In Clemen's book "A Scrapbook of Complex Curve Theory", he discusses in Chapter 2 how the infinite sum giving the period of the Legendre curve matches (mod p) the sum giving the number of points over that same curve in F_p. He then describes in the section "the unity of mathematics" why this is so, using Lefschetz fixed point theorem.
However, this section is very brief, and I was hoping to understand it better. Are there any other good references that you know of? I am trying to understand this connection between the number of points over a finite field for an arbitrary elliptic curve and Picard-Fuchs.