In this paper, we will give a new proof of the Deuring-Shafarevich formula, which asserts a relation between the $p$-ranks of Jacobi varieties. We analyze the zeta functions of global function fields to prove the formula, without using tools of the algebraic geometry.
"On the Deuring-Shafarevich Formula." Tokyo J. Math. 34 (2) 313 - 318, December 2011. https://doi.org/10.3836/tjm/1327931387