Coding Theory & Newton Polygons in Algebraic Geometry: Study of Zeta & L Functions, Papers of Art

Download Coding Theory & Newton Polygons in Algebraic Geometry: Study of Zeta & L Functions and more Papers Art in PDF only on Docsity! DISSERTATION SUMMARY PHONG LE The content of my dissertation is divided into two parts: results in algebraic coding theory and results in Newton polygons of zeta and L functions. The overlap occurs primarily in the use of results in the theory of exponential sums. The results in algebraic coding theory pertain to the utility of an algebraic-geometric trace code as an error correcting code. I also examine a class of functions in order to deter- mine when the corresponding Newton polygons coincides with a combinatorially determined lower bound. 1. Coding Theory My work in coding theory is focused on codes induced by objects found in alge- braic geometry. The study of trace codes in particular benefit greatly from expo- nential sum methods. 1.1. Background. Let q = pr and Fp, Fq and Fqm be fields with p, q and qm elements, respectively. Let X be a non-singular projective curve of genus g defined over Fqm . The function field over Fqm is denoted Fqm(X). We may split a divisor G = ∑ nQQ, defined over Fqm , into two divisors G+ and G− where G+ = ∑ nQ>0 Q and G− = ∑ nQ<0 Q. Hence G = G+ +G−. Define L(G) to be the vector space of functions: L(G) = {f ∈ Fqm(X) | (f) +G ≥ 0} ∪ {0}. Let D be a set of Fqm-rational points away from poles of functions in L(G): D = {P1, . . . , Pn} ⊆ X(Fqm) \ Supp(G+). Now we are in place to define our algebraic geometry code: C := C(D,G) = {(f(P1), . . . , f(Pn)), f ∈ L(G)}. When 2g − 2 < deg(G) < n, by a theorem of Riemann and Roch we have k := dimFqm (L(G)) = deg(G) + 1− g. The dimension of C as an Fqm vector space is also k. This condition allows us to identify a function with its image. Define the trace code of C to be: TrC := Trqm/qC = {(Trqm/q(f(P1)), . . . ,Trqm/q(f(Pn))), f ∈ L(G)}. The trace code TrC defined over a general finite field Fq has been studied exten- sively. However, many of the deeper results have been restricted to the case when q = p is a prime. In [7], Stichtenoth outlines developments in this direction. The general q = pr case is often more complicated. For example, Van Der Vlugt deter- mined an upper bound on the Fp dimension of TrC [8]. He also states conditions where the dimension could be determined exactly. In my dissertation, I generalize his results for arbitrary q = pr. 1 2 PHONG LE 1.2. First Bounds. To determine the dimension of TrC, I first observe that the trace map is Fq-linear. Therefore, the trace map induces an exact sequence: 0→ K → C → TrC → 0, where the kernel K is some subcode of C. Hence mdimFqm (C)− dimFqK = dimFq (Tr(C)). To determine K, I examine a subspace: E = {f ∈ L(G) | f = hq − h, h ∈ Fqm(X)}. Two questions arise: (1) What is the dimension of E? (2) When is K = E? Separate conditions to find the dimension of E and when K = E have been deter- mined. 1.3. The dimension of E. Let bG/qc = ∑ nQ>0 bnQ/qcQ+G−, where b?c is the greatest integer function. Hence for any h ∈ L(bG/qc) we have hq−h ∈ E ⊆ L(G). If #Supp(G−) ≤ 1, there is an exact sequence 0 // Fq ∩ L(bG/qc) // L(bG/qc) φ // E // 0 where φ(h) = hq − h. Let δ = dimFq (Fq ∩ L(bG/qc)). Using this, we can compute the dimension of E and an upper bound for the dimension of TrC. Proposition 1.3.1. If #Supp(G−) ≤ 1 then dimFq (TrC) ≤ m(dimFqm (L(G)) − dimFqm (bG/qc)) + δ. Proposition 1.3.2. Assume #Supp(G−) ≤ 1 and deg(bG/qc) > 2g − 2. Then dimFq (TrC) ≤ m(deg(G)− deg(bG/qc)) + δ. 1.4. A Condition when K = E. When K = E the exact dimension of TrC can be determined. In order to determine this, several tools are required. Our first is Bombieri’s Estimate [3]. Theorem 1.4.1 (Bombieri’s Estimate). Let X be a complete nonsingular curve of genus g, defined over Fqm . Let f ∈ Fqm(X), f 6= hp − h for h ∈ Fq(X), with pole divisor (f)∞ on X. Then∣∣∣∣∣∣ ∑ P∈X(Fqm )\(f)∞ ζ Trqm/p(f(P )) p ∣∣∣∣∣∣ ≤ (2g − 2 + t+ deg(f)∞)qm/2. where t is the number of distinct poles of f on X and ζp is a fixed primitive p-th root of unity in the complex numbers. Assume that f ∈ K \E 6= ∅. Hence f is not of the form hq − h for h ∈ Fqm(X). This allows us to use Bombieri’s Estimate. In my dissertation, I demonstrate that f can be chosen so that f is not of the form gp − g for g ∈ Fq(X). This is a major distinction from the work of Van Der Vlugt. When q = p this is fairly straight- forward. However, to extend the result from Trqm/p to Trqm/q a new approach is necessary. If such an f is chosen, we see that f maximizes the left side of Bombieri’s Estimate. By assuming there are sufficiently many points on X(Fqm) we are led to contradict K 6= E. Hence we achieve our main results: DISSERTATION SUMMARY 5 2.3. Decomposition Theorems. Theorem 2.2.2 was proved using a series of tools all aimed at decomposing 4 into smaller polytopes. The same tools are used for specific families of polytopes and specific polynomials. Assume f is non-degenerate. Let δ1, . . . , δh be all the co-dimension 1 faces of 4 which do not contain the origin. Let fδi be the restriction of f to the face δi. Then 4(fδi) = 4i is n-dimensional. Furthermore f is non-degenerate if and only if each fδi is non-degenerate. Addi- tionally we may decompose the property of ordinarity in the following way: Theorem 2.3.1 (Wan). Let f be non-degenerate and let 4(f) be n-dimensional. This f is ordinary if and only if each fδi is ordinary. Equivalently f is non-ordinary if and only if some fδi is non-ordinary. Several other related theorems were developed from this main decomposition theorem. 2.4. Deligne Polynomials. Let f be a polynomial over Fq with degree d prime to p and its highest degree term, say fd, is a homogeneous form of degree d in n variables which is nonzero, and whose vanishing, if n ≥ 2 defines a smooth hyper- surface in the projective space Pn−1. These are called Deligne polynomials. Katz [5], following results from Browning and Heath-Brown [4], examined polynomials of the form tf(x) + g(x) + 1/t where g is an arbitrary polynomial over Fq in n variables of degree e < d/2. If we fix t ∈ F∗q this is still a Deligne polynomial. Interpreting t as a variable, this is a polynomial in n+ 1 variables. Katz gave sharp complex estimate of the underlying exponential sum using l-adic cohomology. In my dissertation, I use Dwork’s p-adic theory and Wan’s decomposition techniques to study the p-adic Newton polygon of the associated L function. With repeated uses of hyperplane decompositions [9, §7], I obtained the following result: Theorem 2.4.1 (L-). For a fixed g(x) ∈ Fq[x1, . . . , xn] with deg(g) < d/2 the family of Laurent polynomials tf(x) + g(x) + 1/t parameterized by f(x) of degree d is generically ordinary if p ≡ 1 (mod d). In other words, Conjecture 2.2.1 holds for this family. 2.5. Future Directions. My long term goal with this project is to understand completely conditions when Conjecture 2.2.1 holds. I would also like to find better methods of computing D∗(4) in general. The geometric nature of decomposing polytopes makes it difficult to understand Wan’s decomposition theorems beyond dimension 4. Most results using these tools have been applied to single variable polynomials such as in the work of Scholten and Zhu [6] and Yang [11]. However with controlled parameters, such as the degree in the full Deligne polynomials, we can determine high dimensional polytope decompositions in terms of lower dimen- sional behavior. Another direction is the study of relations between the zeta function of a hy- persurface and the zeta function of its mirror. In [10], Wan suggests that these methods can be used to study mirrored zeta functions by decomposing their associ- ated mirror polytopes. He outlines several conjectures in this direction. Such pairs provide many avenues of exploration with these tools. 6 PHONG LE References [1] A. Adolphson and S. Sperber. Newton Polyhedra and the degree of the L-function associated to an exponential sum. Invent. Math., pages 555–569, 1987. [2] A. Adolphson and S. Sperber. Exponential sums and Newton polyhedra: Cohomology and estimates. Ann. Math., pages 367–406, 1989. [3] Enrico Bombieri. Exponential sums in finite fields. Amer. J. Math, 88:71–105, 1966. [4] D Browning, T; Heath-Brown. Integral points on cubic hypersurfaces. preprint, October 2006. [5] Nicholas M. Katz. On a question of Browning and Heath-Brown. preprint, February 2007. [6] Jasper Scholten and Zhu Hui June. The First Slope Case of Wan’s Conjecture. Finite Fields and their Appl., 8:414–419, 2002. [7] Henning Stichtenoth. Algebraic function fields and codes. Springer-Verlag, 1993. [8] Marcel Van Der Vlugt. A New Upper Bound for the Dimension of Trace Codes. Bull. London Math. Soc., 23:395–400, 1991. [9] Daqing Wan. Newton polygons of zeta functions and L functions. Ann. of Math, 2 No. 2:249– 293, 1993. [10] Daqing Wan. Mirror Symmetry for Zeta Functions. Mirror Symmetry V, AMS/IP Studies in Advanced Mathematics, 38:159–184, 2006. [11] Roger Yang. Newton polygons of L-functions of polynomials of the form Xd + λX. Finite fields and Appl., 9:59–88, 2003.