No Arabic abstract
We generalize the classical theory of periodic continued fractions (PCFs) over ${mathbf Z}$ to rings ${mathcal O}$ of $S$-integers in a number field. Let ${mathcal B}={beta, {beta^*}}$ be the multi-set of roots of a quadratic polynomial in ${mathcal O}[x]$. We show that PCFs $P=[b_1,ldots,b_N,bar{a_1ldots ,a_k}]$ of type $(N,k)$ potentially converging to a limit in ${mathcal B}$ are given by ${mathcal O}$-points on an affine variety $V:=V({mathcal B})_{N,k}$ generically of dimension $N+k-2$. We give the equations of $V$ in terms of the continuant polynomials of Wallis and Euler. The integral points $V({mathcal O})$ are related to writing matrices in $textrm{SL}_2({mathcal O})$ as products of elementary matrices. We give an algorithm to determine if a PCF converges and, if so, to compute its limit. Our standard example generalizes the PCF $sqrt{2}=[1,bar{2}]$ to the ${mathbf Z}_2$-extension of ${mathbf Q}$: $F_n={mathbf Q}(alpha_n)$, $alpha_{n}:=2cos(2pi/2^{n+2})$, with integers ${mathcal O}_n={mathbf Z}[alpha_n]$. We want to find the PCFs of $alpha_{n+1}$ over ${mathcal O}_{n}$ of type $(N,k)$ by finding the ${mathcal O}_{n}$-points on $V({mathcal B}_{n+1})_{N,k}$ for ${mathcal B}_{n+1}:={alpha_{n+1}, -alpha_{n+1}}$. There are three types $(N,k)=(0,3), (1,2), (2,1)$ such that the associated PCF variety $V({mathcal B})_{N,k}$ is a curve; we analyze these curves. For generic ${mathcal B}$, Siegels theorem implies that each of these three $V({mathcal B})_{N,k}({mathcal O})$ is finite. We find all the ${mathcal O}_n$-points on these PCF curves $V({mathcal B}_{n+1})_{N,k}$ for $n=0,1$. When $n=1$ we make extensive use of Skolems $p$-adic method for $p=2$, including its application to Ljunggrens equation $x^2 + 1 =2y^4$.
If the $ell$-adic cohomology of a projective smooth variety, defined over a $frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$).
We estimate several probability distributions arising from the study of random, monic polynomials of degree $n$ with coefficients in the integers of a general $p$-adic field $K_{mathfrak{p}}$ having residue field with $q= p^f$ elements. We estimate the distribution of the degrees of irreducible factors of the polynomials, with tight error bounds valid when $q> n^2+n$. We also estimate the distribution of Galois groups of such polynomials, showing that for fixed $n$, almost all Galois groups are cyclic in the limit $q to infty$. In particular, we show that the Galois groups are cyclic with probability at least $1 - frac{1}{q}$. We obtain exact formulas in the case of $K_{mathfrak{p}}$ for all $p > n$ when $n=2$ and $n=3$.
This article is an overview of the vanishing cycles method in number theory over function fields. We first explain how this works in detail in a toy example, and then give three examples which are relevant to current research. The focus will be a general explanation of which sorts of problems this method can be applied to.
It is widely believed that the continued fraction expansion of every irrational algebraic number $alpha$ either is eventually periodic (and we know that this is the case if and only if $alpha$ is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of the present work is to present new families of transcendental continued fractions with bounded partial quotients. Our results are derived thanks to new combinatorial transcendence criteria recently obtained by Adamczewski and Bugeaud.
Let F be a function field in one variable over a p-adic field and D a central division algebra over F of degree n coprime to p. We prove that Suslin invariant detects whether an element in F is a reduced norm. This leads to a local-global principle for reduced norms with respect to all discrete valuations of F.