Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

The behaviour of gauss sums in function fields of one variable over finite fields. The paper discusses their prime factorization, which is analogous to the classical case but different for higher genus fields. Gauss sums play a prominent role in the cyclotomic theory and are used to control their absolute values and for the local constants decomposition. The document also covers drinfeld a-modules, abelian extensions of function fields, and the relation between gauss sums and idele groups.

Typology: Papers

Pre 2010

1 / 11

Download Study of Gauss Sums Behaviour in Function Fields and more Papers Introduction to Sociology in PDF only on Docsity! JOURNAL OF NUMBER THEORY 37, 242 252 (1991) Gauss Sums for Function Fields DINESH S. T HAKUR School of Mathematics, T1FR, Bombay 5, India Communicated b)' D. Goss Received January 2, 1990; revised March 11, 1990 In this paper, we study the behaviour of "Gauss sums" (introduced in [Thl , Th2]) for function fields of one variable over finite fields. We will show (Section 1) that their prime factorization is analogous to the classical case for the rational function field (at least when the infinite place chosen is of degree not more than two) and is quite interesting, though different for higher genus fields. Classically, the factorization of Gauss sums for composite modulus into those for its prime power factors, gives control on their absolute values and is quite important in "the local constants decomposition." In our case, the situation is wildly different (Section 2). All these results should be seen in the light of various analogies established in [Th2]. ~ 1991 Academic Press, Inc. 0. CYCLOTOMIC THEORY FOR FUNCTION FIELDS Gauss sums arise naturally and play a prominent role in the cyclotomic theory, which is essentially a theory of abelian extensions over Q by the Kronecker-Weber theorem. For a function field K of one variable over a finite field, in addition to the usual cyclotomic extensions K(/~,,), which are just constant field extensions, there are, of course, many more abelian extensions of K, like Kummer or Artin-Schreier extensions. Carlitz [Ca] , Drinfeld [Dr] , and Hayes [Hal , Ha2, Ha3] provided families of abelian extensions of K analogous to the cyclotomic extensions over Q. We now describe these and summarize their main properties. For details and more general notions, which we will not need, see the papers of Hayes referred to above. (See also [GR, Go] ) . DEFINITIONS. Let K be a function field of one variable over its field of constants Fq. Let ~ be a place of K and let A be ring of elements of K integral outside ~ . Fix an algebraic closure /~ of K, as our universe. Let K "+ be its under- lying additive group and End(K +) be its ring of endomorphisms. For subring L ~ A of K', let L{F} denote the noncommutative ring generated 242 0022-314X/91 $3.00 Copyright ~" 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. GAUSS SUMS FOR FUNCTION FIELDS 243 by elements of L and by a symbol F, satisfying the commutation relation El= lqF, for all IE L. Then L{F} can be thought of as subring of End(K +) in an obvious fashion. By a Drinfeld A-module D over L (in fact, of "rank one" and of "generic characteristic;" but we will drop these words), we will mean an injective homomorphism D: A ~ L{F}, (a e A ~ D, e L{F}) such that degree of D, as a polynomial in F is the same as a degree of a and the constant term (i.e., coefficient of F ~ in D, is a. Two Drinfeld A-modules D, /3 are considered isomorphic if there is a nonzero k e K* such that kD, = D,k for a~A. For aeA, put A~=: {ueR:Da(u)=O}. This A-module (under D) is nothing but the a-torsion of D. The analogy with the classical case is K~--, Q, A*-+Z a~A ~-+ D. ~ End G.*-~ n ~ Z ~-+ T.=: (x~-+x")~End G m and A, ~ #~, which is the n-torsion of the usual action T of Z on the mul- tiplicative group Gin. Note that 0 is always a-torsion, just like 1 is always the nth root of 1. In fact, 2 e A , is an analogue of both ~e/~n and 1 - ~ . Let h denote the class number of K and 6 denote the degree of the place ~ , so that h6 is the class number of A. EXAMPLES. (a) A = F q [ -T] , h = 6 = 1. T being a generator of A o v e r F q , to give D, it is enough to prescribe a nonzero l and to put Dr= T+ IF. All such D's are isomorphic. In this case, T2-torsion of D is { U ff K : D T2(U ) = I q+ luq2 -~- ( TI + I T q) u q + T 2 u = 0} . (b) [Ha2,11.3] A = F 2 [ x , y ] / y 2 + y = x 3 + x + l . h = 6 = l , D x = x + (x 2 + x) F + F 2, Dy = y + (y2 + y) F+ x(y2+ y)F 2 + F 3. Since x and y generate A and since Dx D y = Dy Dx, D is well defined by this prescription. (c) [Ha2, 11.2] A = F 3 I-x, y ] / y 2 = x - x 2. K is again a rational func- tion field, so that h = 1, but 6 = 2. There is no A-module D over A, but if one considers B = A [ c ] / c 2 = - I and puts ~ = l + x + c y , O = l + x - c y (r/, 0 are units in B), then one has D over B given by D x = x + y~IF+glF 2, D v = y + ( f + x ) qF+c(fF 2. Function Field Cyclotomic Theory [Ha2, Ha3] (1) Let H (notation HA o r H e is also used) denote the maximal abelian unramified extension of K, where oo splits completely. This is an analogue of the Hilbert class field (see [Ro] for more on the notion). Its 246 DINESH S. T H A K U R tends to infinity (see [Hal , Thm. 4.1]). Now, by the Riemann-hypothesis for function fields, the class number h,, is I]7=~ (1 -x i ) , [~i] =w/q, so that (xflq + 1 )2g, ~> h~ >~ ( x / q - 1 )2g~ which implies that, if q > 4 (so that x ~ - 1 > 1), log h, is of the order (in the sense that the ratio of the two sides is bounded between two positive constants) n(NP)", as n tends to infinity. It is not known whether the same estimate holds in the classical case. For the minus part of the class number such estimate holds in the classical case (see [Wa, Thin. 4.20]) and also in function field case (if q >4), by similar considerations. But note that, in function field case, for q = 2 the minus part of h, is just 1 and also that there is a difference between the notions of class numbers of fields and those of their "rings of integers." By the well-known analogy (see, e.g., Weil's "Basic number theory") between q2g 2 (where g is genus of the function field) and the discriminant, the order of log h,, described above is analogous to that predicted by the Braue~Siegel theorem, because the regulator of a field is 1 here. 1. GAUSS SUMS Now we recall ( [Thl ], see also [-Th2]) the definition of Gauss sums. Let D be a Drinfeld A-module. (To obtain gauss sums with good properties, one may normalize D as described below. The effect of normalization is also described below). Let P be a prime of A of degree d. Choose an A-module isomorphism ~0: A / P ~ Ap (an analogue of additive character ~b : Z/p ~ ltp) and let Z~ (J mod d) be Fq-homomorphisms A/P ~ L, where L is a field containing K(Ae), indexed so that gq=~(j ~ ( j m o d d ) (special multiplicative characters which are q J-powers of "teichmuller character"). Then one defines Gauss sums g~=:g(zj) = : - ~ Zj( z 1)~(z)6K(Ae)(Pq a 1). z ~ ( A / P J * One has [Th2] "Fourier expansion" 6(z) = 5Z gj~.j(Z) of ~ in terms of all characters of (A/P)*. Let D be a sgn-normalized A-module. Let L be the compositum of the extensions H1 (/~q~_l) and HI (A e) of H1, which are linearly disjoint by simple considerations (see (1), (2), (3) of Section0). Hence the Galois group of L over HI is canonically isomorphic to the product of the Galois groups of these extensions. So the powers of q6th power Frobenius for the extension HI(]Aqa 1) over HI and elements of (A/P)*=GaI(HI(Ap)/HI) can be thought of as elements of GaI(L/H1). Then just as in [Th2, Thm. I] as ~ is nonzero, gJo is nonzero for some J0 and hence using the Galois GAUSS SUMS FOR FUNCTION FIELDS 247 action gj0+k6 is nonzero for any integer k. In particular, all g /s are non- zero, when (d, 6 ) = 1 (e.g. when 6 = 1). In fact, all gfs are nonzero in general. (See note added in proof.) But just as in [Th2, Prop. I] , because of Fq-linearity of ~,, one obtains only zero "gauss sums," unless one restricts the multiplicative character as in the definition given here. Remark. If one takes D to be just normalized rather than sgn-nor- malized, the same holds if one replaces H~ by H. One also sees that, if h = 6 = 1 then H = Hj -- K and the situation is quite similar to the classical case and to the case considered in [-Th2]. Hence, the "norm" Ge=: 1-Ijmoddgqj I lies in BI (resp. B if D is nor- malized). In [-Th2, Thm. II] we saw that Ge=(- -1 )dp (analogue of g(z) g(Z)= Z ( - 1)q), when A = Fq IT] , P is a monic prime and D is given by Dr= T+ F. In particular, this implies that all gj's lie above P in that case. We will see a different scenario in general. Remark. Change of D to an isomorphic k ~Dk changes Gp to Gp/klq ~)d, SO we assume without loss of generality that D is normalized or sgn-normalized. By Gal(H/K)-action one can move to different isomorphism classes. First note that Eisenstein property ((5), (6) of Section 0) of De(u)/u at P implies that P divides G?. EXAMPLES. (A) If P is a prime of degree one, then using explicit principal ideal theorem of (4), Section 0, one easily sees that G?B = PB. (B) Let q = 2 and P e A be a prime of degree two. Let D e = P + PlF+ P2 F2 and Ae= {2i}. Then a direct calculation shows G? = gogl = ~ 2~ + ~ 2,)~j = 0 + PI/P2 = P~/P2. Since P2 is a unit by our assumption, GeB=P1B. Example (b) in Section 0, with P = x, now shows that G? = x(x + 1), so we see that primes other than P can also enter into the factorizations of Gauss sums. To study this phenomenon in detail, we first focus on the case h = 6 = 1, when the cyclotomic theory is the simplest and analogous to the classical case and to [Th2]. Apart from infinitely many (one for each q) rational function fields, there are only seven other K's of class number one, as established in [-LMQ]. From the list given there, it is easy to check that h = 3 = 1 forces A to be one of the four cases in the theorem below. (These are Examples 11.3-11.6 of [Ha2-1). 248 DINESH S. T H A K U R THEOREM I. When h = 6 = 1, Ge is given by the .following recipe. (1) ([Ha2, Example 11.3] Example (b), Section 0) A = F2 [x, y]/y2 + y = x 3 + x + 1. Gp = PP~ where ~ is the order four automorphism of A given by a ( x ) = x + l, a ( y ) = y + x + l. (2) ([Ha2, Example 11.4]) A = F n [ x , y]/y2+y=x3+w, w2+w+ 1 = O. Ge= p(po)3, where a is the order two automorphism of A given by a(x) =x, or(y)= y + 1 and for a given sign function such that x and y have sign 1, representative P of sign 1 is chosen. (3) ([Ha2, Example 11.5]) A = F 3 [ x , y]/y2 = x 3 - - X - - 1. Ge = ( - I)dp(p~) 2, where a is the order three automorphism of A given by a(x) = x + 1, tr(y)= y and representative P is chosen as in (2). (4) ([Ha2, Example 11.6]) A = F z [ x , y ] / y 2 + y = x S + x 3 + l . Ge = p(pa)2p~2, where ~ is the order four automorphism of A given by t r ( x )=x+ 1, tr(y)= y + x 2. (Remark. (1)-(3) are of genus 1, while (4) is of genus 2). Proof When A has degree 2 element (x in our cases), one can obtain a simple expression for gq_ ~/gj as follows. De = x + x~ F+ x2F 2 gives, as in q2 [Th2, p. 107], ) ~ j ( x ) & = x g j + x l g q ~ +x2g)_2. One uses this relation to successively eliminate higher powers of gfs from a similar relation obtained from Dy with suitable element y of higher degree (y in our cases). Write Xo(X)= 0 and Z0(Y)= fl, so that ;~j(x)= 0 q' and Xj(Y)= flqJ. In our four cases one obtains the following expressions for gq-i/go, respectively: x(x+O)+ y + # x + 0 + l x2(x + O) + y + fl x + O - y ( x - O ) + y - ~ x - - O + l (x W O)(x4 q- X3 + (1 + O)x2) + y + fl X 3 "~ O X 2 " { - (1 + O)x + 02 + O Now note that Gp = FI(gq_ 1 /&). (The claim about the signs immediately follows from this.) Let the product of numerators of the expression (as written, without reducing) be N and the product of denominators be D. Then we claim the following equalities of ideals in our four cases, respectively: GAUSS SUMS FOR FUNCTION FIELDS 251 2. GAUSS SUMS FOR COMPOSITE M O D U L U S We restrict to A = F q F T ] and to D given by D r = T + F for simplicity. One can define Gauss sums for composite modulus a, just as before by replacing P by a and using isomorphism ff : A/a ~ A u. First, let P be a prime of degree d and let a = pn, n > 1. Cardinality of (A/a)* is q d ( n - l ) ( q d 1). As q is a power of the characteristic, a multi- plicative character X factors through (A/P)*. (Order of a multiplicative character Z divides qd_ 1, so that Z qa = g.) Now, f = fn 1 pn 1 + ... + fo, where f,. run through the polynomials of degree less than d and fo is non- zero, are the representatives of reduced residue classes modulo a. We have z ( f ) = )~(fqa) = ~((fo), since f q ~ = f o m o d e . Hence, we have g(z) = ~ )~(fo) ~b ~ f~ , + + f o . l b ' ( A l P ) * fl ...,fn I ~ A I P Using (a) ~, is Fq-linear and (b) if a term in a sum appears a multiple of p number of times, then it contributes zero; it is easy to see PROPOSITION I. When a = P", n > 1, P a prime o f degree d, g()~)=0 unless n = 2 , d = l and q = 2 (i.e., unless P = T or T + I in F 2 [T] , when g(g) = r Now classically (see [Na, p. 252-254]), if n = n l . . . n r is a factorization of composite modulus n into relatively prime factors ni, then ( Z / n ) * = [ I (Z /n i )* and we have the corresponding decomposition of multiplicative character X as ~ = ]-I ~(i in an obvious fashion. Put ~(m) = 0, if m and n are not relatively prime and define the Gauss sum g ( z ) = ~ Z(x) exp(2rHx/n) ,r mod n then (see [Na, p. 253]) one has r g(Z) = [ I zi(n/n~)g(z~). i=1 If one follows the same proof [Na, p. 253], in our case, where we are dealing with additive rather than the multiplicative group, we obtain PROPOSITION II. g()~) = ~ eez~(n/n~) g(z,), i=1 where c~ = ~,x, ~od ~. i r j V I ) ~ i ( x i ) 9 In this wierd additive decomposition, we loose control over the absolute values and even G's or g(x~)'s can become zero. 252 DINESH S. THAKUR ACKNOWLEDGMENTS I thank David Hayes for various conversations regarding the function field cyclotomic theory and David Goss for his suggestion to look at Gauss sums for composite modulus. I am obliged to the Institute for Advanced Study, Princeton, where the main results of this paper were presented in the Drinfeld module seminar in 1987 1988; and to the Tara Institute of fundamental research, Bombay, for their support. Note added in proof. Finally, we prove that gj as defined in Section 1 is never zero: Put flu(z) =: ~,(/tz), ~ 9 A/P. Since the pairing A/P • AlP --, A e defined by (x, y) --. ~b(xy) is nondegenerate bilinear pairing over Fq, any Fq-linear function from AlP, which is a d-dimensional vector space over Fq, to L is a linear combination of ~,/s. Now ~,~, = Z(Zj(l~) gj)zj. If gJo were zero, then O~'s would be in a space generated by less than d of X/s, which is a contradiction. REFERENCES [Ca] [Dr] [GR] [GK] [Go] [ H a l l [Ha2] [Ha3] [LMQ] [Na] [Ro] [Ta] [Th 1 ] [Th2] [Th3 ] [Wa] L. CARLITZ, A Class of polynomials, Trans. Amer. Math. Soc. 43 (1938), 167-182. V. DRINFELD, Elliptic modules, Math. USSR-Sb. 23 (1974), 561-592. [English Translation] S. GALOVICH AND M. ROSEN, The class number of cyclotomic fields J. Number Theory 13 (1981), 363-375. R. GOLD AND H. KISILEVSKY, On geometric Zp extensions of function fields, Manuscripta Math. 62 (1988), 145-161. D. Goss, The arithmetic of function fields, 2, The cyclotomic theory, J. Algebra 81 (1983), 107-149. D. HAYES, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), 77-91. D. HAYES, Explicit class field theory in global function fields, in "Studies in Algebra and Number Theory" (G. C. Rota, Ed.), pp. 173-217, Academic Press, New York, 1979. D. HAYES, Stickelberger elements in function fields, Compositio Math. 55 (1985), 209-239. J. LEITZEL, M. MADAN, AND C. QUEEN, On congruence function fields of class number one, J. Number Theory 7 (1975), 11-27. W. NARKIEWICZ, "Elementary and Analytic Theory of Algebraic Numbers," PWN (Polish Scientific Publishers), Warsaw, 1974. M. ROSEN, The Hilbert class field in function fields, Exposition. Math. 5 (1987), 365-378. T. TAKAHASHI, Good reduction of elliptic modules, J. Math. Soc. Japan 34 (1982), 475487. D. THAKUR, Gamma functions and Gauss sums for function fields and periods of Drinfeld modules, Thesis, Harvard, 1987. D. THAKUR, Gauss sums for Fq[T], Invent. Math. 94 (1988), 105 112. D. THAKUR, Gamma functions for function fields and Drinfeld modules, Ann. of Math., to appear. L. WASHINGTON, "Introduction to Cyclotomic Fields," Springer-Verlag, New York, 19.82. Printed in Belgium