Class Note for SOC 189 at UA
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Date Created: 02/06/15
USING CLASS FIELD THEORY TO CONSTRUCT CURVES WITH MANY POINTS KRISTIN LAUTER AND MICHAEL ZIEVE The Weil bound says that the number of Fq rational points on a genus g curve is at most q 1 2g Various improvements to this bound are due to Stark Manin lhara Serre Drinfeld Vladut Oesterle Stohr Voloch Lauter and others In order to test whether these im proved bounds are best possible it is necessary to construct curves with many points We need a supply of curves for which we can quickly compute the basic invariants genus number of rational points In this project the curves we use are abelian covers of the projective line Class eld theory provides a description of all such covers so the steps of this project are 1 Compute the invariants of curves beginning from the class eld theoretic description 2 Make choices of the class eld theoretic data which will yield curves with many points 3 Compute explicit equations for the corresponding curves by means of Carlitz modules In the second step by curves with many points7 we mean curves whose number of Fq rational points comes close to the best known upper bound for that choice of g and q Of course the closer the better We will split into two groups one group will focus on the rst two steps the other will focus on the third These steps are described in more detail in the next three sections The fourth section discusses further directions we can pursue if time permits 1 CLASS FIELD THEORY For a detailed introduction to class eld theory constructions of curves with many rational points see 8 and In this section we recall the relevant facts from class eld theory For convenience we use the language of function elds for which a basic Date February 17 2000 2 KRISTIN LAUTER AND MICHAEL ZIEVE reference is 10 Let K Fqt and R Fqt We often identify irreducible polynomials in R with the corresponding places of K Theorem For any nonconstant polynomial M E B there epists an abelian eptension LMK with these properties lt1 Ga1ltLMKgt RMWFJ 2 The in nite place ofK splits completely in LM in particular Fq is the full constant eld of LM 3 If P E R is irreducible and coprime to M then the place P is unrami ed in LMK and its decomposition group is generated by the image ofP in RMqu 4 Let P E R be an irreducible factor ofM with multiplicity r and let G P be the n th rami cation group ofP in LMK in the upper numbering cf Ifn gt r 71 then G P I ifO S n S r71 then G P is the subgroup of l MMfF generated by polynomials congruent to 1 mod MPT W In fact the above conditions uniquely determine LM1 Moreover if LK is any nite abelian extension in which the in nite place splits completely there is an M for which LM contains L But we will not need these last two facts in what follows If L is a function eld over Fq let NL denote the number of degree one places on L equivalently the number of Fq rational points on the curve corresponding to L For any nite abelian extension LK we have NL 2 L KnS 71 where nS resp 71 denotes the number of degree one places of K which split completely resp are totally rami ed in LK As a warm up exercise write down an exact formula for n Recall the de nition of decomposition eld if LK is a nite abelian extension and P is a place of K then the maximal subextension of LK in which P splits completely is the sub eld of L xed by the decomposition group of P Our strategy for producing curves with many points is as follows choose a set S of degree one places of K which contains the in nite place and choose a nonconstant polynomial M E B Let L be the sub eld of LM xed by the group generated by the decomposition groups at all the places in S Then NL 2 L K S The genus of L can be computed via the Riemann Hurwitz formula and Hilbert7s theory of rami cation groups cf 9 pp 61776 and 6 Prop 1 Problem for various choices of q S and M compute the genus and number of degree one places on the corresponding eld L 1Terminology LM is the maximal abelian extension of K in which the conductor divides M and the in nite place splits completely CLASS FIELD THEORY AND CURVES WITH MANY POINTS 3 2 CHOOSING THE PARAMETERS In the previous section we constructed certain extensions LIFqz depending on three parameters q7 S and M Now try to nd choices of these parameters for which NL comes close to the known upper bounds for that choice of q and 9 For small 9 and q7 tables of best known upper and lower bounds can be found in Tables ofthe curves obtained by letting S contain all places of degree one except one can be found in 3 CARLITZ MODULES The Carlitz module enables one to write down explicit equations for the elds LM in the above Theorem The basic idea is as follows starting from Mt E R Itht there is a recipe for writing down an associated polynomial IMu 6 RM together with a natural action of RM on the roots of IIM It turns out that this action induces an isomorphism GalfAK RM7 where m is the splitting eld of IM over K Then LM is the sub eld of m xed by Ff The polynomials IM and the action of RM are de ned in 4 p 79 For this part of the project7 begin by reading the rst four sections of In those seven pages7 Hayes gives a self contained proof of most of our Theorem First problem complete the proof of the Theorem Now compute some examples of elds Then compute examples of LM7 and then examples of quotients of LM by decomposition groups over rational places of K Finally7 compute equations for the elds described by the other group of students these will be quotients of certain elds LM 4 FURTHER DIRECTIONS You may have noticed that7 when applying the Theorem for xed q7 it is dif cult to produce curves which have large genus and which have many IFq rational points relative to their genus Try to nd in nite families of curves over IR for xed q7 which have as many points as possible relative to their genus It turns out that7 if we x q and only consider genus g curves which are abelian covers of the projective line7 then as g grows the number of IFq rational points on such a curve cannot grow linearly in 9 whereas all the upper bounds of Weil et al are linear in g 7 in fact7 this number of points is at most cqg logg where 01 is a constant depending only on q This result is due to Erey7 li erret7 and Stichtenoth read their elegant proof in 2 hint rst trace through their proof in case all rami cation is tame7 to see the key 4 KRISTIN LAUTER AND MICHAEL ZIEVE ideas For xed q can you nd abelian covers of the projective line which have any prescribed genus g gt 1 and which have at least glogg rational points over Fq Such covers do exist for every q and g One can also study abelian covers of curves other than the projective line For an explicit treatment of class eld theory in this case see 5 for a non explicit treatment see If there is time and interest carry out any of the above steps in this more general setting REFERENCES l R Auer Ray class elds of global function elds with many rational places mathiAG9803065 2 GI Frey Mi Perret and Hi Stichtenoth On the di erent 0f abelian exten sions of global elds pp 26732 in Coding Theory and Algebraic Geometry Hi Stichtenoth and Mi Al Tsfasman edsi SpringerVerlag New York 1992 3 GI van der Geer Mi van der Vlugt Tables of curves With many points to appear in Mathematics of Computation available at httpwwwwinsuvan1 geerpub1icationshtml 4 Di Hayes Explicit class eld theory for rational function elds Trans Amer Math Soc 189 1974 74791i 5 Di Hayes A brief introduction to Drinfeld modules pp 1732 in The Arithmetic of Function Fields Di Goss et al eds WI de Gruyter Berlin 1992 6 Ki Lauter Deligne Lusztig curves as ray class elds available at http wwwmpimbonnmpgdecgibinpreprintpreprintjearchpl 7 Ki Lauter A Formula for Constructing Curves over Finite Fields with Many Rational Points Journal of Number Theory 74 5672 1999 available at http wwwmpimbonnmpgdecgibinpreprintpreprintjearchpl 8 R Schoof Algebraic curves and coding theory UTM 336 Univ of Trento 1990i 9 JiPi Serre Local Fields SpringerVerlag New York 1979 10 Hi Stichtenoth Algebraic Function Fields and Codes SpringerVerlag Berlin 1993 E mail address k1auter microsoft com E mail address zieve mathusc edu
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