ON STARK’S CLASS NUMBER CONJECTURE AND THE GENERALISED BRAUER–SIEGEL CONJECTURE

Stark conjectured that for any $h\in \Bbb {N}$ , there are only finitely many CM-fields with class number h. Let $\mathcal {C}$ be the class of number fields L for which L has an almost normal subfield K such that $L/K$ has solvable Galois closure. We prove Stark’s conjecture for $L\in \mathcal {C}$ of degree greater than or equal to 6. Moreover, we show that the generalised Brauer–Siegel conjecture is true for asymptotically good towers of number fields $L\in \mathcal {C}$ and asymptotically bad families of $L\in \mathcal {C}$ .

On the Generalized Brauer–Siegel Theorem for Asymptotically Exact Families of Number Fields with Solvable Galois Closure

In 2002, M. A. Tsfasman and S. G. Vlăduţ formulated the generalized Brauer–Siegel conjecture for asymptotically exact families of number fields. In this article, we establish this conjecture for asymptotically good towers and asymptotically bad families of number fields with solvable Galois closure.

Tribonacci numbers and primes of the form p = x2 + 11y2

In this paper we show that for any prime number p not equal to 11 or 19, the Tribonacci number Tp−1 is divisible by p if and only if p is of the form x2 + 11y2. We first use class field theory on the Galois closure of the number field corresponding to the polynomial x3 − x2 − x − 1 to give the splitting behavior of primes in this number field. After that, we apply these results to the explicit exponential formula for Tp−1. We also give a connection between the Tribonacci numbers and the Fourier coefficients of the unique newform of weight 2 and level 11.

LOCALLY ANALYTIC VECTORS AND OVERCONVERGENT -MODULES

Let $p$ be a prime, let $K$ be a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$ , and let $G_{K}$ be the Galois group. Let $\unicode[STIX]{x1D70B}$ be a fixed uniformizer of $K$ , let $K_{\infty }$ be the extension by adjoining to $K$ a system of compatible $p^{n}$ th roots of $\unicode[STIX]{x1D70B}$ for all $n$ , and let $L$ be the Galois closure of $K_{\infty }$ . Using these field extensions, Caruso constructs the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$ -modules, which classify $p$ -adic Galois representations of $G_{K}$ . In this paper, we study locally analytic vectors in some period rings with respect to the $p$ -adic Lie group $\operatorname{Gal}(L/K)$ , in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules, we can establish the overconvergence property of the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$ -modules.