(Japanese page)

Mini-workshop on arithmetic geometry and related topics

Date
April 9 (Mon) - 11 (Wed), 2012
Location
Department of Mathematics, Kyoto University
Falucty of Science Bldg No.3, Lecture Room 110
(Map of the North Campus,  Access Map to Kyoto University)
Organizers
Masataka Chida (Kyoto)
Yoichi Mieda (Kyoto)
Takashi Taniguchi (Kobe)
Tetsushi Ito (Kyoto) (e-mail: my first name at math.kyoto-u.ac.jp)

This workshop is supported by Japan Society for the Promotion of Science Grant-in-Aid for Young Scientists (S) Comprehensive studies on Shimura varieties, arithmetic geometry, Galois representations, and automorphic representations'' (20674001), and Japan Society for the Promotion of Science Grant-in-Aid for Young Scientists (B) Studies on p-adic L-functions and p-adic periods associated to automorphic forms'' (23740015)

### Speakers

Yuichiro Hoshi (Kyoto/RIMS)
Bob Hough (Stanford)
Ming-Lun Hsieh (National Taiwan University)
Yasuhiro Ishitsuka (Kyoto/Dept of Math)
Alan Lauder (Oxford)
Dong Uk Lee (POSTECH)
Yoichi Mieda (Kyoto)
Chung Pang Mok (McMaster)
Jeehoon Park (POSTECH)
Sam Ruth (Princeton)
Kenji Sakugawa (Osaka)
Arul Shankar (Princeton)
Megumi Takata (Kyushu)
Takashi Taniguchi (Kobe)
Yasuhiro Wakabayashi (Kyoto/RIMS)
Kevin Wilson (Princeton)

### Program

10:00-11:00 11:15-12:15 13:20-14:20 14:30-15:30 15:50-16:50
49() Taniguchi Shankar Hsieh Hoshi Wakabayashi
10:00-11:00 11:15-12:15 13:20-14:20 14:30-15:30 15:50-16:50 17:00-18:00
410() Mieda Park Ruth Lauder Ishitsuka Takata
10:00-11:00 11:15-12:15 13:20-14:20 14:30-15:30 15:50-16:50
411() Lee Wilson Hough Sakugawa Mok

 April 9 (Mon) 10:00-11:00 Takashi Taniguchi (Kobe) Counting cubic fields and L-functions for the space of binary cubic forms Abstract: We prove the two main terms of the function counting cubic fields, by using the zeta functions for the space of binary cubic forms. (There is an independent and different proof of the two main terms by Bhargava, Shankar and Tsimerman.) We also prove related generalizations, including to arithmetic progressions, where we discover a curious bias in the secondary term. Partial zeta functions, L-functions and congruence subgroups naturally arise in the analysis. We describe their basic analytic properties. This is a joint work with Frank Thorne. 11:15-12:15 Arul Shankar (Princeton) On the finiteness of the average rank of elliptic curves Abstract: In joint work with Manjul Bhargava, we estimate the number of GL(2,Z)-orbits on integral binary quartic forms having bounded invariants. We use this result to compute the average number of elements in the 2-Selmer group of elliptic curves, showing that the average rank of elliptic curves is finite. In this talk, I shall discuss these results and the methods used to obtain them. 13:20-14:20 Ming-Lun Hsieh (National Taiwan University) Non-vanishing modulo p of anticyclotomic central L-values Abstract: This is a joint work with Masataka Chida. We present a result on the non-vanishing modulo p of certain Rankin-Selberg central L-values with anticyclotomic twists, extending a result of Vatsal to higher weight case. We will also discuss a subtle problem naturally arising from this work on the comparsion between the inner product of modular forms on definite quaternion algebras and the congruence number. 14:30-15:30 Yuichiro Hoshi (Kyoto/RIMS) On the Grothendieck conjecture for hyperbolic polycurves Abstract: The Grothendieck conjecture, in our title, is, roughly speaking, a conjecture to the effect that the arithmetic fundamental group of an anabelian variety completely determines the geometric structure of the variety. After works by Nakamura and Tamagawa, in the mid-1990's, Mochizuki gave a proof of the Grothendieck conjecture for hyperbolic polycurves, i.e., successive families of hyperbolic curves, of dimension less than or equal to two. In this talk, we will discuss some known results on the Grothendieck conjecture for hyperbolic polycurves of higher dimension. 15:50-16:50 Yasuhiro Wakabayashi (Kyoto/RIMS) An explicit formula for the number of dormant indigenous bundles Abstract: The notion of a dormant indigenous bundle is, firstly, appeared in p-adic Teichmüller theory developed by S. Mochizuki. A dormant indigenous bundle is a certain conditional crystal on a hyperbolic curve in positive characteristic, leading more enriched geometry (like as the Poincare metrics on hyperbolic Riemann surfaces). In this talk, we shall discuss this topic and a result on the degree of the moduli classifying dormant indigenous bundles, conjectured by K. Joshi. April 10 (Tue) 10:00-11:00 Yoichi Mieda (Kyoto) Geometric approach to the local Jacquet-Langlands correspondence Abstract: Let F be a p-adic field. The local Jacquet-Langlands correspondence is a natural bijection between irreducible discrete series representations of GLn(F) and irreducible smooth representations of D× where D is a central division algebra over F. In this talk, under the assumption "inv D=1/n", I will explain a geometric approach to construct the bijection. If n is prime, my method provides a purely local proof of the local Jacquet-Langlands correspondence. 11:15-12:15 Jeehoon Park (POSTECH) On the p-adic Weil representation of GL2(Zp) Abstract: The goal of my talk is to find a p-adic analogue (admissible p-adic Banach representation) of GL2(Zp) of the classical smooth finite-dimensional Weil representation of GL2(Zp) associated to a character of a unramified qudratic extension \mathcal{O} of Zp. 13:20-14:20 Sam Ruth (Princeton) Ergodic theory methods and arithmetic statistics Abstract: I will discuss some results of Eskin-McMullen and Eskin-Mozes-Shah about counting integer points on varieties of the type G/H where G is semisimple and H has certain properties. I will also discuss some of their applications to arithmetic problems. 14:30-15:30 Alan Lauder (Oxford) Explicit constructions of rational points on elliptic curves Abstract: I will describe an efficient algorithm for computing certain special values of p-adic L-functions, and present an application to the explicit construction of rational points on elliptic curves. 15:50-16:50 Yasuhiro Ishitsuka (Kyoto/Dept of Math) Complete intersections of two quadrics and Galois cohomology Abstract: Flynn and Skorobogatov studied about a relation between Galois cohomology of hyperelliptic curves of genus 2 and del Pezzo surfaces of degree four defined over Q. In this talk, I speak about a generalization of it for hyperelliptic curves of arbitrary genus. 17:00-18:00 Megumi Takata (Kyushu) Deligne's conjecture on the Lefschetz trace formula for pn-torsion étale cohomology Abstract: Deligne conjectured that, for any correspondence defined over a finite field with characteristic p, the Lefschetz trace formula holds if we twist the correspondence by a sufficiently large power of the Frobenius endomorphism. This conjecture was proved by Kazuhiro Fujiwara under the most general situation. This formula is very important tool in arithmetic geometry. For example, it is applied to various Langlands correspondences. Deligne's conjecture is a statement with respect to l-adic étale cohomology, where l is a prime number which is not equal to p. I will talk about an analogous statement of Deligne's conjecture for pn-torsion étale cohomology. 19:00- (Banquet) April 11 (Wed) 10:00-11:00 Dong Uk Lee (POSTECH) Some new cases of the Mumford-Tate conjecture Abstract: We prove the Mumford-Tate conjecture for abelian varieties when the endomorphism algebra is a totally real field and it has semi-stable reduction of minimal toric rank. This generalizes a recent result of C. Hall. 11:15-12:15 Kevin Wilson (Princeton) The commutative algebra of parameterizations of rings Abstract: After Bhargava painstakingly worked out his parameterization of quartic rings over Z, Deligne suggested that much of his work could be generalized to more general bases using the theory of the Koszul complex. Wood's thesis takes this idea to its limit. One crucial element of Deligne's and Wood's proofs is that the Koszul complex possesses a canonical multiplicative structure. In this talk, we will generalize this multiplicative structure to the free resolution of the "universal rank n ring" through the representation theory of the symmetric group. 13:20-14:20 Bob Hough (Stanford) Torsion in the class group of imaginary quadratic fields Abstract: Recently Tanaguchi and Thorne, and Bhargava, Shankar and Tsimerman have given separate proofs of an asymptotic with a secondary main term for the average number of 3-torsion elements in the class group of imaginary quadratic fields. Their proofs go through the counting function of cubic fields. I will describe a third proof, which considers directly the distribution of 3-torsion Heegner points for quadratic fields, but yields a slightly weaker result. Time permitting, I will also discuss partial results on other torsion in the class group. 14:30-15:30 Kenji Sakugawa (Osaka) Control theorem for non-abelian Galois cohomologies Abstract: In 1970, B.Mazur proved a control theorem for Selmer groups of elliptic curves having the ordinary reduction at p, which describes the behavior of Selmer groups over the cyclotomic Zp extension of finite number fields. Note that Selmer groups of elliptic curves are defined to be a subgroup of the first Galois cohomologies for the Galois modules obtained by torsion points of elliptic curves. In this talk, the speaker will give an analogy of Mazur's theorem for non-abelian analogues of the Selmer groups which are defined to be subsets of the first Galois cohomolodies for non-abelian Galois groups obtained by the unipotent fundamental groups of curves over finite number fields. Since the analogues of Selmer groups are not groups but only objects like pointed sets, we should define some finiteness conditions for these objects. After that, the speaker will give a sketch of the proof of the control theorem for non-abelian analogues. We reduce this theorem to the control theorem for Selmer groups attached to abelian Galois representations which was proved by T.Ochiai (J. Number Theory 82 (2000), no.1, 69-90.). 15:50-16:50 Chung Pang Mok (McMaster) Endoscopic classification for representations on unitary groups Abstract: Arthur has established the endoscopic classification for representations on orthogonal and symplectic groups (modulo the stabilization of the twisted trace formula). In this talk we describe the on-going work of extending Arthur's classification to unitary groups. Time allowed we would also like to mention some applications.