Constant Terms of Coleman Power Series And Euler Systems in Function Field

TL;DR

This paper computes the constant term of Coleman power series and employs Euler systems derived from Drinfeld modules to prove an analogue of the Iwasawa Main Conjecture in characteristic p>0 function fields.

Contribution

It introduces a novel approach to prove the Iwasawa Main Conjecture in function fields using Euler systems from theta functions of Drinfeld modules.

Findings

Established the constant term of Coleman power series in this setting.

Proved an analogue of the Iwasawa Main Conjecture for function fields.

Constructed Euler systems from Drinfeld modules' theta functions.

Abstract

We calculate the constant term of Coleman power series and use it to prove an analogue of Iwasawa Main Conjecture in function fields of characteristic p>0 using Euler systems. This result is proved by a similar method of classical proof of Iwasawa Main Conjecture, go back to Kolyvagin and Rubin. We construct Euler systems from theta function of Drinfeld modules and establish finite-singular comparison equation even when p is not invertible the coefficients.