Small generators of function fields

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1 Journa de Théorie des Nombres de Bordeaux 00 (XXXX), Sma generators of function fieds par Martin Widmer Résumé. Soit K/k une extension finie d un corps goba, donc K contient un éément primitif α, c est à dire K = k(α). Dans cet artice, nous démontrons existence d un éément primitif de petite hauteur en cas d un corps de fonctions. Notre résutat répond à une quéstion de Ruppert en cas d un corps de fonctions. Abstract. Let K/k be a finite extension of a goba fied. Such an extension can be generated over k by a singe eement. The aim of this artice is to prove the existence of a sma generator in the function fied case. This answers the function fied version of a question of Ruppert on sma generators of number fieds. 1. Introduction Let K be a finite extension of a goba fied k where goba fied means finite extension of either Q or of a rationa function fied of transcendence degree one over a finite fied. Such an extension is generated by a singe eement and there exists a natura concept of size on K given by the height. The we-known Theorem of Northcott (originay proved for agebraic numbers but easiy seen to hod aso in positive characteristic) impies that for each rea T there are ony finitey many α K whose height does not exceed T. In particuar there exists a smaest generator. It is therefore natura to ask for ower and upper bounds for the height of a smaest generator. We emphasize the situation where d is fixed and K runs over a extensions of k of degree d. Severa peope proved ower bounds for generators; first Maher [5] for the ground fied k = Q and then Siverman [8] for arbitrary ground fieds (and aso higher dimensions), but see aso [7], [6] and [4] for simper resuts. For an extension K/k of number fieds Siverman s inequaity impies (1.1) h(1, α) og K /(2d(d 1)) og k /(2(d 1)) [k : Q] og d/(2(d 1)) for any generator α of K/k. As shown by exampes of Masser (Proposition 1 [6]) and Ruppert [7], this bound is sharp, at east up to an additive constant depending ony on k and d. A version of Siverman s bound in

2 2 Martin Widmer the function fied case foows quicky from Castenuovo s inequaity. For simpicity et us temporariy assume K and k are finite separabe extensions of the rationa function fied F q (t) both with fied of constants F q. We appy Castenuovo s inequaity as in [9] III.10.3.Theorem with F = K = k(α), F 1 = k and F 2 = F q (α). Writing g k and g K for the genus of k and K we concude [K : F q (α)] g K /(d 1) dg k /(d 1) + 1. From (2.1), (2.2) and the definition of the height in (2.3) we easiy deduce h(1, α) [K : F q (α)]/d and thus (1.2) h(1, α) g K /(d(d 1)) g k /(d 1) + 1/d. The discriminant K = q deg Diff(K/Fq(t)) of K/F q (t) is reated to the genus by the Riemann-Hurwitz formua, more precisey K = q 2g K+2([K:F q(t)] 1). Thus (1.2) matches with (1.1), at east up to an additive constant depending ony on the degrees of k and K. A simiar inequaity as in (1.2) was given by Thunder ([10] Lemma 6). What about upper bounds for the smaest generator? It seems that this probem has not been studied much yet. However, at east for number fieds the probem has been proposed expicity by Ruppert. More precisey Ruppert ([7] Question 2) addressed the foowing question. Question 1 (Ruppert, 1998). Does there exist a constant C = C(d) such that for each number fied K of degree d there exists a generator α of the extension K/Q with h(1, α) og K /(2d) + C? Ruppert used the non-ogarithmic naive height whereas we use the ogarithmic projective absoute Wei height as defined in [2]. However, it is easiy seen that the question formuated here is equivaent to Ruppert s Question 2 in [7]. One can show that there exists aways an integra generator α of K/Q with h(1, α) og K /d. For a proof of this simpe fact see [11]. What is more Ruppert showed that Question 1 has an affirmative answer for K either quadratic or a totay rea fied of prime degree. In fact, using Minkowski s convex body Theorem to construct a Pisot-number generator, it suffices to assume K has a rea embedding and one can drop the prime degree condition. In this specia case the derived sma generator is an agebraic integer. For more detais we refer to [11]. On the other hand, if α is an integra generator of an imaginary quadratic fied K then h(1, α) og K /d og 2. Thus for amost a imaginary quadratic fieds the sma generators are not integra eements. Ruppert s resut for d = 2 reies heaviy on a distribution resut of Duke [3] that does not appear to have an anaogue for higher degrees and is ineffective. As a consequence Ruppert s constant C for d = 2 is ineffective.

3 Sma generators of function fieds 3 In this note we introduce a competey different strategy which appies in the function fied case and the number fied case. However, it reies on the existence of a certain divisor which is guaranteed under GRH but might be rather troubesome to estabish unconditionay. The aim of this short note is to answer positivey Ruppert s question in the function fied case. So et k be an agebraic function fied with finite constant fied k 0 and transcendence degree one over k 0. We have the foowing resut. Theorem 1.1. Let K be a finite fied extension of k. There exists an eement α in K with K = k(α) and a constant C = C(k, [K : k]) depending soey on k and [K : k] such that h(1, α) g K d(k/k) + C where g K denotes the genus of the function fied K with fied of constants K 0 and d(k/k) = [K : k]/[k 0 : k 0 ]. 2. Notation and definitions Throughout this note we fix an agebraic cosure k of k. A fieds are considered to be subfieds of k. For any finite extension F of k we write F 0 for the fied of constants in F ; in other words F 0 is the agebraic cosure of k 0 in F. When we tak of the fied F we impicity mean the fied F with fied of constants F 0. We define the geometric degree d(f/k) of the extension F over k as [F : k] d(f/k) = [F 0 : k 0 ]. Let M(F ) be the set of a paces in F. For a pace in M(F ) et O be the vauation ring of F at ; we can identify with the unique maxima idea in O. We write F = O / for the residue cass fied and F ˆ for the topoogica competion of F at the pace. Write ord for the order function on F ˆ normaized to have image in Z. We extend ord to F ˆ n by defining ord (x 1,..., x n ) = min 1 i n ord x i with the usua convention ord 0 = > 0. Each non-zero eement x of F n gives rise to a divisor (x) over F (x) = ord (x). M(F ) For a divisor A over F we define the Riemann-Roch space in F n L n (A) = {x F n \0; (x) + A 0} {0}.

4 4 Martin Widmer This is a F 0 vector space of finite dimension. Denote its dimension over F 0 by n (A). The degree of a pace in M(F ) is defined by deg F = [F : F 0 ]. Let K be a finite extension of F and et B be a pace in M(K) above. We write f(b/ ) = [K B : F ] for the residue degree of B over. Then we have deg K B = [K B : K 0 ] = [K B : F 0 ] [K 0 : F 0 ] = [K B : F ] [K 0 : F 0 ] [F : F 0 ] (2.1) = f(b/ ) [K 0 : F 0 ] deg F. Writing e(b/ ) for the ramification index we aso have (2.2) e(b/ )f(b/ ) = [K : F ], B see for exampe III.1.11.Theorem in [9]. Each divisor A = a over the smaer fied F naturay defines a divisor A (K) = a e(b/ )B over the arge fied K. (2.3) B As in [10] we define the height h on non-zero x in K n by h(x) = deg K(x) d(k/k). Note that the degree of a principa divisor is zero so that the height defines a function on projective space P n 1 (K) over K of dimension n 1. This shows aso that the height is nonnegative since to evauate the height of x we can assume that one coordinate is 1. Moreover it is absoute in the foowing sense. Suppose x K n and et D be the divisor given by D = (x). Let R be a finite extension of K and view x R n. Let D (R) be the divisor over R given by D (R) = (x). By [1] Chap.15, Thm.9 we have deg R (D (R) ) = d(r/k) deg K (D) and by [1] Chap.15, Thm.2 we have d(r/k) = d(r/k)d(k/k). Thus h(x) remains unchanged if one views x in R n. Therefore the height extends to a projective height on k n. Suppose x L n (A) K n then directy from the definition we see that (2.4) h(x) deg K A d(k/k).

5 Sma generators of function fieds 5 3. The strategy Let S be a finite set of paces in M(K) such that the foowing two properties hod: (i) for each pace p in M(k) there is at most one pace in S that ies above p, (ii)f(b/p) = 1 for a B S and p M(k) with B p. A set S with these two properties wi be caed admissibe. Note that for each fied F with k F K and for a paces B, B S,, M(F ) with B and B we have (3.1) (3.2) B B, f(b / ) = 1. We say the divisor A is admissibe if it can be written in the form (3.3) A = B S 1 B. with an admissibe set S. Lemma 3.1. Suppose A is an admissibe divisor and suppose x = (1, x) with x / K 0 and x L 2 (A). Then k(x) = K and h(x) deg K A/d(K/k). Proof. Suppose k(x) = F K and write (x) = a for the divisor over F. When we consider (x) as a divisor over K we have (x) = B a BB with a B = a e(b/ ). Note that none of the coefficients a is positive and since x / K 0 at east one is negative, say a. Since (x) ies in L 2 (A) and A is an admissibe divisor we concude by (3.1) that there is exacty one pace B in M(K) with B and by (3.2) that f(b / ) = 1. Together with (2.2) we deduce that 1 < [K : F ] = B e(b / )f(b / ) = e(b / ). This means that a B = a e(b / ) < a 1, contradicting the fact A + (x) 0. Thus k(x) = K. The remaining statement comes from (2.4). Lemma 3.2. Suppose A is an admissibe divisor with deg K A > g K. Then there exists α in K with k(α) = K and h(1, α) deg K A d(k/k). Proof. We appy the Theorem of Riemann-Roch to the space L 1 (A) = {x K\0; (x) + A 0} {0} to concude 1 (A) > 1. Therefore we find a α in L 1 (A)\K 0. Now (x) = (1, α) and (α) have the same poe-divisors and since A 0 we see that (x) ies in L 2 (A). Appying Lemma 3.1 proves the emma.

6 6 Martin Widmer 4. Constructing a suitabe divisor In this section we wi prove that K admits an admissibe divisor of degree g K + 1 provided g K is arge enough. Lemma 4.1. Let be a positive integer. The number of paces B M(K) with is deg K B = and f(b/p) = 1 for p M(k) with B p K 0 (2 + 7g K ) K 0 /2 [K 0 : k 0 ][K : k]( K 0 /2 + (2 + 7g k ) K 0 /4 ). Proof. Using Riemann s hypothesis one can obtain a good ower bound for the number of paces B of fixed degree. For instance V.2.10 Coroary (a) in [9] tes us that the tota number of paces B M(K) with deg K B = is K 0 (2 + 7g K ) K 0 /2. From (2.1) we get deg k p = [K 0 : k 0 ]/f(b/p) for p M(k), B p. Suppose f = f(b/p) > 1. Appying V.2.10 Coroary (a) again we get the foowing upper bound for the number of paces p M(k) with deg k p = [K 0 : k 0 ]/f f k 0 [K 0:k 0 ]/f [K 0 : k 0 ] + (2 + 7g k ) f k 0 [K 0:k 0 ]/(2f) [K 0 : k 0 ] = f K 0 /f [K 0 : k 0 ] + (2 + 7g k) f K 0 /(2f) [K 0 : k 0 ] K 0 /2 + (2 + 7g k ) K 0 /4. Above each pace p in M(k) there ie at most [K : k] paces B M(K). Therefore the number of paces B M(K) satisfying deg K B = and f(b/p) = f > 1 is [K : k]( K 0 /2 + (2 + 7g k ) K 0 /4 ). Summing over a divisors f of [K 0 : k 0 ] we find that the number of paces B M(K) satisfying deg K B = and f(b/p) > 1 is [K 0 : k 0 ][K : k]( K 0 /2 + (2 + 7g k ) K 0 /4 ). This in turn impies that the number of paces B M(K) satisfying deg K B = and f(b/p) = 1 is K 0 (2 + 7g K ) K 0 /2 [K 0 : k 0 ][K : k]( K 0 /2 + (2 + 7g k ) K 0 /4 ) and this proves the emma.

7 Sma generators of function fieds 7 5. Proof of Theorem 1.1 Let d be a positive integer. We can find a constant C 1 = C 1 (k, d) such that with = g K + 1 K 0 (2 + 7g K ) K 0 /2 [K 0 : k 0 ][K : k]( K 0 /2 + (2 + 7g k ) K 0 /4 ) > 0 for a extensions K of k satisfying g K > C 1 and [K : k] = d. By virtue of Lemma 4.1 we concude that for each such K there exists a pace B M(K) with deg K B = g K + 1 and f(b/p) = 1 for p M(k) with B p. In particuar there exists an admissibe divisor, namey B, with deg K B = g K + 1. From Lemma 3.2 we concude that if g K > C 1 and [K : k] = d then there exists α K with K = k(α) and h(1, α) (g K + 1)/d(K/k). There are ony finitey many fied extensions K of k of degree d with g K C 1. Hence there exists a constant C as in Theorem 1.1 depending soey on C 1, k, d and thus depending soey on k, d such that the statement of Theorem 1.1 hods for a extensions K of k of degree d. Acknowedgements I woud ike to thank Feipe Vooch for giving me the idea of the proof of Lemma 4.1 and pointing out the connection with Castenuovo s bound mentioned in the Introduction. Moreover I am gratefu to David Masser, Jeffrey Thunder and Jeffrey Vaaer for many deightfu discussions. References 1. E. Artin, Agebraic numbers and agebraic functions, Gordon and Breach, New York, E. Bombieri and W. Guber, Heights in Diophantine Geometry, Cambridge University Press, W. Duke, Hyperboic distribution probems and haf-integra weight Masss forms, Invent. Math. 92 (1988), J. Eenberg and A. Venkatesh, Refection principes and bounds for cass group torsion, Int. Math. Res. Not. no.1, Art. ID rnm002 (2007). 5. K. Maher, An inequaity for the discriminant of a poynomia, Michigan Math. J. 11 (1964), D. Roy and J. L. Thunder, A note on Siege s emma over number fieds, Monatsh. Math. 120 (1995), W. Ruppert, Sma generators of number fieds, Manuscripta math. 96 (1998), J. Siverman, Lower bounds for height functions, Duke Math. J. 51 (1984), H. Stichtenoth, Agebraic function fieds and codes, Springer, J. L. Thunder, Siege s emma for function fieds, Michigan Math. J. 42 (1995), J. D. Vaaer and M. Widmer, On sma generators of number fieds, in preparation (2009). Martin Widmer Institut für Mathematik A Technische Universität Graz Steyrergasse 30/II 8010 Graz Austria E-mai : widmer@tugraz.at