We thank D. Phillips for technical assistance, J. Wagner for the 123.7 cells, J. Boulter for providing nAChR cDNA clones, S. Froehner for use of his peristaltic pump, J. Dunlap for use of his densitometry equipment, and J. Margiotta and J. Wagner for comments on the manuscript, and P.D.G. thanks B.P. for inspiration.

Results on L-functions of Artin-Schreier coverings by Dwork, Bombieri and Adolphson-Sperber are generalized to L-functions of Witt coverings.

We introduce T -adic exponential sums associated to a Laurent polynomial f . They interpolate all classical p m -power order exponential sums associated to f . We establish the Hodge bound for the Newton polygon of L-functions of T -adic exponential sums. This bound enables us to determine, for all m, the Newton polygons of L-functions of p m -power order exponential sums associated to an f that is ordinary for m = 1. We also study deeper properties of L-functions of T -adic exponential sums. Along the way, we discuss new open problems about the T -adic exponential sum itself.

The twisted $T$-adic exponential sum associated to a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the twisted T-adic exponential sum. This bound gives lower bounds for the Newton polygon of the $L$-function of twisted $p$-power order exponential sums

Let ÀDo À 4 denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of Qð ffiffiffiffiffiffiffi ffi ÀD p Þ exists. Let d be a fundamental discriminant prime to D: Let 2k À 1 be an odd natural number prime to the class number of Qð ffiffiffiffiffiffiffi ffi ÀD p Þ: Let w be the twist of the ð2k À 1Þth power of a canonical Hecke character of Qð ffiffiffiffiffiffiffi ffi ÀD p Þ by the Kronecker's symbol n/ð d n Þ: It is proved that the vanishing order of the Hecke L-function Lðs; wÞ at its central point s ¼ k is determined by its root number when jdj5D 1 12 ÀE ; where the constant implied in the symbol 5 depends only on k and E; and is effective for L-functions with root number À1: r