Speaker: Kang Zuo, University of Mainz, Germany

            University of Science and Technology of China

Title: PGL_2-crystalline local systems on the projective line minus 4 points and torsion points on the associated elliptic curve.

Time: 2017年11月17日    上午10:00--11:30

Room: 东区管理科研楼   数学科学学院1518教室

Abstract: In my talk I shall report my recent joint work with R.R. Sun and J.B. Yang. Given an odd prime p we take t to be a number in an unramified extension of the p-adic number ring Z_p such that t (mod p) is not equal to 0 and 1, and C_t  to be the elliptic curve defined by the affine equation y^2=x(x-1)(x-t). For q=p^n we speculate the set of points in C_t(F_q) whose order coprimes to p corresponds to the set of PGL_2(/bar F_q)-crystalline local systems on P^1- { 0, 1, infinity,  t} over some unramified  extension of the p-adic  number field Q_p via periodic Higgs bundles and the p-adic  Simpson correspondence recently established by Lan-Sheng-Zuo for GL-case and Sun-Yang-Zuo for PGL-case. In the arithmetic setting, given an algebraic  number field  K  we introduce  the notion of arithmetic local systems and arithmetic periodic Higgs bundles and speculate the set of torsion points in C_t(K) corresponds to the set of PGL_2-arithmetic local systems on  P^1- { 0, 1, infinity,  t} over K.It looks very mysterious. M. Kontsevich has  already observed that the  K3 surface as  the Kummer surface of the elliptic curve C_t also appears in the construction of the Hecke operators which define the l-adic local systems on P^1- { 0, 1, infinity,  t} over F_q via the GL_2 Langlands correspondence due to V. Drinfeld.