Algebraic Number Theory and Related Topics 2015
Abstracts of talks

November 30 (Mon)
10:00 – 10:45 Minoru Hirose (Kyoto University)
Shintani zeta functions and Gross conjecture
The Gross conjecture is a reﬁnement of Gross-Stark conjecture. The Gross conjecture consists of
two parts: “vanishing order” part and “exact value” part. The second part predicts a mysterious
relation between Stickelberger element and regulator. In 2014, Dasgupta and Spiess proved theﬁrst part of the conjecture under some conditions. In this talk, we construct a “enhanced”
Stickelberger element. By using this, we give an alternative proof of the Dasgupta-Spiess’s theorem
and propose a reﬁnement of the second part of the Gross conjecture.
10:55 – 11:40 Kazuto Ota (Keio University)
Kato’s Euler system and the Mazur-Tate reﬁned conjecture of BSD type
Mazur-Tate proposed a conjecture which compares the Mordell-Weil rank of an elliptic curveto the order of vanishing of Mazur-Tate elements, which are analogues of Stickelberger elements.
In this talk we explain the conjecture, and under some relatively mild assumptions, we give asketch of its proof based on certain divisibility of Kato’s Euler system.
11:50 – 12:40 Florian Sprung (Princeton University/IAS)
Integral Euler systems and Iwasawa theory
For an elliptic curve E over Q and a prime p of good reduction, the Iwasawa main conjecture
is an equality between an analytically constructed ideal, from special values of the Hasse-WeilL-function, and a corresponding algebraic one, from subﬁelds of the cyclotomic Zp -extension Q∞of Q. When p has ordinary reduction, this conjecture has become a theorem via works of Katoand Skinner–Urban. In the supersingular case, one inclusion had been known from works of Kato,Kobayashi, and others. Recently, Wan has proved the reverse inclusion when ap = 0. We sketcha generalization of Wan’s result to the general supersingular case. An essential ingredient in theproof is an integral version of the Euler system constructed by Kings, Loeﬄer, and Zerbes, which
we use in a non-standard way.
14:10 – 14:55 Jun Ueki (Kyushu University/University of Tokyo)
Theory of genera and Iwasawa invariants for 3-manifolds
We ﬁrstly establish an analogue for 3-manifolds of Furuta’s genus formula for number ﬁelds,
as an application of the idele theory by Niibo (and myself) for 3-manifolds. Then, using ourgenus formula, we prove an analogue for branched Zp -covers of Iwasawa’s theorem given in 1973
on µ-invariants. Furthermore, we show an analogue of Kida’s formula on λ-invariants for a coverof degree p-power of branched Zp -covers. Iwasawa gave a proof for Kida’s formula using thecohomology of units in 1981. By the analogy between units and 2-cycles, we give a similar proof
using the cohomology of 2-cycles.
15:10 – 15:55 Kotaro Sugahara (Kyushu University)
Central extensions and reciprocity law for arithmetic surfaces
In 2003, Osipov constructed dimension two central extensions using Parshin-Beilinson adeles
and established reciprocity laws for algebraic surfaces. This theory is related to K2 theory andintersection theory for algebraic surfaces. Since K2 theory does not work for (inﬁnite part of)arithmetic surfaces, we develop a new theory of central extensions based on Arakelov theory, then1

we use our adelic cohomology theory to establish the reciprocity law for arithmetic surfaces. Thisis a joint work with Prof. Weng.
16:10 – 17:00 Takao Komatsu (Wuhan University)
Bernoulli-Carlitz numbers and Cauchy-Carlitz numbers
In 1935 Carlitz introduced Bernoulli-Carlitz numbers as analogues of Bernoulli numbers forthe rational function ﬁeld. In this talk, we introduce Cauchy-Carlitz numbers as analogues of
Cauchy numbers. By using Stirling-Carlitz numbers, we give their arithmetical and combinatorialproperties and relations with Bernoulli-Carlitz numbers for the rational function ﬁeld. Several
new identities are also obtained by using Hasse-Teichmüller derivatives. (This talk is based upona joint work with Hajime Kaneko)

December 1 (Tue)
9:30 – 10:15 Keiji Adachi (Kyushu University)
Distance to cusps and stability
In the study of totally real ﬁelds, Siegel introduced a distance from a modular point to a cusp
and hence constructed corresponding fundamental domains. This distance was generalized towork for all number ﬁelds in Prof. Weng’s study on non-abelian zeta functions. Motivated bythis, working on product of rigid analytic upper half planes, we ﬁrst construct new distances
between modular points and cusps. With the help of the correspondence between modular points
and rank two bundles over curves deﬁned over ﬁnite ﬁelds, we then obtain the following theorem.
Theorem: A rank two bundle on a curve over ﬁnite ﬁelds is Mumford semi-stable if and only if
the distances of its associated modular point to all cusps are no less than one.
10:30 – 11:15 Megumi Takata (Kyushu University)
The inﬁnite base change lifting associated to an APF extension of a mixedcharacteristic local ﬁeld
In the Langlands conjecture, the base change lifting is the counterpart on the automorphic
side of the restriction functor on the Galois side. For a ﬁnite cyclic extension of a number ﬁeld
or a mixed characteristic local ﬁeld, the lifting was constructed by Langlands for GL(2) and by
Arthur-Clozel for GL(n). In this talk, we give such a lifting for a totally ramiﬁed Zp -extensionof a mixed characteristic local ﬁeld. By Kazhdan’s theory of close ﬁelds, we can interpret thisas an operation which maps an automorphic representation of GL(n) over a local ﬁeld of mixed
characteristic to that of positive characteristic.
11:30 – 12:30 Yuichiro Hoshi (RIMS, Kyoto University)
Introduction to inter-universal Teichmüller theory I (Survey)
The purpose of this series of three talks is to give an introductory survey of the inter-universal
Teichmüller theory established by Shinichi Mochizuki. In this talk, which is the ﬁrst talk in theseries, the speaker explains a “rough statement” of the main theorem of inter-universal Teichmüllertheory. Moreover, we discuss why a Diophantine result can be derived from the main theorem.
14:05 – 14:50 Yoshinosuke Hirakawa (Keio University)
On the descent of certain modular Calabi-Yau varieties via the Cynk-Hulek construction
In their paper published in 2007, S. Cynk and K. Hulek introduced a construction of higher
dimensional Calabi-Yau varieties from lower dimensional ones, and constructed some examples ofmodular Calabi-Yau varieties, which are Calabi-Yau varieties whose L-functions are described interms of L-functions of modular forms. In this talk, we combine their idea with the Weil restriction
functors and ﬁxed point free involutions on the so called K3 surfaces (Calabi-Yau surfaces), and2

construct new examples of modular Calabi-Yau varieties whose ﬁeld of deﬁnition is the ﬁeld ofrational numbers. We also explain the relation between these examples and a certain Calabi-Yau
realization problem of Hecke eigen newforms proposed by B. Mazur and D. van Straten.
15:05 – 15:55 Hironori Shiga (Chiba University)
One visualization of Shimura’s complex multiplication theorem via hypergeometricmodular functions (joint work with Atsuhira Nagano)In the work Construction of class ﬁelds and zeta functions of algebraic curves (1967) by Goro
Shimura, it is proved that there exists a modular function (that is called canonical model) thatenables to obtain a certain class ﬁeld of some kind of CM ﬁeld. In our talk we show that for thecase of the CM ﬁeld embedded into the quaternion algebra coming from the compact arithmetic
triangle group we can determine the canonical model as a hypergeometric modular function in anexplicit way. Moreover we give several examples of Hilbert class ﬁelds of such kind of CM ﬁelds
coming from the triangle group ∆(3, 3, 5). For our work, we use Shimura’s reciprocity law andthe existence of the canonical model together with the result by K. Takeuchi (1977). To constructexplicit examples we use the modular function for genus 4 pentagonal curves discovered by K.
Koike (2003).
16:10 – 17:00 Keisuke Arai (Tokyo Denki University)
Points on Shimura curves rational over imaginary quadratic ﬁelds in the non-split case
For a rational indeﬁnite quaternion division algebra B, let M B be the associated Shimura curve.
We consider the behavior of the points on M B rational over a number ﬁeld k when the discriminantd(B) of B grows. Jordan proved that if k is an imaginary quadratic ﬁeld of class number ̸= 1, if ksplits B and if d(B) is suﬃciently large, then M B has no k-rational points. In this talk, I discuss
the case where k does not split B, and give an analogous result by imposing a certain congruent
condition on d(B). I also explain the reason why the non-split case is diﬃcult.

December 2 (Wed)9:00 – 9:45 Kento Yamamoto (Chuo University)
On generators of the Chow group of zero-cycles on diagonal cubic surfaces over3-adic ﬁelds
In this talk, we give an explicit construction of generators of the degree-zero part of the Chowgroup of zero-cycles on some diagonal cubic surfaces over 3- adic ﬁelds. A key point is to checkthat such cycles have non-zero values under the Brauer-Manin pairing. We will show this bycalculating Hilbert symbols.
10:00 – 10:45 Wataru Kai (University of Tokyo)
A moving lemma in the aﬃne space with modulusWe show the contravariance of the additive higher Chow group, introduced by Bloch and Esnault in 2003, with respect to morphisms of smooth aﬃne schemes. It is deﬁned in terms ofalgebraic cycles satisfying the “ modulus condition,” which concerns their behavior at the boundary. Therefore we need a new moving lemma which can manage the “modulus condition.”
11:00 – 11:45 Hiroyasu Miyazaki (University of Tokyo)
On moving algebraic cycles with modulus of bounded degree
Friedlander and Lawson developed the theory of moving algebraic cycles of bounded degree.
The moving lemma of this type plays an important role in Voevodsky’s A1 -homotopy theoryof motives. Recently, a non-homotopical generalization of Voevodsky’s A1 -homotopy theory ofmotives is studied by Kahn-Saito-Yamazaki. The aim of this talk is to present a modiﬁcation of3

the moving lemma of Friedlander-Lawson which might be used in the (future) non-homotopicaltheory of motives.
12:00 – 13:00 Yuichiro Hoshi (RIMS, Kyoto University)
Introduction to inter-universal Teichmüller theory II (Survey)
The purpose of this series of three talks is to give an introductory survey of the inter-universal
Teichmüller theory established by Shinichi Mochizuki. In this talk, which is the second talk in theseries, the speaker introduces the notion of a “mono-theta environment” which plays an essential
role in the nonarchimedean local portion of inter-universal Teichmüller theory, i.e., in the theoryof étale theta functions. The speaker also explains a “multiradial representation of étale thetafunctions” obtained by applying the theory of mono-theta environments.

December 3 (Thu)
9:30 – 10:15 Arata Minamide (RIMS, Kyoto University)
Indecomposability of various proﬁnite groups arising from hyperbolic curves
It is well-known that various proﬁnite groups arising from hyperbolic curves (e.g., the geometric/arithmetic fundamental group of a hyperbolic curve over a number ﬁeld) are center-free. In
this talk, I will introduce the notion of “indecomposability”, which is also a group-theoretic property of proﬁnite groups, and discuss the indecomposability of various proﬁnite groups arising fromhyperbolic curves. Finally, from the point of view of combinatorial anabelian geometry, I willpose the question: Does the Grothendieck-Teichmüller group GT satisfy the indecomposability?
I will give an aﬃrmative answer to a pro-l version of this question by applying a certain anabelian
result over ﬁnite ﬁelds.
10:30 – 11:15 Yu Yang (RIMS, Kyoto University)
On the existence, geometry and p-ranks of vertical ﬁbers of coverings of curves
Let R be a complete DVR with algebraically closed residue ﬁeld of characteristic p > 0, X astable curve over R with smooth generic ﬁber, and f : Y −→ X a morphism of stable curves
over R such that the morphism of generic ﬁbers induced by f is a Galois étale covering. A closed
point x of X is called a vertical point if dimf −1 (x) = 1. In this case, f −1 (x) is called the verticalﬁber associated to x. We study the existence, the geometry and the p-ranks of vertical ﬁbers
under certain assumptions. As an application, we generalize a theorem of M. Raynaud concerning
ordinariness of coverings to the case of stable curves.
11:30 – 12:30 Yuichiro Hoshi (RIMS, Kyoto University)
Introduction to inter-universal Teichmüller theory III (Survey)
The purpose of this series of three talks is to give an introductory survey of the inter-universal
Teichmüller theory established by Shinichi Mochizuki. In order to establish the main theorem ofinter-universal Teichmüller theory, one has to suitably globalize local theories at various primes ofa number ﬁeld. In this talk, which is the third talk in the series, the speaker explains the notionof a “Hodge theater” which is closely related to such a globalization. Finally, the speaker gives abrief sketch of the proof of the main theorem.
14:00 – 14:45 Yasuhiro Wakabayashi (University of Tokyo)
Ordinariness and duality of dormant opers
A dormant oper is a certain principal homogeneous space on a pointed stable curve in positive
characteristic equipped with an integrable logarithmic connection. The study of dormant opers
and their moduli may be linked to various ﬁelds of mathematics, e.g., the p-adic Teichmüller theory
developed by Shinichi Mochizuki, Gromov-Witten theory, combinatorics of rational polytopes
4(and spin networks), etc. The key of the linkages between them is to consider the ordinariness ofdormant opers, which may be thought of as a generalization of the usual ordinariness of curves
in positive characteristic. In this talk, we would like to give a quick review of the deﬁnition of adormant oper and to present some related results, including the ordinariness property of dormant
opers on abelian coverings of the generic curve. Also, we would like to propose a certain duality
between the moduli of dormant opers of rank n(< p) and the moduli of dormant opers of rankp − n.
15:00 – 15:45 Yu Iijima (RIMS, Kyoto University)
Diﬀerence between l-adic Galois representations and pro-l outer Galois representations
associated to hyperbolic curves
Let l be a prime number, k a ﬁeld of characteristic zero, and V an algebraic variety over k. By
considering a natural action of the absolute Galois group of k on the maximal pro-l (respectively,the abelianized maximal pro-l) quotient of the geometric fundamental group of V , we may obtaina pro-l outer Galois representation (respectively, an l-adic Galois representation) associated to V .
In this talk, we explain the nonexistence of an isomorphism between the image of the pro-l outer
Galois representation associated to V and the image of the l-adic Galois representation associatedto V , in the case where V is a hyperbolic curve and k satisﬁes a mild assumption. Also, if time
permits, then we explain a pro-Σ analog of the above result, where Σ is a nonempty set of prime
numbers.
16:00 – 17:00 Hidekazu Furusho (Nagoya University)
Galois action on knots (Invited)
I will show that the absolute Galois group of the rational number ﬁeld acts non-trivially on ‘thespace of proﬁnite knots’ in a non-trivial way.

December 4 (Fri)9:30 – 10:15 Ippei Nagamachi (University of Tokyo)
On a good reduction criterion for proper polycurves with sections
Oda and Tamagawa showed that whether a hyperbolic curve over a discrete valuation ﬁeld,
whose residue character is p ≥ 0, has good reduction can be determined by its pro-p′ outer Galois
representation. In this talk, I give a higher dimensional version of this criterion, i.e., a goodreduction criterion for proper polycurves with sections (successive extensions of family of proper
curves of genus ≥ 1 with section), under some condition about genera and p. In the proof, I will
construct a homotopy exact sequence of ﬁbration for certain Tannaka duals, which plays a crucialrole.
10:30 – 11:20
Yuji Odaka (Kyoto University)
Canonical Kähler metrics and arithmetics - Generalising Faltings height
In a recent e-print (1508.07716), we introduced an extension of the Faltings height (1983) for
general arithmetic varieties, which was originally deﬁned for abelian varieties. We show that theextension inherits most of basic properties observed for the original Faltings height, especially for“K-stable” case, and depicts my broader conjectural picture involving e.g. the Minimal ModelProgram. Indeed this “K-stability/ canonical Kähler metrics” explains naturality of the original
deﬁnition for abelian varieties case.

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11:35 – 12:25 Chris Hall (University of Wyoming)
Specialization and big monodromy
Let k be a number ﬁeld and U/k be a smooth geometrically-connected curve. We consider a“compatible system” of lisse Fℓ -sheaves on U and compare the monodromy Gt,ℓ of special ﬁbers
with the geometric monodromy Ggeomof the generic ﬁber. We explain how to show that “most”ℓ
special ﬁbers have “big” monodromy, e.g., that Gt,ℓ contains the derived subgroup of Ggeom.
ℓ13:50 – 14:40 Haoyu Hu (University of Tokyo)
Semi-continuity of total dimension divisors for l-adic sheaves
Let F be a lisse ℓ-adic sheaf on an open dense subset U of a smooth variety X of characteristicp ̸= ℓ. Its non-logarithmic ramiﬁcation along generic points of X − U gives its total dimension
divisor supported on X − U . In this talk, we will show that, in a smooth ﬁbration, the total
dimension divisor of the restriction of an ℓ-adic sheaf on ﬁbers is lower semi-continuous. It is a
generalization of the same property for relative curves obtained by G. Laumon in 1970’s in thegeometric case. This is a joint work with E. Yang.
14:50 – 15:35 Kazuki Tokimoto (University of Tokyo)
Aﬃnoids in the Lubin-Tate perfectoid space and some cases of the local Langlands
correspondence
The non-abelian Lubin-Tate theory states that the cohomology of the Lubin-Tate tower realizes
the local Langlands correspondence and the local Jacquet-Langlands correspondence. Motivatedby this theory, Boyarchenko-Weinstein and Imai-Tsushima recently constructed families of aﬃnoids in a certain limit space of the tower (the Lubin-Tate perfectoid space) and showed that thecohomology of the reduction of (a formal model of) each aﬃnoid realizes the two correspondences
for particular kinds of representations. In this talk, we give a systematic construction of a familyof aﬃnoids generalizing that of Imai-Tsushima in the ”essentially tame” case and discuss thecohomology of the reductions.
15:45 – 16:40 Yoichi Mieda (University of Tokyo)
Non-abelian Lubin-Tate theory and Kloosterman sheaves
Let F be a non-archimedean local ﬁeld. By the local Langlands correspondence, irreduciblesupercuspidal representations of GLn (F ) are parameterized by irreducible n-dimensional representations of the Weil group of F . This correspondence is constructed by using the ℓ-adic étale
cohomology of the universal deformation space of a 1-dimensional formal OF -module of heightn (non-abelian Lubin-Tate theory). If F has equal characteristic, the parameter attached to asimple supercuspidal representation can also be constructed by using the Kloosterman sheaf. In
this talk, I will compare these two geometric constructions. If time permits, I will also give aperspective on the case of general reductive groups.
Program Committee: Hiroki Sumida-Takahashi (Tokushima Univ.), Yasuo Ohno (Tohoku Univ.),Takahiro Tsushima (Chiba Univ.)

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