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• W594382950 abstract "Let K=F(x, y) be an algebraic function field over a finite prime field F defined by an equation y2 = x5+ a (a≠0 a∈F) . Then, under the assumption P≡1 mod 5, the L-function of K is computed by relating it to the Hasse-Witt matrix of K. 1.lntroduction.Let F=GF(P)be a finite prime field of characteristic P≠2. Let K=F(x,y)be an algebrajc funCtjon field over F defjned by an equation y2=X5+a(a≠0,a∈F). Wewish to study the numerator エ(〟)=1+α1α+α2リットル2+クα1祝3+ク2乙‘4 0f the zeta-functjon of K. We have already discussed the particular case of p≡2,3,4mod5jn[4], [5].In fact,jf p...2,3mod5,then エ(α)=1+ク2〟4, andif p≡4mod5,then エ(〟)=1+2♪α2+ク2〟4. Thus,in thjs note,We wi11go furtherto discuss the remainlng CaSe,that js,P...l mod5. Let Nlbe the number of prlme divisors of degree one of K. Moreover,Wewi11denote a constant field extension of K of degree two by K2 and also the number of prime divisors of degree one of K2by Nl(2).Applying the general theoryin Hasse[1]to our case,We Canimmediately obtain the followlng formulae IV1=ク+1+cl,IV1(2)=♪2+1+C2, where cl and c2mean the so-Called error terms and they satisfy theinequalities (1) lcll≦4レク,lc21≦4ク. Then,the coefficients al and a2are glVen by (2) dl=Cl,2α2=C12+C2. 16 Tadashi WASHIO On the other hand, Iet A be the Hasse-Witt matrix of K. Then, we have already proved that f Trace A = l-N1' Trace Ap+1 = lN1 (2) where the notation ~; means the residue class modulo p represented by an integer m. ([ 3]). It follows that { ~~= ~= _Trace A (3) Trace A2. Thus, in order to determine error terms, we will use Informatron about the Hasse-Witt matrix, which is carried out in 2 . In 3, we give explicit expressions for the coefficlents of the L-fllnction. 2 . Hasse-Witt matrices. The Hasse Witt matrices of a hyperelliptic function field has been discussed by Miller [ 2 1 . In this note, we limite ourselves only to the case where a hyperelliptic function field K=F(x, y) is defined by y x +a, (a~0 ,a~EF), over a finite prime field F=GF(p) with characteristic p . Throughout this note we will always assume that p~~~ I mod 5 . Let A ( O ~u ,v~ I ) be the coefficient of x'1 in the followlng polynonual , p-* lg~((x5+a) 2 x'1)=~!( ~ 2 ~i 5i'~'1) a x O ,i~ p2 1 i where W means the p~1-1inear operator satisfying O if (p, w)= I W (x) = ~ x p otherwise . Then, the square matrix A= (A~ .) is called the Hasse-Witt matnx of K Because of the fact that the solutions (u, v, i) of the equation p+1 51 +u+ I p (v+ l), ( O~~<ru, v~1, O~i~) 2 are glven by (O, O p-1 ) and (1, l, 2p2), we have p-1 3p-3 p-* p-* Ao'*= a A*,*= a A1'0=0, AQ,*= p-* 2,-..~p-2 This implies that L-Functons of Algebraic Function Fields defined by y2=x'+a over GF(~) 17 ~~~L 3 b -3 p -1 p -1 ro lo Trace A=Ao,0+Ao'o a a + p I ~~:.~_ (4) __ ~ 11 2 p-1 2 3p 3 J:-1 Trace A2 =AO '02 + A1 I a a + _ p~:~1 _ ~~ ~:L THEOREM I . We will conveniently denote the representative in the same letter a . Let sl ' s2 be respectively the integers satisfying Z of a~F by (5) p-1 ~ S1 =: ( p-1 5" @default.
• W594382950 created "2016-06-24" @default.
• W594382950 creator A5008867767 @default.
• W594382950 date "1983-02-28" @default.
• W594382950 modified "2023-10-17" @default.
• W594382950 title "L-Functions of Algebraic Function Fields defined by y2=x5+a over GF(p)" @default.
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