\(T\)-adic exponential sums over affinoids (Q2079491)

Fix a prime \(p\). The work [\textit{C. Liu} and \textit{D. Wan}, Algebra Number Theory 3, No. 5, 489--509 (2009; Zbl 1270.11123)] defined \(T\)-adic exponential sums that interpolate classical \(p\)-adic exponential sums associated to a Laurent polynomial \(f\) (i.e.,~loosely speaking, a regular function on an algebraic torus). Roughly speaking, the author partially generalizes [Liu and Wan, loc. cit.] from the torus to a finitely-punctured line, allowing \(f\) to have an arbitrarily large finite number of poles (not just two as in the torus case), but must work \((\pi^{1/c}, p)\)-adically instead of purely \(T\)-adically (or equivalently, \(\pi\)-adically) for reals \(c>0\) in a limited range. (Here \(T\) and \(\pi\) are related by the change of variables \(T = E(\pi)-1\), where \(E\) denotes the Artin-Hasse exponential function.) There are two main results. The first, Theorem~1.1, is an entireness (analytic continuation) result, proven using a generalization of the Dwork-Monsky-Reich trace formula (see \S7 and [\textit{D. Reich}, Am. J. Math. 91, 835--850 (1969; Zbl 0213.47502)]). The second, Theorem~1.2, is a ``Newton over Hodge'' bound in the spirit of [\textit{H. J. Zhu}, Int. Math. Res. Not. 2004, No. 30, 1529--1550 (2004; Zbl 1089.11044)], and may well have applications in the vein of the latter. For the precise results and technical details, we refer the reader to the paper.