p-adic variation of L functions of one variable exponential sums, I

For a polynomial f(x) in ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]) of degree d ≥ 3 let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] be the L function of the exponential sum of f mod p. Let NP [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] denote the Newton polygon of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]. Let HP ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]) denote the Hodge polygon of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /], which is the lower convex hull in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /] of the points (n, n(n+1)/2d) for 0 ≤ n ≤ d - 1. Let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="08i" /] be the space of degree-d monic polynomials parameterized by their coefficients. Let GNP [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="09i" /] := [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10i" /] NP [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11i" /] be the lowest Newton polygon over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12i" /] if exists. We prove that for p large enough GNP [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13i" /] exists and we give an explicit formula for it. We also prove that there is a Zariski dense open subset [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14i" /] defined over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15i" /] in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16i" /] such that for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17i" /] and for p large enough we have NP [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18i" /] = GNP [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19i" /]; furthermore, as p goes to infinity their limit exists and is equal to HP [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20i" /]. Finally we prove analogous results for the space of polynomials f(x) = x^d + ax with one parameter. In particular, for any nonzero a ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21i" /] we show that lim_p→∞ NP ((x^d + ax) ⊗ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22i" /]) = HP [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23i" /].