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

Friday, 22 December, 2017 - 10:30 - 11:30
LSB 222
Seminar Type: 
Speaker Name: 
Prof. Kang ZUO
Johannes Gutenberg-Universität

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 Zp such that t (mod p) is not equal to 0 and 1, and Ct to be the elliptic curve defined by the affine equation y 2 = x (x – 1)(x – t).

For q = pn we speculate the set of points in Ct (Fq) whose order coprimes to p corresponds to the set of PGL2 (Fq)-crystalline local systems on P1 – {0, 1, ∞} over some unramified extension of the p-adic number field Qp 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 Ct (K) corresponds to the set of PGL2-arithmetic local systems on P1 – {0, 1, ∞} over K.

It looks very mysterious. M. Kontsevich has already observed that the K3 surface as the Kummer surface of the elliptic curve Ct also appears in the construction of the Hecke operators which define the l-adic local systems on P1 – {0, 1, ∞} over Fq via the GL2 Langlands correspondence due to V. Drinfeld.