Kiran S. Kedlaya
We survey some recent applications of p-adic cohomology to machine computation of zeta functions of algebraic varieties over finite fields of small characteristic, and suggest some new avenues for further exploration.
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We give a proof of double shuffle relations for $p$-adic multiple zeta values by developing higher dimensional version of tangential base points and discussing a relationship with two (and one) variable $p$-adic multiple polylogarithms.
The cyclotomic $p$-adic multi-zeta values are the $p$-adic periods of $π_{1}(\mathbb{G}_{m} \setminus μ_{M},\cdot),$ the unipotent fundamental group of the multiplicative group minus the $M$-th roots of unity. In this paper, we compute the cyclotomic $p$-adic multi-zeta values at all depths. This paper generalizes the results in [6] and [7]. Since the main result gives explicit formulas we expect it to be useful in proving non-vanishing and transcendence results for these $p$-adic periods and also, through the use of $p$-adic Hodge theory, in proving non-triviality results for the correspondin…
We establish a tannakian formalism of $p$-adic multiple polylogarithms and $p$-adic multiple zeta values introduced in our previous paper via a comparison isomorphism between a de Rham fundamental torsor and a rigid fundamental torsor of the projective line minus three points and also discuss its Hodge and etale analogues. As an application we give a way to erase log poles of $p$-adic multiple polylogarithms and introduce overconvergent $p$-adic multiple polylogarithms which might be $p$-adic multiple analogue of Zagier's single-valued complex polylogarithms.
These notes give a basic introduction to the theory of $p$-adic and motivic zeta functions, motivic integration, and the monodromy conjecture.
In this brief note, we will investigate the number of points of bounded (twisted) height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the $p$-adic analytic properties of the height zeta function. In particular, we will show that for a large class of projective varieties this function is $p$-adic meromorphic.
Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of K^n. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than -1/2 for n=2 and less than -1 for n=3 which occur as the real part of a pole.
In this paper we compute the values of the p-adic multiple polylogarithms of depth two at roots of unity. Our method is to solve the fundamental differential equation satisfied by the crystalline frobenius morphism using rigid analytic methods. The main result could be thought of as a computation in the p-adic theory of higher cyclotomy. We expect the result to be useful in proving non-vanishing results since it gives quite explicit formulas.
This paper pursues positive characteristic analogues of the results of Furusho, Komori, Matsumoto and Tsumura on $p$-adic multiple $L$-functions. We consider $\infty$-adic and $v$-adic multiple zeta functions concerned by Anglès, Ngo Dac and Tavares Ribeiro. Our main results in this paper consist of: (1) integral expressions of special values of $\infty$-adic multiple zeta functions, (2) integral expression of $v$-adic multiple zeta functions themselves similar to those of $p$-adic multiple $L$-functions, (3) Kummer-type congruence for the special values of $v$-adic multiple zeta functions at …
We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in Arizona winter school 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactification. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at infinity and definition of a special tangential base point on the moduli space.
In this paper, we show the non-existence of finitely generated integral $p$-adic cohomology which satisfies finite étale descent and the associated rational cohomology coincides with rigid cohomology.
We prove a mixed characteristic analog of the Beilinson-Lichtenbaum Conjecture for p-adic motivic cohomology. It gives a description, in the stable range, of p-adic motivic cohomology (defined using algebraic cycles) in terms of differential forms. This generalizes a result of Geisser from small Tate twists to all twists and uses as a critical new ingredient the comparison theorem between syntomic complexes and p-adic nearby cycles proved recently in Colmez-Niziol.
This work is a study of $p$-adic multiple zeta values at roots of unity ($p$MZV$μ_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $(\mathbb{P}^{1} - \{0,μ_{N},\infty\})/ \mathbb{F}_{q}$. The main tool is new objects which we call $p$-adic pro-unipotent harmonic actions. In this part IV we define and study $p$-adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of $p$MZV$μ_{N}$'s involve sequences of positive integers ; in this IV-1, by consid…
We prove that the algebra of p-adic multi-zeta values are contained in another algebra which is defined explicitly in terms of series.
We construct p-adic period map using derived de Rham cohomology of Illusie and give a simple proof of Fontaine's C_{dR} conjecture.
In this short paper, we give a $p$-adic analogue of the Hard Leftschetz Theorem.
We study a refinement of the symmetric multiple zeta value, called the $t$-adic symmetric multiple zeta value, by considering its finite truncation. More precisely, two kinds of regularizations (harmonic and shuffle) give two kinds of the $t$-adic symmetric multiple zeta values, thus we introduce two kinds of truncations correspondingly. Then we show that our truncations tend to the corresponding $t$-adic symmetric multiple zeta values, and satisfy the harmonic and shuffle relations, respectively. This gives a new proof of the double shuffle relations for $t$-adic symmetric multiple zeta value…
We compute the Archimedean doubling zeta integrals which appear in the interpolation formulas for the p-adic L-functions of Siegel modular forms, and verify that they agree with the modified Archimedean Euler factors for p-adic interpolation conjectured by Coates--Perrin-Riou.
The higher rank Lefschetz formula for p-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a p-adic group on its Bruhat-Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions
The fact that the double zeta values at n and m can be written as a sum of products of two zeta values and of zeta value at m+n, whenever n+m is odd is due to Euler. We shall show a weak version of this result for the Galois l-adic realization.
The article is dedicated to the memory of George Voronoi. It is concerned with ($p$-adic) $L$-functions (in partially ($p$-adic) zeta functions) and cyclotomic ($p$-adic) (multiple) zeta values. The beginning of the article contains a short summary of the results on the Bernoulli numbers associated with the studies of George Voronoi. Results on multiple zeta values have presented by D. Zagier, by P. Deligne and A.Goncharov, by A. Goncharov, by F. Brown, by C. Glanois and others. S. Ünver have investigated p-adic multiple zeta values in the depth two. Tannakian interpretation of p-adic multiple…
We compute the integral $p$-adic étale cohomology of Drinfeld symmetric spaces of any dimension. This refines the computation of the rational $p$-adic étale cohomology from Colmez-Dospinescu-Nizioł. The main tools are: the computation of the integral de Rham cohomology from CDN and the integral $p$-adic comparison theorems of Bhatt-Morrow-Scholze and Česnavičius-Koshikawa which replace the quasi-integral comparison theorem of Tsuji used in CDN.
We present a detailed overview of the construction of the A_inf-cohomology theory from the preprint "Integral p-adic Hodge theory", joint with B. Bhatt and P. Scholze. We focus particularly on the p-adic analogue of the Cartier isomorphism via relative de Rham--Witt complexes.
This is a survey of some recent developments concerning the p-adic cohomology of algebraic varieties over fields of positive characteristic and local fields of mixed characteristic, plus some related areas like p-adic Hodge theory.
In this short note, we give a proof of the Riemann hypothesis for Goss $v$-adic zeta function $ζ_{v}(s)$, when $v$ is a prime of $\mathbb{F}_{q}[t]$ of degree one.
On a lifting of $t$-adic symmetric multiple zeta values
本文提出带新参数 $s$ 的 $(s,t)$-进对称多重 zeta 值，建立其双重洗牌关系、对偶性与循环和公式，并讨论有限对应物。
Our main aim in this paper is to give a foundation of the theory of $p$-adic multiple zeta values. We introduce (one variable) $p$-adic multiple polylogarithms by Coleman's $p$-adic iterated integration theory. We define $p$-adic multiple zeta values to be special values of $p$-adic multiple polylogarithms. We consider the (formal) $p$-adic KZ equation and introduce the $p$-adic Drinfel'd associator by using certain two fundamental solutions of the $p$-adic KZ equation. We show that our $p$-adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the $p$-adic KZ…
We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether-Lefschetz divisor in the sense of Hassett.
In this short note, we show a p-adic analogue of Beilinson's equivalence comparing two derived categories: the derived category of holonomic modules and derived category of modules whose cohomologies are holonomic.
$p$-adic multiple zeta values and binomial multiple harmonic sums
本文提出了一种方法，用于推导$p$-adic多重ζ值的显式公式，引入了二项多重调和和作为关键工具。
For primes $p>3$ we produce a new derivation of the universal $p$-adic sigma function and $p$-adic Weierstrass zeta functions of Mazur and Tate for ordinary elliptic curves by a method that highlights congruences among coefficients in Laurent expansions of elliptic functions, and works simultaneously for generalized elliptic curves defined by Weierstrass equations.
We define and apply a method to study the non-vanishing of $p$-adic cyclotomic multiple zeta values. We prove the non-vanishing of certain cyclotomic multiple harmonic sums, and, via a formula proved in another paper, which expresses a cyclotomic multiple harmonic sums as an infinite sum of products of $p$-adic cyclotomic multiple zeta values, this implies the non-vanishing of certain $p$-adic cyclotomic multiple zeta values.
$p$-adic multiple zeta values of integer indices
本文研究整数索引的$p$-adic多重ζ值，引入了可接受性和可正规化条件，定义了有限有理线性组合的$p$-adic多重ζ值，并证明了双洗关系成立。
In this article we initiate the study of the heat traces and spectral zeta functions for certain p-adic Laplacians. We show that the heat traces are given by p-adic integrals of Laplace type, and that the spectral zeta functions are p-adic integrals of Igusa-type. We find good estimates for the behaviour of the heat traces when the time tends to infinity, and for the asymptotics of the function counting the eigenvalues less than or equal to a given quantity.
Mishiba's Conjecture on the Coaction of $\infty$-adic Multiple Zeta Values
本文在正特征中研究多重zeta值，构造了模$\zeta_A(q-1)$的$\infty$-adic多重zeta值代数上的余作用，证明其与Mishiba的构造一致，并确定了$\star$-逆值上的余作用及对映，从而证明了Mishiba的猜想。
We establish duality results for the cohomology of the Weil group of a $p$-adic field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for discrete Weil modules, which implies Tate-Nakayama Duality. We define Weil-smooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of $\G_m$ on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves.
We show that in positive characteristic special loci of deformation spaces of rank one $\ell$-adic local systems are quasilinear. From this we deduce the Hard Lefschetz theorem for rank one $\ell$-adic local systems and a generic vanishing theorem. Last version: a few typos corrected
The definition for the $p$-adic Hurwitz-type Euler zeta functions has been given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing the values of this kind of $p$-adic zeta function at negative integers, we show that it interpolates the Euler polynomials $p$-adically. Many properties are provided for the $p$-adic Hurwitz-type Euler zeta functions, including the convergent Laurent series expansion, the distribution formula, the functional equation, the reflection formula, the derivative formula, the $p$-adic Raabe formula and so on. The definition for the $p$-adic Euler $L$…
This paper studies the derived de Rham cohomology of F_p and p-adic schemes, and is inspired by Beilinson's recent work. Generalising work of Illusie, we construct a natural isomorphism between derived de Rham cohomology and crystalline cohomology for lci maps of such schemes, as well logarithmic variants. These comparisons give derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory. Placing these ideas in the skeleton of Beilinson's construction leads to a new proof of Fontaine's crystalline conjecture and Fontaine-Jannsen's semistable conjecture.
Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n>1 variables and let χbe a character of R^{\times}. Let M_i(u) be the number of solutions of f=u in (R/P^i)^n for i \in \mathbb{Z}_{\geq 0} and u \in R/P^i. These numbers are related with Igusa's p-adic zeta function Z_{f,χ}(s) of f. We explain the connection between the M_i(u) and the smallest real part of a pole of Z_{f,χ}(s). We also prove that M_i(u) is divisible by q^{\ulcorner(n/2)(i-1)\urcorner}, where the corners indicate that we …
Slopes and weights of $\ell$-adic cohomology of rigid spaces
研究p进局部域上刚性空间的ℓ进上同调，证明其弗罗贝尼乌斯特征值是代数整数并给出p进赋值界，推导权重界证明相关猜想，还给出特定单值纯反常层例子回答问题。
The $t$-adic symmetric multiple zeta values were defined Jarossay, which have been studied as a real analogue of $\boldsymbol{p}$-adic finite multiple zeta values. In this paper, we consider the star analogues based on several regularization processes of multiple zeta-star values: harmonic regularization, shuffle regularization, and Kaneko-Yamamoto's type regularization. We also present the cyclic sum formula for $t$-adic symmetric multiple zeta(-star) values, which is the counterpart of that for $\boldsymbol{p}$-adic finite multiple zeta(-star) values obtained by Kawasaki. The proof uses our …
We consider the symmetric multiple zeta values in $\mathcal{S}_m$ without modulo $π^2$ reduction for indices in which $1$ and $3$ appear alternately. We investigate those values that can be expressed as a polynomial of the Riemann zeta values, and give a conjecturally complete list of explicit formulas for such values.
We describe how a systematic use the deep methods from $\ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz, Laumon allow to make progress on various classical questions from analytic number theory. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 Arizona Winter School.
We construct a cohomology theory with compact support H^i_c(X_ar,Z(n))$ for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds and rational and numerical equivalence agree up to torsion, then the groups H^i_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.
With every path on a projective line minus zero, one and infinity there is associated a measure. We are studying a sum of two such measures associated to paths from the tangential point at zero to roots of one. We show that the obtained measure can be defined very elementary. Integrating agaist this measure we get p-adic Hurwitz zeta functions constructed previously by Shiratani.
$t$-adic symmetric multiple zeta values for indices with alternating $1$ and $3$, starting with $1$ and ending with $3$
本文针对Hirose等人提出的部分$t$-进对称多重zeta值相关猜想，证明了以1开头、以3结尾且1和3交替的指标对应的这类值可表示为Riemann zeta值的多项式，推进了相关数论问题的研究。
For p prime, we give an explicit formula for Igusa's local zeta function associated to a polynomial mapping f=(f_1,...,f_t): Q_p^n -> Q_p^t, with f_1,...,f_t in Z_p[x_1,...,x_n], and an integration measure on Z_p^n of the form |g(x)||dx|, with g another polynomial in Z_p[x_1,...,x_n]. We treat the special cases of a single polynomial and a monomial ideal separately. The formula is in terms of Newton polyhedra and will be valid for f and g sufficiently non-degenerated over F_p with respect to their Newton polyhedra. The formula is based on, and is a generalization of results of Denef - Hoornaer…
On the torsion in the cohomology of the integral structure sheaf of affinoid adic spaces
本文证明了特征零非阿基米德域上正规仿射阿迪空间的整结构层上同调一致挠，源于Bartenwerfer的注记，部分回答了Hansen和Kedlaya的问题及《非阿基米德苏格兰书》相关问题。
Let k be a finite field of characteristic p>0. We construct a theory of weights for overholonomic complexes of arithmetic D-modules with Frobenius structure on varieties over k. The notion of weight behave like Deligne's one in the l-adic framework: first, the six operations preserve weights, and secondly, the intermediate extension of an immersion preserves pure complexes and weights.
We discuss the formalism of Iwasawa theory descent in the setting of the localized K_1-groups of Fukaya and Kato. We then prove interpolation formulas for the `leading terms' of the global Zeta isomorphisms that are associated to certain Tate motives and of the p-adic L-functions that are associated to certain critical motives.
The p-cohomology of an algebraic variety in characteristic p lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for p-cohomology in characteristic p.
We prove an analogue for $p$-adic coefficients of the Deligne--Laumon theorem on local acyclicity for curves. That is, for an overconvergent $F$-isocrystal $E$ on a relative curve $f:U\rightarrow S$ admitting a good compactification, we show that the cohomology sheaves of $\mathbf{R}f_!E$ are overconvergent isocrystals if and only if $E$ has constant Swan conductor at infinity.
$p$-adic cyclotomic multiple zeta values depend on the choice of a number of iterations of the crystalline Frobenius of the pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,μ_{N},\infty\}$. In this paper we study how the iterated Frobenius depends on the number of iterations, in relation with the computation of $p$-adic cyclotomic multiple zeta values in terms of cyclotomic multiple harmonic sums. This provides new results on that computation and the definition of a new pro-unipotent harmonic action.
Let $K$ be a finite extension of ${\mathbb Q}_p$ and let $X$ be Drinfel'd's symmetric space of dimension $d$ over $K$. Let $Γ\subset {\rm SL}_{d+1}(K)$ be a cocompact discrete (torsionfree) subgroup and let ${X}_Γ=Γ\backslash {X}$, a smooth projective ${K}$-variety. In this paper we investigate the de Rham and log crystalline (log convergent) cohomology of local systems on $X_Γ$ arising from $K[Γ]$-modules. (I) We prove the monodromy weight conjecture in this context. To do so we work out, for a general strictly semistable proper scheme of pure relative dimension $d$ over a cdvr of mixed chara…
This paper aims to study the zero distribution of $v-$adic multiple zeta values over function fields. We show that the interpolated $v-$adic MZVs at negative integers only vanish at what we call the ''trivial zeros'', for degree one prime over rational function fields. And we conjecture that this result can be generalized to all primes.
Pro-unipotent harmonic actions and a computation of $p$-adic cyclotomic multiple zeta values
本文引入 pro-unipotent 调和作用，建立 $p$-进分圆多重 zeta 值与分圆多重调和和的互推公式，证明并推广相关猜想，同时给出有限分圆多重 zeta 值空间的维数界。
Under the assumption that Galois representations associated to Siegel modular forms exist (it is known only for genus at most 2), we show that the cohomology with p-adic integral coefficients of Siegel Varieties, when localized at a non-Eisenstein maximal ideal of the Hecke algebra, is torsion-free, provided the prime p is large with the respect to the weight of the coefficient system. The proof uses p-adic Hodge theory, the dual BGG complex modulo p in order to compute the Hodge-Tate weights for the mod p cohomology. We apply this result to the construction of Hida p-adic families for symplec…
We compute the $p$-adic étale and the pro-étale cohomologies of the Drinfeld half-space of any dimension. The main input is a new comparison theorem for the $p$-adic pro-étale cohomology of $p$-adic Stein spaces.
We study analytic properties of multiple zeta-functions of generalized Hurwitz-Lerch type. First, as a special type of them, we consider multiple zeta-functions of generalized Euler-Zagier-Lerch type and investigate their analytic properties which were already announced in our previous paper. Next we give `desingularization' of multiple zeta-functions of generalized Hurwitz-Lerch type, which include those of generalized Euler-Zagier-Lerch type, the Mordell-Tornheim type, and so on. As a result, the desingularized multiple zeta-function turns out to be an entire function and can be expressed as…
We extend the definition of the Chen-Ruan cohomology ring to smooth, proper, tame, Deligne-Mumford stacks over fields of positive characteristic and prove that a modified version of the Frobenius action preserves the product.
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