- 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)

- 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)

10:00-11:00 | 11:15-12:15 | 13:20-14:20 | 14:30-15:30 | 15:50-16:50 | ||
---|---|---|---|---|---|---|

4月9日(月) | 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 | |

4月10日(火) | 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 | ||

4月11日(水) | 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 GL_{n}(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 GL_{2}(Z_{p}) | |

Abstract: The goal of my talk is to find a p-adic analogue (admissible p-adic Banach representation) of GL_{2}(Z_{p}) of the classical smooth finite-dimensional Weil representation of GL_{2}(Z_{p}) associated to a character of a unramified qudratic extension \mathcal{O} of Z_{p}. | |

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 p-torsion étale cohomology^{n} | |

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 p-torsion étale cohomology.^{n} | |

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 Z_{p} 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. |

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Last modified: April 4, 2012