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.

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