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Register a new userConiveau over $p$-adic fields and points over finite fields
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Authors
Hel`ene Esnault
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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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If the $ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For $K$ a $p$-adic field, this is math/0405318, Theorem 1.1. If the model $sX$ is regular, one has a congruence $|sX(k)|equiv 1 $ modulo $|k|$ for the number of $k$-rational points 0704.1273, Theorem 1.1. The congruence is violated if one drops the regularity assumption.
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 $n=3$.
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 $x^2 + 1 =2y^4$.
We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.
In 2005, Kayal suggested that Schoofs algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then explain details of a generalization involving Pilas algorithm for abelian varieties.
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