arXiv subjects
Publications and source records attributed to David Jarossay.
We present a concise method for deriving an explicit formula for $p$-adic multiple zeta values. The formula features a variant of multiple harmonic sums, termed binomial multiple harmonic sums.
Given a finite set $S$ of distinct primes, we propose a method to construct polylogarithmic motivic Chabauty-Kim functions for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$ using resultants. For a prime $p\not\in S$, the vanishing loci of the images of such functions under the $p$-adic period map contain the solutions of the $S$-unit equation. In the case $\vert S\vert=2$, we explicitly construct a non-trivial motivic Chabauty-Kim function in depth 6 of degree 18, and prove that there do not exist any other Chabauty-Kim functions with smaller depth and degree. The method, inspired by work of Dan-Cohen and the first author, enhances the geometric step algorithm developed by Corwin and Dan-Cohen, providing a more efficient approach.
We prove that an irreducible projective complete intersection of dimension at least two with isolated singularities has trivial fundamental group. As an application, the surface $\Upsilon$ parametrizing cuboids and its minimal resolution of singularities are simply connected. By an independent argument we also show that the surface $V$ parametrizing face cuboids and its resolution are simply connected as well. We then introduce two smooth open subvarieties $S_{1}$ and $S_{2}$ of the surface parametrizing face cuboids, show that each has fundamental group isomorphic to $\mathbb{F}_{3}\ltimes \mathbb{Z}^{2}$, and prove that their Malcev completions reduce to the free pro-unipotent group on three generators. In an appendix we treat the corresponding real loci, whose fundamental groups, in contrast, are far from trivial.
If Z is an open subscheme of Spec ZZ, X is a sufficiently nice Z-model of a smooth curve over QQ, and p is a closed point of Z, the Chabauty-Kim method leads to the construction of locally analytic functions on X(ZZ_p) which vanish on X(Z); we call such functions "Kim functions". At least in broad outline, the method generalizes readily to higher dimensions. In fact, in some sense, the surface M_{0,5} should be easier than the previously studied curve M_{0,4} since its points are closely related to those of M_{0,4}, yet they face a further condition to integrality. This is mirrored by a certain "weight advantage" we encounter, because of which, M_{0,5} possesses new Kim functions not coming from M_{0,4}. Here we focus on the case "ZZ[1/6] in half-weight 4", where we provide a first nontrivial example of a Kim function on a surface. Central to our approach to Chabauty-Kim theory (as developed in works by S. Wewers, D. Corwin, and the first author) is the possibility of separating the geometric part of the computation from its arithmetic context. However, we find that in this case the geometric step grows beyond the bounds of standard algorithms running on current computers. Therefore, some ingenuity is needed to solve this seemingly straightforward problem, and our new Kim function is huge.
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.
We prove that for any associator, two specific families of coefficients of the associator can be expressed in terms of coefficients of lower depth. Combining these results to our notions of adjoint $p$-adic multiple zeta values and multiple harmonic values, we obtain a new point of view on the question of relating $p$-adic and finite multiple zeta values, and a few other application to the study of $p$-adic multiple zeta values via explicit formulas.
$p$-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.
We introduce adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values. They are two variants of cyclotomic multiple zeta values, closely related to each other. They arise as key tools for the study of $p$-adic cyclotomic multiple zeta values. Moreover, cyclotomic multiple harmonic values provide an adelic lift to a cyclotomic generalization of finite multiple zeta values. We establish certain standard properties of these two objects. We consider two types of properties : some related to double shuffle relations, and some related to associator and Kashiwara-Vergne relations.
We define subvarieties of $\mathcal{M}_{0,n}$ equipped with algebraic functions that are solutions to the generic double shuffle equations satisfied by multiple polylogarithms on $\mathcal{M}_{0,n}$.
We show that the special values at tuples of positive integers of the $p$-adic multiple $L$-function introduced by the first-named author et al. can be expressed in terms of the cyclotomic multiple harmonic values introduced by the second-named author.
We generalize the definition of overconvergent $p$-adic multiple polylogarithms and of $p$-adic cyclotomic multiple zeta values and we prove a bound on their norm. A byproduct of the proof is a characterization of these objects in terms of certain regularized $p$-adic iterated integrals. The generalization of the definition consists in replacing the underlying Frobenius structure by its iterations. The bound on the norms of overconvergent $p$-adic multiple polylogarithms that we obtain is a prerequisite for our subsequent papers on $p$-adic cyclotomic multiple zeta values.
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 considering an operation which we call localization (inverting certain integration operators) in the pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,μ_{N},\infty\}$, and by using $p$-adic pro-unipotent harmonic actions, we extend the definition of $p$MZV$μ_{N}$'s to indices for which these integers can be negative, and we study these generalized $p$MZV$μ_{N}$'s.
This is a review on the two first parts of our work on $p$-adic multiple zeta values at $N$-th roots of unity ($p$MZV$μ_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,μ_{N},\infty\}$ (where $N$ and $p$ are coprime). We restrict for simplicity the review to the case of $N=1$, i.e. the case of $p$-adic multiple zeta values ($p$MZV's). The main tools are new objects which we call $p$-adic pro-unipotent harmonic actions. These are continuous group actions on a space containing the non-commutative generating series of weighted multiple harmonic sums, they are related to the motivic Galois action on $π_{1}^{\un}(\mathbb{P}^{1} - \{0,1,\infty\})$ and to the Poisson-Ihara bracket, and interrelated by some maps. They are defined in \cite{J2} and \cite{J3} ; the definition relies on a simplification of the differential equation of the Frobenius, proved as a preliminary technical fact by \cite{J1}. Part I (\cite{J1},\cite{J2},\cite{J3}) is an explicit computation of the Frobenius of $π_{1}^{\un,\crys}(\mathbb{P}^{1} - \{0,1,\infty\})$, and in particular of $p$MZV's. We give formulas which keep a track of the motivic Galois action. Part II (\cite{J4},\cite{J5},\cite{J6}) is a study of the algebraic properties of $p$MZV's brought together with the formulas of part I. We state an explicit elementary version of the Galois theory of $p$MZV's.
Let $p$ a prime number. For all $N \in \mathbb{N}^{\ast}$ prime to $p$, let $k_{N}$ be a finite field of characteristic $p$ containing a primitive $N$-th root of unity. Let $X_{k_{N},N}=\text{ }\mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }k_{N}$. This work is an explicit theory of the crystalline pro-unipotent fundamental groupoid $(π_{1}^{\un,\crys})$ of $X_{k_{N},N}$. In the parts I to IV, we have considered each possible value of $N$ separately. The purpose of part V is to study the role of the morphisms relating $π_{1}^{\un}(\mathbb{P}^{1} - \{0,μ_{N_{1}},\infty\})$ and $π_{1}^{\un}(\mathbb{P}^{1} - \{0,μ_{N_{2}},\infty\})$ when $N_{1}$ divides $N_{2}$. In V-1, we specify this question to the theme of part I, the computation of the Frobenius. For any $N \in \mathbb{N}^{\ast}$, let $K_{N}=\mathbb{Q}_{p}(ξ_{N})$ where $ξ_{N}\in \overline{\mathbb{Q}_{p}}$ is a primitive $N$-th root of unity, and $X_{K_{N},N} = \mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }K_{N}$. For $N$ prime to $p$, we are used to view the Frobenius of $π_{1}^{\un,\crys}(X_{k_{N},N})$ as a structure on $π_{1}^{\un,\DR}(X_{K_{N},N})$. In V-1, we show that the Frobenius of $π_{1}^{\un,\DR}(X_{K_{N},N})$, iterated $α\in \mathbb{N}^{\ast}$ times, can be extended canonically as a structure of $π_{1}^{\un,\DR}(X_{K_{p^αN},p^αN})$. This allows to define generalizations of adjoint $p$-adic multiple zeta values associated with roots of unity of order $p^αN$, and several related objects. This also gives a canonical framework to relate to each other the direct method of computation of the Frobenius of I-1 and the indirect methods of computation of the Frobenius of I-2 and I-3.
Let $X_{0}=\mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p>0$ and containing a primitive $N$-th root of unity. We establish an explicit theory of the crystalline pro-unipotent fundamental groupoid of $X_{0}$. In part I, we have computed explicitly the Frobenius, and in particular cyclotomic $p$-adic multiple zeta values. In part II, we use part I to understand the algebraic relations of cyclotomic $p$-adic multiple zeta values via explicit formulas ; this is in particular a study of the harmonic Ihara actions and the maps of comparisons between them introduced in I-2 and I-3. In II-1, we have developed the basics of algebraic theory of cyclotomic sequences of prime weighted multiple harmonic sums and adjoint cyclotomic multiple zeta values viewed as variants of those of the algebraic theory of cyclotomic multiple zeta values. In this II-2, we use part I and II-1 to show that one can read some standard algebraic relations of cyclotomic $p$-adic multiple zeta values via the explicit formulas and via the standard algebraic relations of sequences of multiple harmonic sums. This amounts to say that the harmonic Ihara actions and the comparison maps are compatible with algebraic relations. The two main results are two "harmonic" versions of Besser-Furusho-Jafari's theorem that $p$-adic multiple zeta values satisfy the regularized double shuffle relations. This gives two different answers to what we could call "the adjoint variant" of a question of Deligne and Goncharov about reading the quasi-shuffle relation of $p$-adic multiple zeta values via explicit formulas.
Let $X=\text{ }\mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }W(\mathbb{F}_{q})$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p$ prime to $N$ and containing a primitive $N$-th root of unity. We establish an explicit theory of the crystalline Frobenius of the pro-unipotent fundamental groupoid of $X$. In part I, we have computed explicitly the Frobenius action. In part II, we use this computation to understand explicitly the algebraic relations of cyclotomic $p$-adic multiple zeta values. We have used the ideas and the vocabulary of the Galois theory of periods, and in our framework, certain sequences of prime weighted multiple harmonic sums have been dealt with as if they were periods. In this II-3, we define three notions which essentialize our three types of computations, respectively : a "continuous" groupoid $π_{1}^{\un,\DR}(X_{K})^{\hat{\text{cont}}}$, a "localization" $π_{1}^{\un,\DR}(X_{K})^{\loc}$ of $π_{1}^{\un,\DR}(X_{K})$, and a "rational counterpart at zero" $π_{1}^{\un,\RT,0}(X_{K})$ of $π_{1}^{\un,\DR}(X_{K})$. As an application, and as a conlcusion of this part II, we justify and clarify our Galois-theoretic point of view, in particular, we construct period maps and state period conjectures for sequences of prime weighted multiple harmonic sums.
We obtain formulas relating $p$-adic cyclotomic multiple zeta values and cyclotomic multiple harmonic sums. In particular, we obtain a series formula for $p$-adic cyclotomic multiple zeta values, and conversely a formula for certain cyclotomic multiple harmonic sums in terms of $p$-adic cyclotomic multiple zeta values. Our formulas are related to the motivic framework via a new notion which we call pro-unipotent harmonic actions, which are ad hoc $p$-adic byproducts of the Ihara action. As an application, we prove a conjecture of Akagi, Hirose and Yasuda on the relation between $p$-adic multiple zeta values and multiple harmonic sums, and we generalize it to the cyclotomic case. We also deduce bounds on the dimension of the spaces of finite cyclotomic multiple zeta values.