We describe an algorithm to compute the zeta-function of a proper, smooth curve over a finite field, when the curve is given together with some auxiliary data. Our method is based on computing the matrix of the action of a semi-linear Frobenius on the first cohomology group of the curve by means of Serre duality. The cup product involved can be computed locally, after first computing local expansions of a globally defined lift of Frobenius. The resulting algorithm's complexity is softly cubic in the field degree, which is also the case with Kedlaya's algorithm in the hyperelliptic case.
In differential algebra, a proper scheme $X$ defined over an algebraically closed field $K$ with a derivation $\partial$ on it descends to the field of constants $K^\partial$ if $X$ itself lifts the derivation $\partial$. This is a result by A. Buium. Now in the arithmetic case, the notion of a derivation is replaced by the notion of a $π$-derivation $δ$ or equivalently in the flat case, a lift of Frobenius $ϕ$. We will show an analogous result in the arithmetic case of equal characteristic. We show our results using the arithmetic analogue of Taylor expansion using Witt vectors.
For curves of genus bigger than one we prove that Buium's first arithmetic jet spaces (p-jet spaces) admit the structure of a torsor under some line bundle. This result lifts a known constructions in characteristic p where the first $p$-jet space modulo p is a sheaf under the Frobenius tangent sheaf (parametrizing Frobenius linear derivations). In particular we show there is a natural family of lifts of the Frobenius tangent bundle so that the first $p$-jet space (and hence higher order lifts of the Frobenius) form torsor a under this bundle. The Cech cohomology classes associated to this tors…
Let $X$ be a smooth scheme over a perfect field $k$ of positive characteristic. M.V.~Nori and V.~Srinivas studied infinitesimal liftings of Frobenius morphism $F:X\to X$. Namely, given a Frobenius lifting $F_Y: Y\to Y$ over $W_{n-1}(k)$ the obstruction of lifting $F_Y$ over $W_n(k)$ is a class in $H^1(X, T_X\tensor B_1Ω^1_{X/k})$. We give a modified version of their theory and applications, in which we consider lifting $(n-1)$th iterated Frobenius $F^{n-1}:X\to X$ directly over $W_n(k)$.
We study the algbraic dynamics for endomorphisms of projective spaces with coefficients in a p-adic field whose reduction in positive characteritic is the Frobenius. In particular, we prove a version of the dynamical Manin-Mumford conjecture and the dynamical Mordell-Lang conjecture for the coherent backward orbits for such endomorphisms. We also give a new proof of a dynamical version of the Tate-Voloch conjecture in this case. Our method is based on the theory of perfectoid spaces introduced by P. Scholze. In the appendix, we prove that under some technical condition on the field of definiti…
We prove that if $K$ is a complete algebraically closed non-archimedian valuation field of positive characteristic and $f$ is an endomorphism of $\mathbb{P}_{K}^{N}$ which is totally inseparable and behaves as the Frobenius on the special fiber, then $f$ satisfies the dynamical Mordell-Lang (DML) property. We also discuss some corollaries and generalizations.
We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using $p$-adic cohomology.
We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite étale cover, admit a toric fibration over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wiśniewski, which states that in characteristic zero a smooth image of a projective toric variety is a toric variety. To this end we analyse the behaviour of toric varieties in families showing some generization and specialization results. Furthermore, we prove a pos…
We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with Frobenius and other endomorphisms over $\dbF_{p^q}$, whose $p$-divisible groups in mixed characteristic $(0,p)$ are with complex multiplication.
We show that Aomoto's $q$-deformation of de Rham cohomology arises as a natural cohomology theory for $Λ$-rings. Moreover, Scholze's $(q-1)$-adic completion of $q$-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in mixed characteristic equipped with a suitable lift of Frobenius. If we attach $p$-power roots of $q$, the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.
If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if $\sD_X$ is the sheaf of differential operators on $X$ [EGAIV], it is well known that giving an action of $\sD_X$ on an $\sO_X$-module $\sE$ is equivalent to giving an infinite sequence of $\sO_X$-modules descending $\sE$ via the iterates of the Frobenius endomorphism of $X$. We show that this result can be generalized to any infinitesimal deformation $f : X \to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it extends to adic formal schemes such that $p$ belongs to an…
Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over k have the same zeta function (i.e., the same number of points over all extensions of k) if and only if their corresponding Jacobians are isogenous. We remedy this situation by showing that if, instead of just the zeta function, all Dirichlet L-series of the two curves are equal via an isomorphism of their Dirichlet character group…
This article is the third one of a series of three articles devoted to direct images of isocrystals: here we consider overconvergent isocrystals with Frobenius structure. For a liftable proper smooth morphism we establish the overconvergence of direct images, owing to the first article and the existence of lifts of Frobenius. This result partially answers a conjecture of Berthelot on the overconvergence of direct images of overconvergent F-isocrystals under a proper smooth morphism.
We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends previous work of ours to all curves for which a good lift to characteristic zero is known. We develop all the necessary bounds, analyse the complexity of the algorithm and provide a complete implementation.
Let f be a polynomial of degree n in ZZ[x_1,..,x_n], typically reducible but squarefree. From the hypersurface {f=0} one may construct a number of other subschemes {Y} by extracting prime components, taking intersections, taking unions, and iterating this procedure. We prove that if the number of solutions to f=0 in \FF_p^n is not a multiple of p, then all these intersections in Å^n_{\FF_p} just described are reduced. (If this holds for infinitely many p, then it holds over \QQ as well.) More specifically, there is a_Frobenius splitting_ on Å^n_{\FF_p} compatibly splitting all these subschemes…
This article is the second one of a series of three articles devoted to direct images of isocrystals: here we consider convergent isocrystals with Frobenius structure. Let V be a complete discrete valuation ring, with residue field k = V/m of characteristic p > 0 and fraction field K of characteristic 0. Firstly we characterize convergent F-isocrystals on a smooth affine k-scheme. Secondly, for perfect k and after a detailed exposition of the Teichmûller liftings, especially for the affine rigid line, we derive the existence of Frobenius isomorphisms on the direct images of convergent F-isocry…
Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring of an Abhyankar valuation of a function field over a perfect ground field of prime characteristic is Frobenius split. We show that a Frobenius splitting of a sufficiently well-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigate properties of valuations centered on arbitrary Noetherian domains of prime characteristic. In contrast to [arXiv:1507.06009], this paper emphasizes the role of centers in controlling Frobenius properties of valuations rings in prime characteri…
本文给出了一个计算有限基数域$q$上一般普通超椭圆曲线雅可比簇上有理点的算法，其时间复杂度为$O(n^{2+o(1)})$，空间复杂度为$O(n^2)$，其中$n=\log(q)$ .在后者的复杂性估计的属和特征被假定为固定的。我们的算法形成了两者的推广，J. - F. Mestre和T.佐藤我们正则地提升了超椭圆曲线的雅可比矩阵的一个关于θ常数的算术不变量。θ空值是相对于级别$2^νp$的半规范θ结构计算的，其中$ν>0$是整数，并且$p=\mathrm{char}(\F_q)>2$。本文的结果提出了一个整体的肯定回答的问题，是否存在一个拟二次时间算法的数量的有理点的一般普通阿贝尔簇上定义在有限域上的计算。
Suppose $(X_n)$ is a sequence of positive-dimensional smooth projective complete intersections over $\mathbb{F}_q$ with dimensions bounded from above and with characteristic zero lifts $(\tilde{X}_n)$ to smooth projective geometrically connected varieties. Suppose each complex variety $\tilde{X}^{an}_n$ has (underlying real manifold equipped with) a Riemannian metric $g_n$ of sectional curvature at least $-κ_n^2$, $κ_n\geq 0$, and diameter at most $D_n$. In this note, we show that if \[\lim_{n\rightarrow\infty}\min\{d:X_n(\mathbb{F}_{q^d})\neq\emptyset\}=+\infty,\] then $κ_nD_n\rightarrow+\inf…
