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

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*to be the elliptic curve defined by the affine equation

_{t }*y*

^{2}*= x*(

*x – 1*)(

*x – t*)

*.*

For *q *= *p ^{n}* we speculate the set of points in

*C*(

_{t }*F*) whose order coprimes to

_{q}*p*corresponds to the set of

*PGL*

_{2}(

*F*)-crystalline local systems on P

_{q}^{1}– {0, 1, ∞} over some unramified extension of the

*p*-adic number field

*Q*via periodic Higgs bundles and the

_{p}*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, ∞} 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, ∞} over

*F*via the

_{q}*GL*

_{2}Langlands correspondence due to V. Drinfeld.