Japan - Taiwan Joint Workshop on Multiple Zeta Values 2020———————————

Abstracts———————————
February 8. (Saturday)
Jing Yu(National Taiwan University, Taiwan)

Title: Motivic Galois Groups and MultiZeta Values in Characteristic p
Abstract: We report our recent studies of MZV in positive characteristic. Let Γs be the motivic
Galois group of the t-motive Ms associated to the depth r MZV ζA (s), following Anderson-Thakur.
Assuming a conjecture of Lara Rodriguez-Thakur on ”slicing”, we show that ζA (s) is Eulerian if
and only if Γs ∼= Gm . Also if the depth r MZV ζA (s) is “zetalike ”, then Γs is an extension of Gmby a vector group of dimension ≤ r. The dimension of this vector group can be explicitly given interms of the tuple s.

Yen-Tsung Chen(National Tsing Hua University, Taiwan)

Title: On the linear independence of θ-adic multiple zeta values over function ﬁelds
Abstract: In this talk, we will talk about Chang-Mishiba v-adic multizeta values over function
ﬁelds. We prove an analogue of the Goncharov’s direct sum conjecture when v is of degree one.
The same result is also valid for v-adic Carlitz multiple polylogarithms at algebraic point when v
is of degree one.

Ryotaro Harada(Nagoya University, Japan)

Title: A positive characteristic analogue of alternating multizeta values
Abstract: In 2004, Thakur introduced and studied a positive characteristic analogue of multizeta
values. It is known that they have properties including non-vanishing property, sum-shuﬄe relation,
period interpretation and linear independence by the work of Anderson, Chang and Thakur. In
this talk, we will introduce and study alternating multizeta values in positive characteristic whichgive a certain generalization of Thakur’s multizeta values. We will describe these objects havean alternating version of the above four properties. These results enable us to settle Goncharov’s
conjecture for alternating multizeta values in positive characteristic.
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February 9. (Sunday)
Masataka Ono(Kyushu University, Japan)

Title: On truncated versions of t-adic symmetric multiple zeta values
Abstract: The t-adic symmetric multiple zeta values (t-adic SMZVs), which was introduced byHirose, Jarossay, and Rosen independently, is a symmetric counterpart of the p-adic ﬁnite multiple
zeta values. In this talk, we give 2 types of truncated version of t-adic SMZVs and present theirproperties. This talk is based on joint works with Shin-ichiro Seki and Shuji Yamamoto.
Minoru Hirose(Kyushu University, Japan)

Title: Motivic Galois group over Z[1/2] and linear relations among motivic alternating multiple
zeta values
Abstract: Motivic alternating multiple zeta values are signed analogues of motivic multiple zetavalues. In this talk, we introduce alternating analogues of the conﬂuence relations, and show thatthey give all linear relations among motivic alternating multiple zeta values. Furthermore we explainthat this result gives a complete answer to a Z[1/2] analogue of a well-known open conjecture thatthe motivic Galois group of mixed Tate motives over Z coincides with Grothendieck-Teichmullergroup. This is a joint work with Nobuo Sato at Kyushu University.
Yen-Liang Kuan

(National Center for Theoretical Sciences, Taiwan)

Title: The Mordell-Weil theorem for t-modules
Abstract: For each positive characteristic multiple zeta value (deﬁned by Thakur) ζA (s), ChangPapanikolas-Yu constructed the t-module Es deﬁned over A and integral points vs , us ∈ Es (A).
They proved that ζA (s) is Eulerian (resp. zeta-like) if and only if vs is an Fq [t]-torsion point inEs (A) (resp. vs , us are Fq [t]-linearly dependent in Es (A)). In this talk, we are interested in thestructure theory of the t-module Es (A). Poonen proved an analogue for Drinfeld modules of theMordell-Weil theorem. We shall generalize his results to the case of speciﬁc families of t-modules.
In particular, we prove that the t-module Es (A) is the direct sum of its torsion submodule, which
is ﬁnite, with a free Fq [t]-module of rank ℵ0 .
Oguz Gezmis(National Tsing Hua University, Taiwan)

Title: Deformation of Multiple Zeta Values and Their Logarithmic Interpretation in Positive
Characteristic
Abstract: Pellarin introduced the deformation of multiple zeta values of Thakur as elements
over Tate algebras. In this talk, we discuss to relate these values to a certain coordinate of ahigher dimensional Drinfeld module over Tate algebras. Furthermore we introduce the multiple
polylogarithms and represent the deformation of multiple zeta values as a liner combination ofmultiple polylogarithms.
2Henrik Bachmann(Nagoya University, Japan)

Title: q-double zeta values and modular forms
Abstract: In their 2006 article, Gangl, Kaneko and Zagier established a duality between the spaceof even period polynomials of modular forms for SL2 (Z) and the space of formal double shuﬄerelations among double zeta values. We will discuss a variant of this duality for certain q-analoguesof double zeta values. As one consequence, we obtain explicit rational solutions to the q-double
shuﬄe equations in depth two generalizing Gangl–Kaneko–Zagier’s “Bernoulli realization”. This is
joint work in progress with Ulf Kühn (Hamburg) and Nils Matthes (Oxford).
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February 10. (Monday)
Tomoya Machide(National Institute of Informatics, Japan)

Title: Computational results on ranks of matrices of extended double shuﬄe relations for modulo
2 (joint work with Tomohiro Sonobe)
Abstract: Extended double shuﬄe relations of multiple zeta values are linear relations with integer
coeﬃcients, which are obtained by shuﬄe and stuﬄe products and renormalization. It is conjectured
that these relations suﬃce to give all linear relations for a ﬁxed weight, and the conjecture is
conﬁrmed up to weight 20 by computer. In this talk, I introduce computational results which
update record to weight 21. Our method uses ranks of matrices corresponding to these relations,where the entries of the matrices are taken modulo 2.

Yoshitaka Sasaki(Osaka University of Health and Sport Sciences, Japan)

Title: The tangent symmetry for the multiple zeta-function
Abstract: Special values of the multiple zeta-function at non-positive integers depend on limiting
processes except for some exceptions. Akiyama, Egami and Tanigawa, and Akiyama and Tanigawa
introduced and evaluated multiple zeta values at non-positive integers for certain limiting processes,
and showed relations among such multiple zeta values, called the tangent symmetry. In this talk,we show that the multiple zeta-function satisﬁes the similar symmetric relation. Then, tangent
symmetries of multiple zeta values for several limiting processes are obtained via such formula.

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Yuichiro Taguchi(Tokyo Institute of Technology, Japan)

Title: Mordell-Weil groups over moderately large global ﬁelds
Abstract: A perfect ﬁeld K is said to be Kummer-faithful if, for any ﬁnite extension L of K and any
semi-abelian variety A over L, the Mordell-Weil group A(L) has no non-trivial divisible subgroup.
By the Mordell-Weil theorem, an algebraic number ﬁeld of ﬁnite degree is Kummer-faithful. Itis known that sub-p-adic ﬁelds are also Kummer-faithful. We construct a family of Kummerfaithful number ﬁelds which are not sub-p-adic. We also consider a function ﬁeld analogue of thisconstruction. This is a joint work with Yoshiyasu Ozeki.
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February 11. (Tuesday)
Yoshihiro Takeyama(University of Tsukuba, Japan)

Title: Finite multiple harmonic q-series at a root of unity and Kaneko-Zagier conjecture
Abstract: Kaneko and Zagier introduce two variants of multiple zeta value: the ﬁnite multiple
zeta value (FMZV) and the symmetric multiple zeta value (SMZV). They conjecture that there
exists a Q-algebra isomorphism which sends FMZVs to SMZVs. In this talk we introduce a ﬁnite
multiple harmonic q-series at a root of unity, which reproduces the FMZV and the SMZV throughan algebraic operation and an analytic one, respectively. This property oﬀers an explanation,
though not a proof, for the conjecture due to Kaneko and Zagier. We also discuss recent progressin our framework. This talk is based on a joint work with H. Bachmann (Nagoya Univ.) and K.
Tasaka (Aichi Prefectural Univ.).

Shuji Yamamoto(Keio University, Japan)

Title: A variant of multiple polylogarithms and relations on them
Abstract: Let Lik (z1 , . . . , zr ) be the multiple polylogarithm of shuﬄe type. We are mainly interested in the functions of one variable z obtained by substituting 1 or z to each zi , and linear
relations on them, such as Li2 (z) = Li1,1 (1, z) − Li1,1 (z, z). It turns out that some relations (duality, sum formula, ...) can be nicely formulated in terms of certain linear combinations of thosepolylogarithms. In this talk, we will introduce these functions and present known relations.

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Ryota Umezawa(Nagoya University, Japan)

Title: Relations between iterated log-sine integrals, multiple zeta values and multiple polylogarithms
Abstract: Iterated log-sine integrals introduced by the speaker are related to multiple zeta values
and multiple polylogarithms. In this talk, we discuss relations between iterated log-sine integrals,
multiple zeta values and multiple polylogarithms. In particular, we give an explicit evaluation of
iterated log-sine integrals in terms of multiple zeta values and multiple polylogarithms as the maintheorem. We also give some conjectures on iterated log-sine integrals and multiple zeta values.

Yoshinori Mishiba(University of the Ryukyus, Japan)

Title: Logarithmic interpretation of Taylor coeﬃcients of t-motivic multiple zeta values
Abstract: For each index, we construct a t-module such that the ∞-adic (resp. v-adic) multiple
zeta value at the index appears in a certain coordinate of the ∞-adic (resp. v-adic) logarithmic
vector of this t-module at a speciﬁc point, where v is a ﬁnite place of a rational function ﬁeld
over a ﬁnite ﬁeld. This is used to show the existence of a linear map from the space of ∞-adic
MZV’s to that of v-adic MZV’s, which is a function ﬁeld analogue of a conjecture of Furusho incharacteristic zero. In ∞-adic setting, we give a generalization of this logarithmic interpretation.
More precisely, we determine all of the coordinates of the logarithmic vector in terms of Taylor
coeﬃcients of (∞-adic) t-motivic MZV’s and t-motivic Carlitz multiple star polylogarithms. Thisis a joint work with Chieh-Yu Chang and Nathan Green.

Chieh-Yu Chang(National Tsing Hua University, Taiwan)

Title: Algebra structure of multiple zeta values in positive characteristic
Abstract: In this talk, we will talk about MZV’s over function ﬁelds in positive characteristic.
We show that for each ﬁnite place v of the base function ﬁeld, the v-adic MZV’s form an algebrawith multiplication law given by Thakur’s sum-shuﬄe product for ∞-adic MZV’s. We then use
Yu’s sub-t-module theorem to show that there is an algebra homomorphism from ∞-adic MZV’sto v-adic MZV’s. Joint work with Yen-Tsung Chen and Yoshinori Mishiba.
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