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Register a new userProbabilistic Galois Theory over $P$-adic Fields
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Benjamin L. Weiss
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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 $n=3$.
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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 $p>0$).
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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 $x^2 + 1 =2y^4$.
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