Generic A-family of exponential sums

作者：Zhu Hui June^*

来源：Journal of Number Theory, 2014, 143: 82-101.

DOI：10.1016/j.jnt.2014.03.013

• Exponential sums
• Artin-Schreier curves
• L functions
• Dwork theory
• Frobenius problem

Let (s) over right arrow: = (s(1), s(2), . . . s(m)) with s(1) < . . . < s(m) being positive integers. Let A((s) over right arrow) be the space of all 1-variable polynomials f (x) = Sigma(m)(l=1) a(l)x(sl) parameterized by coefficients (a) over right arrow = (a(1), . . . , a(m)) with a(m) not equal 0. We study the p-adic valuation_ of the roots of the L-function of exponential sum of (f) over bar for modulo p reduction of any generic point f is an element of A((s) over right arrow)((Q) over bar). Let NP((f) over bar) be the normalized p-adic Newton polygon of the L function of exponential sums of (f) over bar. Let GNP(A((s) over bar), (F) over bar (p)) be the generic Newton polygon for A((s) over right arrow) over (F) over bar (p), and let HP(A((s) over right arrow)) := N P-p(Pi(d-1)(i=1)(1 - p(1/d)T)) be the absolute lower bound of NP(A(g)). One knows that NP((f) over bar) < GNP(A(<(s)over right arrow>); (F) over bar (p)) < HP(<(f)over bar>) for all prime p and for all (f) over bar is an element of A((s) over right arrow)((Q) over tilde), and these equalities hold only when p equivalent to 1 mod d. In the case (s) over right arrow = (s, d) with s < d coprime we provide a computational method to determine GNP(A(s, d), <(F)over bar>(p)) explicitly by constructing its generating polynomial H-r is an element of Q[X-r,X-1 X-r,X-2, . . . X-r,(d-1)] for each residue class p equivalent to r mod d. For p equivalent to r mod d (with 2 <= r <= d - 1 coprime to d) large enough H-r = Sigma(d-1)(n=1)h(r,nk,r,n) X-r,n(kr,n) with Pi(d-1)(n=1)h(r,nk,r,n) not equal 0 if and only if GNP(A(s, d), (F) over bar (p)) has its breaking points after the origin at <br xmlns:set="http://exslt.org/sets">((n, n(n + 1)/2d + (1 - s/d)k(r,n)/p-1)) n=1,2, . . , d-1
If a not equal 0 then for any f = x(d) + ax(s) is an element of A(s,d)((Q) over bar) and for any prime p equivalent to r mod d large enough we have that NP((f) over bar) = GNP(A(s, d), (F) over bar (p)) and
lim(p ->infinity) NP((f) over bar) = HP (A(s,d)).
Our method applies to compute the generic Newton polygon of Artin-Schreier family y(P) - y = x(d) + ax(s) parameterized by a for p large enough.

• 出版日期2014-10