Daqing Wan
In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Professor Bernard Dwork's unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p.
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The aim of this article is to illustrate, on the example of Dwork hypersurfaces, how the study of the representation of a finite group of automorphisms of a hypersurface in its etale cohomology allows to factor its zeta function.
Zeta functions over finite fields- An exposition of Dwork's methods
本文系统阐述Dwork在有限域Zeta函数研究中使用的p进解析与余调方法，并将其应用于Dwork超曲面族的一个具体例子。
We consider the one-parameter family of hypersurfaces in $\Pj^5$ with projective equation (X_1^5+X_2^5+X_3^5+X_4^5+X_5^5) = 5λX_1 X_2... X_5, (writing $λ$ for the parameter), proving that the Galois representations attached to their cohomologies are potentially automorphic, and hence that the zeta function of the family has meromorphic continuation throughout the complex plane.
This paper proves the general rank one case of Dwork's conjecture over the affine space. It generalizes and improves the method of ANT-0141 "Dwork's conjecture on unit root zeta functions" (Ann. Math., 150(1999), 867-929). In addition, explicit information about the zeros and poles (along the Gouvêa-Mazur conjecture direction) for the unit root zeta function is obtained. The paper is to appear in JAMS.
Let $\mathbb{F}_q$ be a finite field with $q$ elements, $ψ$ a non-zero element of $\mathbb{F}_q$, and $n$ an integer $\geq 3$ prime to $q$. The aim of this article is to show that the zeta function of the projective variety over $\mathbb{F}_q$ defined by $X_ψ\colon x_1^n+...+x_n^n - n ψx_1... x_n=0$ has, when $n$ is prime and $X_ψ$ is non singular (i.e. when $ψ^n \neq 1$), an explicit decomposition in factors coming from affine varieties of odd dimension $\leq n-4$ which are of hypergeometric type. The method we use consists in counting separately the number of points of $X_ψ$ and of some vari…
In his work studying the Zeta functions of families of hypersurfaces, Dwork came upon a one-parameter family of hypersurfaces (now known as \emph{the} Dwork family). These examples were not only useful to Dwork in his study of his deformation theory for computing Zeta functions of families, but they have also proven to be extremely useful to physicists working in mirror symmetry. A startling result is that these families are very closely linked to hypergeometric functions. This phenomenon was carefully studied by Dwork and Candelas, de la Ossa, and Rodríguez-Villegas in a few special cases. Dw…
We prove that the partial zeta function introduced in [9] is a rational function, generalizing Dwork's rationality theorem.
This is an expository paper which gives a quick introduction to Dwork's conjecture about p-adic meromorphic continuation of his unit root zeta function arising from algebraic geometry. Special emphasis is given to the case of elliptic family and its relation to p-adic modular forms such as Gouvêa-Mazur conjecture and some sort of p-adic Ramanujuan-Peterson conjecture. There is also a section describing some of my personal interactions with Dwork.
We study the variation of unit roots of the Dwork families of Calabi-Yau varieties over a finite field by the method of Dwork-Katz and also from the point of view of formal group laws. A p-adic analytic formula for the unit roots away from the Hasse locus is obtained.
In this article we review the observation, due originally to Dwork, that the zeta-function of an arithmetic variety, defined originally over the field with p elements, is a superdeterminant. We review this observation in the context of a one parameter family of quintic threefolds, and study the zeta-function as a function of the parameter ϕ. Owing to cancellations, the superdeterminant of an infinite matrix reduces to the (ordinary) determinant of a finite matrix, U(ϕ), corresponding to the action of the Frobenius map on certain cohomology groups. The parameter-dependence of U(ϕ) is given by a…
By restricting the variables running over various (possibly different) subfields, we introduce the notion of a partial zeta function. We prove that the partial zeta function is rational in an interesting case, generalizing Dwork's well known rationality theorem. In general, the partial zeta function is probably not rational. But a theorem of Faltings says that the partial zeta function is always nearly rational.
The goal of this article is to develop BV (Batalin-Vilkovisky) formalism in the $p$-adic Dwork theory. Based on this formalism, we explicitly construct a $p$-adic dGBV algebra (differential Gerstenhaber-Batalin-Vilkovisky algebra) for a smooth projective complete intersection variety $X$ over a finite field, whose cohomology gives the $p$-adic Dwork cohomology of $X$, and its cochain endomorphism (the $p$-adic Dwork Frobenius operator) which encodes the information of the zeta function $X$. As a consequence, we give a modern deformation theoretic interpretation of Dwork's theory of the zeta fu…
Dwork crystals I
本文对 Dwork 关于环面超曲面 zeta 函数的显式 p-进极限公式思想进行了初等阐述。
We prove that if two Calabi-Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard--Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard--Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.
Starting from a classical generating series for Bessel functions due to Schlomilch, we use Dwork's relative dual theory to broadly generalize unit-root results of Dwork on Kloosterman sums and Sperber on hyperkloosterman sums. In particular, we express the (unique) p-adic unit root of an arbitrary exponential sum on the torus in terms of special values of the p-adic analytic continuation of a ratio of A-hypergeometric functions. In contrast with the earlier works, we use noncohomological methods and obtain results that are valid for arbitrary exponential sums without any hypothesis of nondegen…
We study in detail the family of Calabi-Yau hypersurfaces defined by x_1+...+x_n+1/(x_1...x_n)=t over a finite field k. We determine its local and global monodromy and the trivial factors of its moment zeta function and Dwork's unit root zeta function.
This is an expository paper which gives a simple arithmetic introduction to the conjectures of Weil and Dwork concerning zeta functions of algebraic varieties over finite fields. A number of further open questions are raised.
We demonstrate that Wan's alternate description of Dwork's unit root L-function in the rank one case may be modified to give a proof of meromorphy that is classical, eliminating the need to study sequences of uniform meromorphic functions.
The purpose of this article is to give an explicit description, in terms of hypergeometric functions over finite fields, of zeta function of a certain type of smooth hypersurfaces that generalizes Dwork family. The point here is that we count the number of rational points employing both character sums and the theory of weights, which enables us to enlighten the calculation of the zeta function.
As a consequence of Wan's theorem about Dwork's conjecture, the unit root $L$-functions of the $n$-dimensional Kloosterman family are $p$-adic meromorphic. By studying the symmetric power $L$-functions associated to the Kloosterman family, we prove that for each $0\le j\le n$, the unit root $L$-function coming from slope $j$ has a unique unit root which equals 1.
Klein Quartic Curve and its Modularity
本文通过将克莱因四次曲线实现为 7 次费马曲线的商来计算其局部 Zeta 函数，并借助伽罗瓦表示给出相应的模形式。
We compute a basis for the p-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give p-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.
We investigate relations among Schur multiple zeta functions and zeta-functions of root systems attached to semisimple Lie algebras. Schur multiple zeta functions are defined as sums over semi-standard Young tableaux. Then, assuming the Young tableaux is of anti-hook shape, we show that they can be written in terms of modified zeta-functions of root systems of type $A$. Our proof is quite computational, but we also give a pictorial interpretation of our argument in terms of Young tableaux. It is also possible to understand that one of our theorems gives an expression of Schur multiple zeta fun…
This work contributes to the study of the non-trivial roots of the Riemann zeta function. In view of the Hilbert-Polya conjecture a series of self-adjoint operators on a Hilbert space is constructed whose eigenvalues approximate these roots.
Expressing Weierstrass type infinite products in terms of Stieltjes integrals is discussed. The asymptotic behavior of particular types of infinite products is compared against the asymptotic behavior of the entire function Xi(s), well-known in Riemann zeta function theory. An approximate formula for the distribution of the non-trivial roots of Riemann's zeta function is obtained.
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 define the notion of {\it Dwork family of logarithmic $F$-crystals}, a typical example of which is the family of Gauss hypergeometricdifferential systems, viewed as parametrized by their exponents of algebraic monodromy. The $p$-adic analytic dependence of the Frobenius operation upon those exponents, is Dwork's "Boyarsky Principle". We discuss, in favorable cases, the $p$-adic analytic continuation of the unit root $F$-subcrystal in the open tube of a singularity, uniformly w.r.t. the exponents. We obtain a conceptual proof of the Koblitz-Diamond formula $p$-adically analog to Gauss' evalu…
We study multiple zeta values (MZVs) from the viewpoint of zeta-functions associated with the root systems which we have studied in our previous papers. In fact, the $r$-ple zeta-functions of Euler-Zagier type can be regarded as the zeta-function associated with a certain sub-root system of type $C_r$. Hence, by the action of the Weyl group, we can find new aspects of MZVs which imply that the well-known formula for MZVs given by Hoffman and Zagier coincides with Witten's volume formula associated with the above sub-root system of type $C_r$. Also, from this observation, we can prove some new …
The shuffle product plays an important role in the study of multiple zeta values. This is expressed in terms of multiple integrals, and also as a product in a certain non-commutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of the shuffle product. In fact, we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of multiple zeta values of root systems. As an application, we give a proof of extended double shuffle relations without using Drinfel'd integral expressions for mu…
Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well-suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces and also can be used to compute the L-function of an exponential sum.
We give an overview of the theory of functional relations for zeta-functions of root systems, and show some new results on functional relations involving zeta-functions of root systems of types $B_r$, $D_r$, $A_3$ and $C_2$. To show those new results, we use two different methods. The first method, for $B_r$, $D_r$, $A_3$, is via generating functions, which is based on the symmetry with respect to Weyl groups, or more generally, on our theory of lattice sums of certain hyperplane arrangements. The second method for $C_2$ is more elementary, using partial fraction decompositions.
In this note we shall study the Witten multiple zeta function associated to the exceptional Lie algebra g_2. Our main result shows that its special values at nonnegative integers can always be expressed as rational linear combinations of the multi-polylogs evaluated at 12th roots of unity, except for two irregular cases.
We consider a certain linear combination $S(\mathbf{s},\mathbf{y};I;Δ)$ of zeta-functions of root systems, where $Δ$ is a root system of rank $r$ and $I\subset\{1,2,\ldots,r\}$. Showing two different expressions of $S(\mathbf{s},\mathbf{y};I;Δ)$, we find that a certain signed sum of zeta-functions of root systems is equal to a sum involving Bernoulli functions of root systems. This identity gives a non-trivial functional relation among zeta-functions of root systems, if the signed sum does not identically vanish. This is a genralization of the authors' previous result proved in \cite{KMTLondon…
We prove certain general forms of functional relations among Witten multiple zeta-functions in several variables (or zeta-functions of root systems). The structural background of those functional relations is given by the symmetry with respect to Weyl groups. From those relations we can deduce explicit expressions of values of Witten zeta-functions at positive even integers, which is written in terms of generalized Bernoulli numbers of root systems. Furthermore we introduce generating functions of those Bernoulli numbers of root systems, by which we can give an algorithm of calculating Bernoul…
In this paper we provide a new series representation for the values of Riemann zeta function at integer arguments, namely: $ ζ(m)=\sum_{n=1}^{\infty}\frac{m(-1)^{n-1}Γ(1-ω_{m}n)...Γ(1-ω_{m}^{m-1}n)}{n!n^m}$, where $n$ is an integer that lager than $1$ and $ω$ is the $m$-th root of unity. This series converges quite fast. It's derived by some technique of infinite partial fraction decomposition. With this technique we also establish other useful formulas related to gamma function.
Dwork's conjecture, now proven by Wan, states that unit root L-functions "coming from geometry" are p-adic meromorphic. In this paper we study the p-adic variation of a family of unit root L-functions coming from a suitable family of toric exponential sums. In this setting, we find that the unit root L-functions each have a unique p-adic unit root. We then study the variation of this unit root over the family of unit root L-functions. Surprisingly, we find that this unit root behaves similarly to the classical case of families of exponential sums. That is, the unit root is essentially a ratio …
The L-function of symmetric powers of classical Kloosterman sums is a polynomial whose degree is now known, as well as the complex absolute values of the roots. In this paper, we provide estimates for the p-adic absolute values of these roots. Our method is indirect. We first develop a Dwork-type p-adic cohomology theory for the two-variable infinite symmetric power L-function associated to the Kloosterman family, and then study p-adic estimates of the eigenvalues of Frobenius. A continuity argument then provides the desired p-adic estimates.
The Kaneko-Zagier conjecture states that finite and symmetric multiple zeta values satisfy the same relations. In the previous work with H.~Bachmann and Y.~Takeyama, we proved that the finite and symmetric multiple zeta value are obtained as an `algebraic' and `analytic' limit at $q\rightarrow 1$ of a certain truncated multiple harmonic $q$-series, and studied its relations in order to give partial evidence of the Kaneko-Zagier conjecture. In this paper, we start with truncated multiple harmonic $q$-series of level $N$, which is a $q$-analogue of the truncated colored multiple zeta value. We i…
A connection between the poles of the zeta function of a recurrence sequence and the module of relations of its roots
本文研究整数线性递推序列zeta函数极点枚举的重复计数问题，证明其等价于根的关系统模非平凡，并给出使枚举为单射的充分条件。
In this paper we propose different notions of F_zeta-geometry, for zeta a root of unity, generalizing notions of F_1-geometry (geometry over the "field with one element") based on the behavior of the counting functions of points over finite fields, the Grothendieck class, and the notion of torification. We relate F_zeta-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as counting functions of certain ind-varieties. We investigate the existence of F_zeta-structures in examples arising from general linear groups, matrix equations over finite fields, and some qu…
We introduce the notion of finite multiple harmonic q-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetrized multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we obtain families of linear relations among these series which induce linear relations among FMZVs and SMZVs of the same form. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vec…
Some explicit values of a $q$-multiple zeta-star function at roots of unity
本文给出 $q$-多重 zeta-star 值在单位根处的显式公式，利用行列式或 Bell 多项式表达，并借助生成函数递推关系推广至其他类型值。
In this article, we consider the expressions for content-parametrized Schur multiple zeta-functions in terms of multiple zeta-functions of Euler-Zagier type and their star-variants, or in terms of modified zeta-functions of root systems. First of all, we focus on the Schur multiple zeta-function of hook type. And then, applying the Giambelli formula and induction argument, we obtain the expressions for general content-parametrized Schur multiple zeta-functions.
The conical zeta values are a generalization of the multiple zeta values which are defined by certain multiple sums over convex cones. In this paper, we present a relation between the values of the Dedekind zeta functions for totally real fields and the conical zeta values for certain algebraic cones. More precisely, we show that the values of the partial zeta functions for totally real fields can be expressed as a rational linear combination of the conical zeta values associated with certain algebraic cones up to the square root of the discriminant.
Some explicit values of a $q$-multiple zeta function at roots of unity
针对由 Stirling 数推广自然产生的某类 $q$-重 zeta 函数，论文利用 Bell 多项式、行列式和 $r$-Stirling 数等工具，给出其在单位根处取值的显式公式。
We define zeta-functions of weight lattices of compact connected semisimple Lie groups. If the group is simply-connected, these zeta-functions coincide with ordinary zeta-functions of root systems of associated Lie algebras. In this paper we consider the general connected (but not necessarily simply-connected) case, prove the explicit form of Witten's volume formulas for these zeta-functions, and further prove functional relations among them which include their volume formulas.
Let K/k be an Abelian extension of number fields, S be a set of places of k, and p be an odd prime number. We continue an earlier investigation of the author into the values at zero of the S-imprimitive partial zeta functions of K/k. An earlier result provides, under the assumption that the p-power roots of unity in K are cohomologically trivial, a criterion for the values to have larger than expected p-valuation. The present paper provides such a criterion for a special class of degree 2p extensions for which the p-power roots of unity are not cohomologically trivial. For such extensions, new…
Trivial zeros of zeta functions of type $\mathrm{A}_r$
本文研究类型 A_r 根系 zeta 函数在非正整数点的取值，引入有序极限值定义奇点处值，给出显式或递推公式，证明三个有序极限值在特定条件下消失（多变量平凡零点类比），并探讨其他零点与经典 Eisenstein 级数恒等式的可能联系。
We consider generalised root identities for zeta functions of curves over finite fields, ζ_{k}, and compare with the corresponding analysis for the Riemann zeta function. We verify numerically that, as for ζ, the ζ_{k} do satisfy the generalised root identities and we investigate these in detail for the special cases of μ=0,-1\:\&\:-2. Unlike for ζ, however, we show that in the setting of zeta functions of curves over finite fields the μ=-2 root identity is consistent with the Riemann hypothesis (RH) proved by Weil. Comparison of this analysis with the corresponding calculations for ζilluminat…
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.
In this paper, we introduce iterated integrals associated with colored rooted trees and give proofs for the shuffle relations for $\boldsymbol{p}$-adic finite and $t$-adic symmetric polylogarithms. This method generalizes the theory of the finite multiple zeta values associated with $2$-colored rooted trees introduced by Ono.
Inspired by the theory of Hodge correlators due to Goncharov and by the plectic principle of Nekovář and Scholl, we construct higher plectic Green functions and give a higher order generalization of Hecke's formula for abelian $L$-functions over arbitrary number fields. We hence provide a potential method to generalize multiple zeta values over number fields. We recover classical multiple zeta values and multiple polylogrithms evaluated at roots of unity, when the number field in consideration is the rational field $\mathbb{Q}$.
We study the values of the zeta-function of the root system of type $G_2$ at positive integer points. In our previous work we considered the case when all integers are even, but in the present paper we prove several theorems which include the situation when some of the integers are odd. The underlying reason why we may treat such cases including odd integers is also discussed.
The exceptional group $G_2$ has two maximal parabolic subgroups $P_{long}$, $P_{short}$ corresponding to the so-called long root and short root. In this paper, the second author introduces two zeta functions associated to $(G_2,P_{long})$ and $(G_2,P_{short})$ respectively, and the first author proves that these zetas satisfy the Riemann Hypothesis.
We introduce an "$L$-function" $\mathcal{L}$ built up from the integral representation of the Barnes' multiple zeta function $ζ$. Unlike the latter, $\mathcal{L}$ is defined on a domain equipped with a non-trivial action of a group $G$. Although these two functions differ from each other, we can use $\mathcal{L}$ to study $ζ$. In fact, the transformation formula for $\mathcal{L}$ under $G$-transformations provides us with a new perspective on the special values of both $ζ$ and its $s$-derivative. In particular, we obtain Kronecker limit formulas for $ζ$ when restricted to points fixed by eleme…
Motivated by the work of Candelas, de la Ossa and Rodriguez-Villegas [6], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [23] on higher Hasse-Witt matrices for toric hypersurfaces following Katz's method of local expansion [14, 15]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz's congruence relations in terms of expansion coefficients [15] to Dwork's congruence relations [8] about peri…
We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also introduce generating functions of special values of those lattice sums, and study their properties by virtue of the theory of convex polytopes. Consequently we evaluate special values of those lattice sums, especially certain special values of zeta-functions of root systems and their affine analogues. In some special cases it is possible to treat sums running o…
We show that the analytic continuations of Helson zeta functions $ ζ_χ(s)= \sum_1^{\infty}χ(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole critical strip $ 1/2 < \Re s <1 $ under Riemann Hypothesis for the function $ χ$ taking values in cubic roots of unity. If the sets are symmetric with respect to the real axis, the same can be achieved with $ χ$ taking values $ \pm 1 $.
Shuffle product for multiple zeta functions
该研究推广部分分式分解公式并给出两种证明，借助根系统zeta函数给出归纳步骤，推导多重zeta函数的洗牌乘积及函数双重洗牌关系。
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…
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