Weil: Bourbaki exp 16
Ask Irene for other references if needed
Conference volume: Theta functions, Bowdoin 1987, Proc Symp Pure Math 49, maybe this is part I only.
A key author might be Barsotti. E.g. in Asterisque 63: Theta functions in positive char.
Review of Faltings: The determinant of cohomology in the etale topology Theta functions can be viewed as the canonical section of the determinant of cohomology. The corresponding notion in the etale topology would be L-series. “All known constructions of p-adic L-functions seem to be related to some situation where some etale cohomology has vanishing Euler char.
Theta functions: Last chapter of Shimura: Abelian Varieties with CM
nLab page on Theta functions