RATIONAL POINTS ON HIGHER DIMENSIONAL VARIETIES:PROGRAM• Dec 2nd (Mon)– 9:30 - 10:30 Emanuel Peyre– 11:00 - 12:00 Shihoko Ishii– 12:00 - 14:00 Lunch– 14:00 - 15:00 Tony Várilly-Alvarado– 15:30 - 16:30 Kazuhiro Ito• Dec 3rd (Tue)– 9:30 - 10:30 Masato Kuwata– 11:00 - 12:00 David McKinnon– 12:00 - 14:00 Lunch– 14:00 - 15:00 Bianca Viray– 15:30 - 16:30 Kevin Destagnol• Dec 4th (Wed)– 9:00 - 10:00 François Loeser– 10:15 - 11:15 Yuta Kusakabe– 11:30-12:30 Takashi Suzuki– 12:30 - 14:30 Lunch– 14:30 - 17:00 Free discussion• Dec 5th (Thu)– 9:30 - 10:30 Kanetomo Sato
– 11:00 - 12:00 Khoa Nguyen– 12:00 - 13:30 Lunch– 13:30 - 14:30 Hideo Kojima– 14:45 - 15:45 Kaoru Sano
– 16:00 - 17:00 Takahiro Shibata• Dec 6th (Fri)1

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RATIONAL POINTS ON HIGHER DIMENSIONAL VARIETIES: PROGRAM– 9:00 - 10:00 Hiromu Tanaka– 10:15 - 11:15 Yuri Yatagawa– 11:30 - 12:30 Kazuhiko Yamaki– 12:30 - 14:00 LunchTitles and Abstracts• Emanuel Peyre
Title: Rational points of bounded height.
Abstract: In the 80’s Manin’s program lead to several breakthroughs in
the understanding of the behaviour of rational points of bounded height onvarieties. In recent years, after the discovery of new phenomena, this fieldhas known significant progress. In particular, the expected behaviour wasproven on the complement of accumulating thin subsets in several cases
and the work of S. Tanimoto, B. Lehmann and A. Sengupta gave a better
understanding of the accumulating thin subsets on Fano varieties. At thesame time, new questions were raised regarding extensions of the program,beyond Fano varieties, and when using other invariants. This talk will givea survey of the state of the art in this program.
• Shihoko Ishii
Title: Prime divisor with a good liftability of functions.
Abstract: We consider a point 0 of a smooth variety A over a field of characteristic p > 0. We conjecture that every prime divisor E with the center at
0 has a good property: “E can be lifted to a prime divisor Ẽ with the centerat 0̃ on a variety Ã over the field of characteristic 0 with the good property with respect to the liftability of a function.” If the conjecture holds,
then some statements in characteristic 0 induces the statements in positive
characteristic. In the talk, I will give some evidences of the conjecture.
• Tony Várilly-Alvarado
Title: Quasi-hyperbolicity via explicit symmetric differentials.
Abstract: We report on a joint project with Nils Bruin. A surface X isalgebraically quasi-hyperbolic if it contains finitely many curves of genus
0 or 1. In 2006, Bogomolov and de Oliveira used asymptotic computations to show that sufficiently nodal surfaces of high degree in projective
three-space carry symmetric differentials, and they used this to prove quasihyperbolicity of these surfaces. We explain how a granular analysis of theirideas, combined with computational tools and insights, yield explicit results
for the existence of symmetric differentials, and we show how these results
can be used to give constraints on the locus of rational curves on surfaces
like the Barth Decic, Buechi’s surface, and certain complete intersectionsof general type.
• Kazuhiro Ito
Title: On a torsion version of the weight-monodromy conjecture and itsapplications

RATIONAL POINTS ON HIGHER DIMENSIONAL VARIETIES: PROGRAM

3Abstract: I will talk about a torsion version of the weight-monodromy
conjecture for a proper smooth scheme over a local field. I will prove it
for surfaces and for the equal characteristic case. In the equal characteristic case, the proof relies on an ultraproduct variant of Weil II recently
established by A.Cadoret. As applications, I will discuss some finitenessproperties of the Brauer group and the Chow group of codimension two ofa proper smooth scheme over a local field.
• Masato Kuwata
Title: Rational points on generalized Kummer varieties
Abstract: Let E be an elliptic curve over Q. We show that finding points
defined over some cyclic extensions of fixed degree d is related to find Qrational points on a generalized Kummer variety defined as the quotientof E d−1 by a certain cyclic group action. Conjecturally, the existence ofpoints on E defined over some cyclic extensions is related to the vanishingof the twist of the L-function of E by a Dirichlet character. If d = 3, we canshow there are many rational points by finding a section of certain elliptic
fibration on the generalized Kummer surface. In this talk we discuss thecase d = 5, where we run into a subtle arithmetic question.
• David McKinnon
Title: Curves of best approximation
Abstract: Imagine a lonely rational point, sitting on a smooth projective
variety somewhere. It longs for company, but the other rational points keep
their distance ? or else they are complicated, and who wants a complicatedrelationship? Then the point notices a beautiful rational curve, passing
right through it (hard to imagine how the point managed to miss this),with lots of other rational points on it that aren?t that complicated at all!
Does this story always have a happy ending like this? Come to my talk so
I can tell you that we don’t know.
• Bianca Viray
Title: On the index of an intersection of quadrics
Abstract: Not available• Kevin Destagnol
Title: Counting points of given degree via the height zeta function
Abstract: Let X = Symd Pn := Pn × · · · × Pn /Sd where the symmetricd-group acts by permuting the d copies of Pn . Manin’s conjecture gives
a precise prediction for the number of rational points on X of boundedheight in terms of geometric invariants of a resolution of X and the studyof Manin’s conjecture for X can be derived from the geometry of numbersin the cases n > d and for n = d = 2. In this talk, I will explain how one
can use the fact that Pn is an equivariant compactification of an algebraic
group and the height zeta function machinery in order to study the rational
points of bounded height on X in new cases that are not covered by the4

RATIONAL POINTS ON HIGHER DIMENSIONAL VARIETIES: PROGRAM

geometry of numbers techniques. This might in particular be an interesting
testing ground for the latest refinements of Manin’s conjecture.
• François Loeser
Title: Non-archimedean parametrizations, points counting and applications
Abstract: We will provide a general overview on some recent work on
non-archimedean parametrizations and their applications. We will startby presenting our work with Cluckers and Comte on the existence of good
Yomdin-Gromov parametrizations in the non-archimedean context and a padic Pila-Wilkie theorem. We will then explain how this is used in our workwith Chambert-Loir to prove bialgebricity results in products of Mumford
curves. We will conclude with our recent work with Cluckers and Foreyproviding uniform bounds for the number of Fq [t]-points of bounded degreeon algebraic varieties.
• Yuta Kusakabe
Title: Dense entire curves and Oka manifolds
Abstract: It is a fundamental problem whether a given complex manifold
admits a dense entire curve. Oka manifolds are defined by the validity of
the Oka principle which implies the existence of dense entire curves. Our
result states that Oka manifolds are characterized as complex manifolds
which admit dense entire curves not only in themselves but also in spacesof holomorphic maps from Stein manifolds. If time permits, we also discuss
the relation between Oka manifolds and Gromov?s ellipticity.
• Takashi Suzuki
Title: Iwasawa theory for abelian varieties over function fields and HodgeWitt elliptic surfaces
Abstract: I will discuss Iwasawa theory (in particular, Iwasawa mu-invariants)
for abelian varieties over function fields under constant Zp -extensions, mainly
from the side of algebraic geometry in positive characteristic. We give
formulas for the mu-invariant in terms of the dimensions of the TateShafarevich scheme and the Brauer scheme. When the abelian variety issemistable, we also give a formula for mu in terms of the L-function using
Lai-Longhi-Tan-Trihan’s Iwasawa main conjecture. The vanishing of mu
is related to the Hodge-Witt property of algebraic surfaces. Even thoughsupersingular K3 surfaces provide plenty of elliptic curves with non-zero
mu-invariant, we show that elliptic curves with mu=0 form a dense open
subvariety in the moduli of elliptic surfaces. This means that generic elliptic curves satisfy an analogue of Iwasawa’s mu=0 conjecture. Joint with K.
F. Lai, I. Longhi, K.-S. Tan and F. Trihan.
• Kanetomo Sato
Title: Étale cohomology and a zeta value of arithmetic schemes
Abstract: (j/w Takao Yamazaki) Let X be an integral proper regular arithmetic scheme of dimension d. I will start this talk with a result that theétale cohomology groups of X with ‘Qp (d)’-coefficients are isomorphic tothe Selmer groups of Bloch-Kato. This comparison result is based on a certain vanishing of étale 2-extensions, and another comparison result relating

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the order of Tate-Shafarevich groups with p-adic Abel-Jacobi mappings andlocal factors (of the L- and the zeta functions) when d = 2. I will further
give a residue formula of the zeta function ζ(X, s) at s = 2 modulo a rational number prime to p, when X is a proper regular model of a CM elliptic
curve over Q and p is a good and regular prime number.
• Khoa Nguyen
Title: The Hermite-Joubert problem and a conjecture of Brassil-Reichstein
Abstract: The Hermite-Joubert problem asks for which integers n ≥ 5 suchthat for every characteristic zero field extension E/F of degree n thereexists δ ∈ E ∗ with TrE/F (δ) = TrE/F (δ 3 ) = 0. We prove that every n of
the form 3k1 + 3k2 + 3k3 for integers k1 > k2 > k3 ≥ 0 does not satisfy theabove property in the Hermite- Joubert problem. This settles a conjectureof Brassil-Reichstein. The proof involves solving a system of diophantineequations that is an instance of the Mordell-Lang conjecture (i.e. Faltings’theorem).
• Hideo Kojima
Title: Rational open surfaces of log Kodaira dimension ≤ 1
Abstract: Let V be a smooth projective rational surface defined over analgebraically closed field. Let C be a connected curve on V such that
S = V − C has log Kodaira dimension ≤ 1. We show that, if the numberof irreducible components of C is larger than the Picard number of V , thenS contains a simple affine surface. We also give some results on the curve
C when V = P2 .
• Kaoru Sano
Title: Kawaguchi-Silverman conjecture for semi-abelian varieties
Abstract: The Weil height function measures the number-theoretical complexity of rational points. For a dominant rational self-map on a smoothprojective variety, the asymptotic behavior of the height along the forward
orbit is related to the theory of canonical heights. Kawaguchi and Silverman defined the arithmetic degree of the rational point, which measures
the asymptotic behavior of the height. On the other hand, the dynamical degree is a classical invariant of a dominant rational self-map on asmooth projective variety.Kawaguchi and Silverman conjectured that thearithmetic degree of a rational point whose forward orbit is Zariski-dense isequal to the dynamical degree. In this talk, we prove Kawaguchi-Silverman
conjecture for self-morphisms on semi-abelian varieties. This is joint workwith Yohsuke Matsuzawa.
• Takahiro Shibata
Title: Non-density of points of small arithmetic degrees
Abstract: Given a surjective self-morphism f : X → X on a projective
variety over a number field, we can define the arithmetic degree αf (x) off at a point x ∈ X. The Kawaguchi–Silverman conjecture predicts thatany forward f -orbit of a point x ∈ X at which arithmetic degree is strictly
smaller than the first dynamical degree of f is not dense. We give a more

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detailed structure of the locus of points of small arithmetic degree as aconjecture, and verify it in some cases. This is a joint work with Yohsuke
Matsuzawa, Sheng Meng, and De-Qi Zhang.
• Hiromu Tanaka
Title: Rational points and minimal model program in positive characteristic
Abstract: It is known that smooth Fano varieties have rational points if thebase field is either a finite field or the function field of a curve defined over analgebraically closed field of characteristic zero. In this talk, we summarise
some related results from the viewpoint of minimal model program. We
also discuss some open problems on this topic.
• Yuri Yatagawa
Title: Wild ramification and characteristic cycle
Abstract: The characteristic cycle of a constructible sheaf on a smoothvariety is an algebraic cycle on the cotangent bundle of the variety, and,
when the variety is projective, it computes the Euler characteristic of thesheaf. In this talk, we give a sufficient condition for two sheaves having thesame characteristic cycle, which is a weaker one than Deligne-Illusie’s fortwo sheaves having the same Euler characteristic. This is the joint workwith T. Saito.
• Kazuhiko Yamaki
Title: Non-density of small points on divisors on abelian varieties and theBogomolov conjecture
Abstract: In the paper of the same title, we proved that the Bogomolov
conjecture over function fields of any characteristic holds by showing thepositivity of the canonical height of an effective divisor on a nowhere degenerate abelian variety with trivial trace. In this talk, we will discuss theproof of this positivity result.