 
Summary: A twovariable refinement of the Stark conjecture in
the function field case
Greg W. Anderson
Abstract
We propose a conjecture refining the Stark conjecture St(K/k,S) in the function field
case. Of course St(K/k,S) in this case is a theorem due to Deligne and independently
to Hayes. The novel feature of our conjecture is that it involves two rather than one
variable algebraic functions over finite fields. Because the conjecture is formulated in the
language and framework of Tate's thesis we have powerful standard techniques at our
disposal to study it. We build a case for our conjecture by (i) proving a parallel result in
the framework of adelic harmonic analysis which we dub the adelic Stirling formula, (ii)
proving the conjecture in the genus zero case and (iii) explaining in detail how to deduce
St(K/k,S) from our conjecture. In the genus zero case the class of twovariable algebraic
functions thus arising includes all the solitons over a genus zero global field previously
studied and applied by the author, collaborators and others. Ultimately the inspiration
for this paper comes from striking examples given some years ago by R. Coleman and D.
Thakur.
Contents
1 Introduction 1
2 General notation and terminology 5
