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.
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.
Using the combinatorial description of shuffle product, we prove or reformulate several shuffle product formulas of multiple zeta values, including a general formula of the shuffle product of two multiple zeta values, some restricted shuffle product formulas of the product of two multiple zeta values, and a restricted shuffle product formula of the product of $n$ multiple zeta values.
Multiple Dedekind zeta values were recently defined by the second author. In a separate paper, the second author constructed double shuffle relations in some cases as a response to questions asked by Richard Hain and Alexander Goncharov. In this paper, we develop the technique for obtaining more shuffle relations and produce many examples of shuffle products over an imaginary quadratic field. We also define the notion of self shuffle of a (multiple) Dedekind zeta value and use it at many instances. We define a refinement of the multiple Dedekind zeta values. Our key examples are self shuffles …
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…
In this paper, we investigate the ``shuffle-type'' formula for special values of desingularized multiple zeta functions at integer points. It is proved by giving an iterated integral/differential expression for the desingularized multiple zeta functions at integer points.
We treat desingularized multiple zeta-functions introduced by Furusho, Komori, Matsumoto and Tsumura. In this paper, we prove functional relations, which are shuffle type product formulae, between desingularized multiple zeta-functions and desingularized values.
In this paper, we settle the problem posed by Singer which is on a comparison problem between the renormalized values of shuffle type and of harmonic type of multiple zeta functions.
This paper contains examples of shuffle relations among multiple Dedekind zeta values. Dedekind zeta values were defined by the author in his paper "Multiple Dedekind zeta functions". Here we concentrate on the cases of real or imaginary quadratic fields with up to double iteration. We give examples of integral shuffle relation in terms of iterated integrals over membranes and of infinite sum shuffle relation, sometimes called stuffle relation. Using both types of shuffles for the product $ζ_{K,C}(2)ζ_{K,C}(2)$, we find relations among multiple Dedekind zeta values for real and for imaginary q…
We prove a sum-shuffle formula for multiple zeta values in Tate algebras (in positive characteristic), introduced in \cite{PEL3}.This follows from an analog result for double twisted power sums, implying that an ${\mathbb{F}\_p$-vector space generated by multiple zeta values in Tate algebras is an ${\mathbb{F}\_p$-algebra.
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…
In 1998, Borwein, Bradley, Broadhurst and Lisoněk posed two families of conjectural identities among multiple zeta values, later generalized by Charlton using his alternating block notation. In this paper, we prove a new class of identities among multiple zeta values that simultaneously resolve and generalize these conjectures.
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 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.
Extended double shuffle relations for multiple zeta values are obtained by the fact that any product of regularized multiple zeta values has two different representations, and the case of two-fold product is considered in general. In this paper, we give two formulas for the generating function of the triple zeta values of any fixed weight by the use of the extended double shuffle relations obtained by not only two-fold products of double and single zeta values but also three-fold products of single zeta values. As applications of the formulas, we also obtain some parameterized, weighted and re…
Symmetric multiple zeta values (SMZVs) are elements in the ring of all multiple zeta values modulo the ideal generated by $ζ(2)$ introduced by Kaneko-Zagier as counterparts of finite multiple zeta values. It is known that symmetric multiple zeta values satisfy double shuffle relations and duality relations. In this paper, we construct certain lifts of SMZVs which live in the ring generated by all multiple zeta values and $2πi$ as certain iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ along a certain closed path. We call this lifted values as refined symmetric multiple zeta value…
It is conjectured that the regularized double shuffle relations give all algebraic relations among the multiple zeta values, and hence all other algebraic relations should be deduced from the regularized double shuffle relations. In this paper, we provide as many as the relations which can be derived from the regularized double shuffle relations, for example, the weighted sum formula of L. Guo and B. Xie, some evaluation formulas with even arguments and the restricted sum formulas of M. E. Hoffman and their generalizations.
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.
The formal multiple zeta space we consider with a computer is an $\mathbb{F}_2$-vector space generated by $2^{k-2}$ formal symbols for a given weight $k$, where the symbols satisfy binary extended double shuffle relations. Up to weight $k=22$, we compute the dimensions of the formal multiple zeta spaces, and verify the dimension conjecture on original extended double shuffle relations of real multiple zeta values. Our computations adopt Gaussian forward elimination and give information for spaces filtered by depth. We can observe that the dimensions of the depth-graded formal multiple zeta spa…
We study a certain class of q-analogues of multiple zeta values, which appear in the Fourier expansion of multiple Eisenstein series. Studying their algebraic structure and their derivatives we propose conjectured explicit formulas for the derivatives of double and triple Eisenstein series.
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.
It is known that there are infinitely many singularities of multiple zeta functions and the special values at non-positive integer points are indeterminate. In order to give a suitable rigorous meaning of the special values there, Furusho, Komori, Matsumoto and Tsumura introduced desingularized values by using their desingularization method to resolve all singularities. On the other hand, Ebrahimi-Fard, Manchon and Singer introduced renormalized values by the renormalization method à la Connes and Kreimer and they showed that the values fulfill the shuffle-type product formula. In this paper, …
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.
Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multizeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborifed zeta values are algebra morphisms for these new products on trees.
Partial fraction methods play an important role in the study of multiple zeta values. One class of such fractions is related to the integral representations of MZVs. We show that this class of fractions has a natural structure of shuffle algebra. This finding conceptualizes the connections among the various methods of stuffle, shuffle and partial fractions in the study of MZVs. This approach also gives an explicit product formula of the fractions.
The algebraic and combinatorial theory of shuffles, introduced by Chen and Ree, is further developed and applied to the study of multiple zeta values. In particular, we establish evaluations for certain sums of cyclically generated multiple zeta values. The boundary case of our result reduces to a former conjecture of Zagier.
A large family of relations among multiple zeta values may be described using the combinatorics of shuffle and quasi-shuffle algebras. While the structure of shuffle algebras have been well understood for some time now, quasi-shuffle algebras were only formally studied relatively recently. In particular, Hoffman gives a thorough discussion of the algebraic structure, including a choice of algebra basis, and applies his results to produce families of relations among multiple zeta values and their generalisations. In a recent preprint, Hirose and Sato establish a family of relations coming from …
In their seminal paper "Double zeta values and modular forms" Gangl, Kaneko and Zagier defined a double Eisenstein series and used it to study the relations between double zeta values. One of their key ideas is to study the formal double space and apply the double shuffle relations. They also proved the double shuffle relations for the double Eisenstein series. More recently, Kaneko and Tasaka extended the double Eisenstein series to level 2, proved its double shuffle relations and studied the double zeta values of level 2. Motivated by the above works, we define in this paper the correspondin…
Regularised double shuffle relations for planar Arborified Zeta Values
Université de Montpellier, IMAG, Institut Montpelliérain Alexander Grothendieck（蒙彼利埃大学，IMAG，亚历山大·格罗滕迪克蒙彼利埃研究所）；Department of Mathematics, IRIMAS, Université de Haute Alsace（阿尔萨斯大学高等研究院数学系）；IRMA, Université de Strasbourg（斯特拉斯堡大学数学与离散数学研究所）；Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）
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 …
Quasi-shuffle algebras have been a useful tool in studying multiple zeta values and related quantities, including multiple polylogarithms, finite multiple harmonic sums, and q-multiple zeta values. Here we show that two ideas previously considered only for multiple zeta values, the interpolated product of S. Yamamoto and the symmetric sum theorem, can be generalized to any quasi-shuffle algebra.
In this paper we obtain a recursive formula for the shuffle product and apply it to derive two restricted decomposition formulas for multiple zeta values (MZVs). The first formula generalizes the decomposition formula of Euler and is similar to the restricted formula of Eie and Wei for MZVs with one strings of 1's. The second formula generalizes the previous results to the product of two MZVs with one and two strings of 1's respectively.
In this paper we present some of the recent progresses in multiple zeta values (MZVs). We review the double shuffle relations for convergent MZVs and summarize generalizations of the sum formula and the decomposition formula of Euler for MZVs. We then discuss how to apply methods borrowed from renormalization in quantum field theory and from pseudodifferential calculus to partially extend the double shuffle relations to divergent MZVs.
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 study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms of level 2.
Quasi-shuffle products, introduced by the first author, have been useful in studying multiple zeta values and some of their analogues and generalizations. The second author, together with Kajikawa, Ohno, and Okuda, significantly extended the definition of quasi-shuffle algebras so it could be applied to multiple zeta q-values. This article extends some of the algebraic machinery of the first author's original paper to the more general definition, and uses this extension to obtain various algebraic formulas in the quasi-shuffle algebra in a transparent way. Some applications to multiple zeta va…
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.
Interpolated multiple zeta values can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values. In this paper, we give some algebraic relations of interpolated multiple zeta values, such as the symmetric sum formula, the shuffle regularized sum formula, a weighted sum formula and some evaluation formulas with even arguments. All the algebraic relations provided in this paper are deduced from the extended double shuffle relations.
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 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.
We consider the problem of deducing the duality relation from the extended double shuffle relation for multiple zeta values. Especially we prove that the duality relation for double zeta values and that for the sum of multiple zeta values whose first components are 2's are deduced from the derivation relation, which is known as a subclass of the extended double shuffle relation.
We study a variant of multiple zeta values of level 2, which forms a subspace of the space of alternating multiple zeta values. This variant, which is regarded as the `shuffle counterpart' of Hoffman's `odd variant', exhibits nice properties such as duality, shuffle product, parity results, etc., like ordinary multiple zeta values. We also give some conjectures on relations between our values, Hoffman's values, and multiple zeta values.
In this paper, we are going to perform the shuffle products of $Z_-(n) = \sum_{a+b=m} (-1)^{b} ζ(\{1\}^{a},b+2)$ and $Z_+^\star(n) = \sum_{c+d=n} ζ^{\star}(\{1\}^{c},d+2)$ with $m+n = p$. The resulted shuffle relation is a weighted sum formula given by \begin{equation*} \frac{(p+1)(p+2)}{2} ζ(p+4) =\sum_{m+n=p} \sum_{|\boldsymbolα|=p+3} ζ(α_{0}, α_{1}, \ldots, α_{m}, α_{m+1}+1) \sum_{a+b+c=m} \Bigl( W_{\boldsymbolα}(a,b,c) + W_{\boldsymbolα}(a,b,c=0) + W_{\boldsymbolα}(a=0,b,c) + W_{\boldsymbolα}(a=0,b=m,c=0) \Bigr), \end{equation*} where $W_{\boldsymbolα}(a,b,c) = 2^{σ(a+b+1)-σ(a)-(b+1)} (1-2…
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.
We define an elliptic generating series whose coefficients, the elliptic multizetas, are related to the elliptic analogues of multiple zeta values introduced by Enriquez as the coefficients of his elliptic associator; both sets of coefficients lie in $\mathcal{O}(\mathfrak{H})$, the ring of functions on the Poincaré upper half-plane $\mathfrak H$. The elliptic multizetas generate a $\mathbb Q$-algebra $\mathcal{E}$ which is an elliptic analogue of the algebra of multiple zeta values. Working modulo $2πi$, we show that the algebra $\mathcal{E}$ decomposes into a geometric and an arithmetic part…
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.
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 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.
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 …
