n))_{n\in\mathbb{N}}$, where $p_n$ denotes the $n$-th prime. This extends the main result of [4] to the weighted setting and leads to new applications in uniform distribution theory, ergodic theory, and additive combinatorics.

math.NT↗
arXiv2026

A Lie algebra associated with adjoint multiple zeta values

Jarossay (arXiv math.NT1412.5099) introduced adjoint multiple zeta values and, by using Racinet's dual formulation of the generating series of multiple zeta values, found $\mathbb{Q}$-algebraic relations among them, referred to as the \textit{adjoint double shuffle relations}. Additionally, Jarossay defined the affine scheme $\mathrm{AdDMR}_0$ determined by the adjoint double shuffle relations and posed a question whether $\mathrm{AdDMR}_0$ is isomorphic to Racinet's double shuffle group $\mathrm{DMR}_0$ (Publ. Math. Inst. Hautes Études Sci. (2002), no. 95). In this paper, we refine Jarossay's question by introducing the condition referred to as the adjoint conditions, and, based on this refinement, we study the corresponding Lie algebraic aspect. Within this framework, we construct the Lie algebra associated with the adjoint double shuffle relations by imposing Hirose's parity results.

math.NT↗