EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION ...
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS OVER FINITE FIELDS DAVID A. CARDON AND M. RAM MURTY Abstract. We nd a lower bound on the number of quadratic exten-
sions of the function eld Fq ( ) whose class groups have an element of a xed order. More precisely, let be a power of an1 odd prime and let be a 1 xed positive integer 3. There are `( 2 + g ) polynomials 2 Fq [ ] with deg( p ) such that the class groups of the quadratic extensions ) have an element of order . Fq ( T
q
g
q
D
T;
D
T
`
D
g
1. Introduction In a recent paper Murty [11] showed that if g is a xed integer 3 then the number of imaginary quadratic elds whose absolute discriminant is x and whose class group has an element of order g is x + g . He also showed that the number of real quadratic elds whose discriminant is x and whose class group has an element of order g is x g . In this paper we prove the analogous result for function elds rather than number elds in the analog of the imaginary quadratic case. The problem of divisibility of class numbers for number elds has been studied extensively. Gauss studied the case g = 2. The case g = 3 was studied by Davenport and Heilbronn [4]. For any g the in nitude of such elds was established by Nagell [12], Honda [9], Ankeny and Chowla [2], Hartung [8], Yamaomoto [14], and Weinberger [13]. Assuming the ABC Conjecture Murty [10] obtained a quantitative lower bound on the number of such elds. More recently Murty [11] improved the technique to give the result mentioned earlier without assuming the ABC conjecture. A conjecture of Cohen and Lenstra [3] predicts that as x increases a positive fraction of discriminants x produce quadratic extensions whose class number is divisible by any xed g. Interest in function elds was stimulated by the doctoral thesis of E. Artin [1] and the class number problem for function elds has been studied. For example, if g is not divisible by q then Friesen [7] constructed in nitely many polynomials M 2pF q [T ] of even degree such the class groups of the quadratic extensions F q (T; M ) of the function eld F q (T ) have an element of order g. Friedman and Washington [6] have studied the Cohen-Lenstra conjecture in the function eld case, and Yu [15] has established the Cohen-Lenstra 1 2
1 2
Research partially supported by NSERC.
1
1
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
2
conjecture as the characteristic p tends to in nity for xed discriminantal degree. We now state the main result of this paper: Theorem. Let q be a power of an odd prime and let g be ap xed positive integer 3. There are q`( + g ) quadratic extensions F q (T; D) of F q (T ) with deg(D) ` whose class group has an element of order g. The remainder of this paper will present the proof as a series of lemmas. The main outline is as follows. We show that if n and m are monic elements of F q [T ], if a 2 F q is not a square, if deg(mg )p> deg(n2 ), and if D = n2 ?amg is squarefree, then the class group of F q (T; D) has an element of order g. Using sieve methods and by letting m and n vary we are able to give a lower bound on the number of m and n such that D is squarefree. Finally, we show that as m and n vary there are relatively few duplicated values of D. 1 2
1
2. Preliminaries F q will denote the nite eld with q elements where q is a power of an odd prime. R = F q [T ] is the polynomial ring with coecients in F q over the indeterminate T and the function eld F q (T ) is the eld of fractions of R. We will assume that g is an odd integer that is relatively prime to q. The symbol p will always represent a monic irreducible polynomial in R. The symbols n and m will also be monic (but not necessarily irreducible) P polynomials in R of degrees j and k respectively. The expression m f (m) would mean to sum f (m) over all monic polynomials m of xed degree k. If a and b are elements of R, then (a; b) represents the greatest common (monic) divisor of a and b. If a and b are ordinary integers then (a; b) will denote the greatest common divisor in the usual sense. 3. Class groups with elements of order g In the following lemma we construct quadratic extensions of F q (T ) whose class groups contain elements of order g. Lemma 1. Let g be a positive integer 3. Assume n; m 2 R are monic, ?a 2 F q is not a square, deg(mg ) >pdeg(n2), and D = n2 ? amg is squarefree. Then the class group for F q (T; D) has an element of order g. Proof. First we note that (n; m) = 1 because if there were a common factor of n and m then D would not be squarefree. We factor amg as p p amg = n2 ? D = (n + D)(n ? D): p p p p Suppose I j(n + D) and I j(n ? D). Then n + D 2 I and n ? Dp2 I which implies p that n; D 2 I and I = R = F q [T ]. Thus the ideals (n + D) and (n ? D) are relatively prime. Therefore p p (m)g = (n + D)(n ? D) = ag a0 g
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
p
p
3
where a and a0 are ideals such that (n + D) = ag and (n ? D) = a0 g . Taking norms we nd that N (m) = q2 deg(m) = N (a)2 so that N (a) = qdeg(m) . Now suppose that ar is principal for some r < g: p ar = (u + v D ): Then p N (a)r = qr deg(m) = N (u + v D) = qdeg(u ?v D) 2
2
and because the leading coecient of v2 D is not a square this is
qdeg(D) = qdeg(n ?am ) = qdeg(m ) g
2
g
= N (a)g : This is a contradiction unless r = g.
4. How often is D = n2 ? amg squarefree? In light of Lemma 1 we would like to construct a lower bound on the number of squarefree expressions D = n2 ? amg as n and m vary such that deg(n) = j and deg(m) = k and deg(mg ) > deg(n2 ). Regarding k as the independent parameter we will maximize the number of possible values of D by choosing j to be the optimally large value j = bgk=2c if gk is odd or j = bgk=2c ? 1 if gk is even. Let s(h) be 1 or 0 according as h is squarefree or not. Also let( 2 sz (h) = 1 if d does not divide h whenever 1 deg(d) z 0 otherwise. We would like estimate the sum X s(n2 ? amg ): deg(m)=k deg(n)=j
Lemma 2. By counting expressions n2 ? amg that are squarefree in the small factors we obtain the following sieving inequality: X 2 X X X sz (n ? amg ) s(n2 ? amg ) sz (n2 ? amg ) ? 1: m;n
m;n
m;n
m;n;p deg(p)>z n ?amg 0(p2 ) 2
With an appropriate choice of z (depending on k) we will show that for large k #fdistinct squarefree values of n2 ? amg g X X s(n2 ? amg ) sz (n2 ? amg ) m;n qj+k :
m;n
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
4
Several auxiliary functions will be useful for estimating the terms in Lemma 2. De ne the Mobius function on the nonzero elements of R. If h 2 R has factorization ap 1 p t t where a 2 F q and the pi are irreducible monic polynomials in R then 8 > if h 2 F q ; (?1)t if i = 1 for all i; :0 otherwise. For z 1 let Y P (z) = p 1
irreducible p deg(p)z
and let
Nm;z (j ) =
X deg(n)=j
sz (n2 ? amg ):
For xed m; h 2 R let m (h) = #fn 2 R=hR : n2 ? amg 0(h)g: Thus m (h) is the number of n 2 R=hR satisfying the congruence n2 ?amg 0 (mod h). We will use the following elementary estimate several times: Lemma 3. If (u) represents the number of irreducible polynomials in Fq [T ] of degree u > 0, then (u) qu =u. P Proof. Since qu = dju d (d), the upper bound is clear.
Lemma 4.
m (d1 d2 ) = m (d1 )m (d2 ) if d1 and d1 are coprime. m (p2 ) = qdeg(p) if p is irreducible and p divides m. m (p2 ) 2 if p is irreducible and p does not divide m. m (d2 ) 2(d) qdeg(m) for squarefree d where (d) is the number of distinct monic irreducible polynomials of degree 1 that divide d. Proof. The multiplicativity of m is an immediate consequence of the Chi1. 2. 3. 4.
nese remainder theorem. Suppose that n satis es n2 ? amg 0(p2 ) with p dividing m. Then p divides n. There are exactly qdeg(p) multiples of p modulo p2 . Suppose that n satis es n2 ? amg 0(p2 ) but that p does not divide m. The solution n must be a `lift' of a solution modulo p. That is n = n1 + pt where n21 ? amg 0(p). We know there are at most two solutions of the congruence modulo p. Then 0 (n1 + pt)2 ? amg (n21 ? amg ) + 2n1 pt (mod p2 )
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
implies
5
0 n1 ?pam + 2n1 t (mod p): g
2
When p does not divide n and (2; q) = 1 there is a unique t (mod p) satisfying the last congruence. Thus the solution n1 (mod p) gives rise to a unique solution n (mod p2 ). Therefore, in this case, n (p) 2. Now let (d) represent the number of distinct nonconstant monic polynomials dividing d where d is squarefree. Then Y Y Y Y m (d) = m(p2 ) m (p2) qdeg(p) 2 2 (d) qdeg(m) : pjd pjm
pjd (p;m)=1
pjd pjm
pjd (p;m)=1
The following lemma tells us a choice of z that allows the sieve in Lemma 2 to yield interesting information. Lemma 5. Given any > 0 we can choose (independently of m) so that if z = log(k) then
Nm;z (j ) = qj
Y
(1 ? m (p2 )q? deg(p ) ) + O(q(1+)k ): 2
deg(p)z
P Proof. Let Nm;z (j ) = deg(n)=j sz (n2 ? amg ). Then X X X Nm;z (j ) =
n
d monic d2 j(n2 ?amg ;P (z))
If j deg(d2 ) then
X
Nm;z (j ) =
X
X n n2 ?amg 0(d2 )
1:
1 = m (d2 )qj ?deg(d ) ;
X deg(n)=j n2 ?amg 0(d2 )
Thus
d d2 jP (z)
(d)
2
deg(n)=j n2 ?amg 0(d2 )
while if j deg(d2 ) then
(d) =
1 m (d2 ):
(d)fm (d2 )qj?deg(d ) + O(m (d2 ))g 2
djP (z) Y X (1 ? m (p2 )q? deg(p2 ) ) + O(m (d2 )): = qj deg(p)z djP (z)
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
Now
X djP (z)
n (d2 ) qdeg(m) = qdeg(m)
X djP (z)
6
2 (d)
Y
deg(p)z deg( m ) q 3qz :
(1 + 2)
Given any > 0 we can choose such that if z = log(k) then the last expression is bounded by qk for suciently large k. Therefore for suciently large k we have Y (1 ? m (p2 )q? deg(p ) ) + O(q(1+)k ): Nm;z (j ) = qj 2
deg(p)z
Lemma 6. We have X the 2lower bound X sz (n ? amg ) = Nm;z (j ) qj+k : m;n
m
Proof. We notice that Y Y Y (1 ? m (p2 )q? deg(p ) ) = (1 ? q? deg(p) ) (1 ? m (p2 )q? deg(p ) ) 2
2
deg(p)z
Then we sum m X overY
X X
Y
pjm deg(p)z
Y
pjm
(1 ? q? deg(p) )
(1 ? q? deg(p) ) =
(p;m)=1 deg(p)z
Y
(1 ? 2q? deg(p ) ) 2
all p
X djm
(d)q? deg(d) :
(1 ? m (p2 )q? deg(p ) ) 2
deg(m)=k deg(p)z
pjm deg(p)z
(d)q? deg(d) =
deg(m)=k djm X (d)q?2 deg(d) = qk deg(d)k
X
(d)q? deg(d) qk?deg(d)
deg(d)k = qk f(1 ? q?1 ) + O(q?k )g qk :
Summing the expression in Lemma 5 as m varies P such that deg(m) = k and apply the last inequality gives the lemma: m Nm;z (j ) qj +k . Lemma 7. Pm (m) log(k)qk Proof. X X k?deg(p) k X ?u qu (m) q q q u log(k)qk : m
p deg(p)k
uk
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
Lemma 8.
X
Proof. We may write
m;n;p deg(p)>z n2 ?amg 0(p2 )
X
m;n;p deg(p)>z 2 n ?amg 0(p2 )
where
1=
Mm;p(j ) =
7
1 = o(qi+j ):
X X m
p deg(p)>z
Mm;p (j )
X n n2 ?amg 0(p2 )
1:
Because Mm;p (j ) = m (p2 )qj ?deg(p ) if j deg(p2 ) and Mm;p (j ) m (p2 ) if j < deg(p2 ) we obtain an upper bound on Mm;p(j ) ( j?deg(p ) Mm;p(j ) 2(j?q deg(p) + 1) if (p; m) = 1, q if pjm. Summing over irreducible p results in X X X j?deg(p) Mm;p (j ) 2(qj ?deg(p ) + 1) + q 2
2
2
z<deg(p)j
z<deg(p)j (p;m)=1 qj?z qj
z<deg(p)j pjm
z + j + (m)qj?z :
Then summing the last expression over m gives
XX m p
j +k?z
j +k
X
Mm;p (j ) q z + q j + qj?z (m) m j + k j + k ? z q z + q j + log(k)qj+k?z k) qj+k ( zq1z + 1j + log( qz ) = o(qj +k ):
We have now shown (Lemmas 2, 6, and 8) that the number of squarefree values of n2 ? amg as m and n vary is X 2 s(n ? amg ) qj+k : m;n
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
8
It remains to be shown that there is not too much duplication among the expressions n2 ? amg . Lemma 9. The number of squarefree elements of the form n2 ? amg with deg(n) = j and deg(m) = k that are representable in more than one way is o(qj+k ). Proof. Let m1 and m2 be xed unequal polynomials such that D = n21 ? amg1 = n22 ? amg2 for some m1 and m2 . Then a(mg1 ? mg2 ) = n21 ? n22 = (n1 ? n2 )(n1 + n2 ): For xed m1 and m2 the choices for n1 and n2 are derived from the divisors of a(mg1 ? mg2 ) which cannot exceed deg(mg1 ? mg2 ) gk. The number of possible values of m1 and m2 is O(q2k ) and therefore the total number of squarefree elements of the form n2 ? amg that are duplicated is O(gkq2k ) which is o(qj +k ). We have now shown that there are qj +k distinct values of D = n2 ?amg . Since j = bgk=2c or j = bgk=2c ? 1 there are qgk( + g ) distinct p values of + ) ` ( g quadratic extensions F q (T; D) of F q (T ) D. Therefore there are q such that deg(D) `. This completes the proof of the theorem stated at the beginning. Acknowledgment. We wish to thank Yu-Ru Liu for her willingness to go over the details of this paper with the rst author. 1 2
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1
References [1] E. Artin, Quadratische Korper im Gebiet der hoheren Kngruenzen I, II, Math. Zeitschrift 19 (1924) 153-246. [2] N. Ankeny and S. Chowla, On the divisibility of the class numbers of quadratic elds, Paci c Journal of Math., 5 (1955) p. 321-324. [3] H. Cohen and H.W. Lenstra Jr., Heuristics on class groups of number elds, Springer Lecture Notes, 1068 in Number Theory Noordwijkerhout 1983 Proceedings. [4] H. Davenport and H. Heilbronn, On the density of discriminants of cubic elds, II, Proc. Royal Soc., A 322 (1971), p. 405-420. [5] R. Gupta and M. Ram Murty, Class groups of quadratic functions elds, in preparation. [6] Eduardo Friedman and Lawrence C. Washington, On the distribution of divisor class groups of curves over nite elds, in Theorie des nombres (Quebec, PQ 1987), p. 227-239, de Gruyter, Berlin, 1989. [7] Christian Friesen, Class number divisibility in real quadratic function elds, Canad. Math. Bull., Vol. 35(3), 1992, p. 361-370. [8] P. Hartung, Proof of the existence of in nitely many imaginary quadratic elds whose class number is not divisible by 3, J. Number Theory, 6 (1974), 276-278. [9] T. Honda, A few remarks on class numbers of imaginary quadratic elds, Osaka J. Math., 12 (1975), 19-21. [10] M. Ram Murty, The ABC conjecture and exponents of class groups of quadratic elds, Contemporary Mathematics, Volume 210, 1998, pages 85-95. [11] M. Ram Murty, Exponents of class groups of quadratic elds, (to appear).
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
9
die Klassenzahl imaginar quadratischer Zahlkorper, Abh. Math. [12] T. Nagell, Uber Seminar Univ. Hamburg, 1 (1992), p. 140-150. [13] P. Weinberger, Real Quadratic Fields with Class Numbers Divisible by , Journal of Number Theory, 5 (1973) p. 237-241. [14] Y. Yamaomoto, On unrami ed Galois extensions of quadratic number elds, Osaka J. Math., 7 (1970) 57-76. [15] Jiu-Kang Yu, Toward the Cohen-Lenstra conjecture in the function eld case, preprint. n
David A. Cardon, Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA
E-mail address :
[email protected]
M. Ram Murty, Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6, CANADA
E-mail address :
[email protected]
sions of the function eld Fq ( ) whose class groups have an element of a xed order. More precisely, let be a power of an1 odd prime and let be a 1 xed positive integer 3. There are `( 2 + g ) polynomials 2 Fq [ ] with deg( p ) such that the class groups of the quadratic extensions ) have an element of order . Fq ( T
q
g
q
D
T;
D
T
`
D
g
1. Introduction In a recent paper Murty [11] showed that if g is a xed integer 3 then the number of imaginary quadratic elds whose absolute discriminant is x and whose class group has an element of order g is x + g . He also showed that the number of real quadratic elds whose discriminant is x and whose class group has an element of order g is x g . In this paper we prove the analogous result for function elds rather than number elds in the analog of the imaginary quadratic case. The problem of divisibility of class numbers for number elds has been studied extensively. Gauss studied the case g = 2. The case g = 3 was studied by Davenport and Heilbronn [4]. For any g the in nitude of such elds was established by Nagell [12], Honda [9], Ankeny and Chowla [2], Hartung [8], Yamaomoto [14], and Weinberger [13]. Assuming the ABC Conjecture Murty [10] obtained a quantitative lower bound on the number of such elds. More recently Murty [11] improved the technique to give the result mentioned earlier without assuming the ABC conjecture. A conjecture of Cohen and Lenstra [3] predicts that as x increases a positive fraction of discriminants x produce quadratic extensions whose class number is divisible by any xed g. Interest in function elds was stimulated by the doctoral thesis of E. Artin [1] and the class number problem for function elds has been studied. For example, if g is not divisible by q then Friesen [7] constructed in nitely many polynomials M 2pF q [T ] of even degree such the class groups of the quadratic extensions F q (T; M ) of the function eld F q (T ) have an element of order g. Friedman and Washington [6] have studied the Cohen-Lenstra conjecture in the function eld case, and Yu [15] has established the Cohen-Lenstra 1 2
1 2
Research partially supported by NSERC.
1
1
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
2
conjecture as the characteristic p tends to in nity for xed discriminantal degree. We now state the main result of this paper: Theorem. Let q be a power of an odd prime and let g be ap xed positive integer 3. There are q`( + g ) quadratic extensions F q (T; D) of F q (T ) with deg(D) ` whose class group has an element of order g. The remainder of this paper will present the proof as a series of lemmas. The main outline is as follows. We show that if n and m are monic elements of F q [T ], if a 2 F q is not a square, if deg(mg )p> deg(n2 ), and if D = n2 ?amg is squarefree, then the class group of F q (T; D) has an element of order g. Using sieve methods and by letting m and n vary we are able to give a lower bound on the number of m and n such that D is squarefree. Finally, we show that as m and n vary there are relatively few duplicated values of D. 1 2
1
2. Preliminaries F q will denote the nite eld with q elements where q is a power of an odd prime. R = F q [T ] is the polynomial ring with coecients in F q over the indeterminate T and the function eld F q (T ) is the eld of fractions of R. We will assume that g is an odd integer that is relatively prime to q. The symbol p will always represent a monic irreducible polynomial in R. The symbols n and m will also be monic (but not necessarily irreducible) P polynomials in R of degrees j and k respectively. The expression m f (m) would mean to sum f (m) over all monic polynomials m of xed degree k. If a and b are elements of R, then (a; b) represents the greatest common (monic) divisor of a and b. If a and b are ordinary integers then (a; b) will denote the greatest common divisor in the usual sense. 3. Class groups with elements of order g In the following lemma we construct quadratic extensions of F q (T ) whose class groups contain elements of order g. Lemma 1. Let g be a positive integer 3. Assume n; m 2 R are monic, ?a 2 F q is not a square, deg(mg ) >pdeg(n2), and D = n2 ? amg is squarefree. Then the class group for F q (T; D) has an element of order g. Proof. First we note that (n; m) = 1 because if there were a common factor of n and m then D would not be squarefree. We factor amg as p p amg = n2 ? D = (n + D)(n ? D): p p p p Suppose I j(n + D) and I j(n ? D). Then n + D 2 I and n ? Dp2 I which implies p that n; D 2 I and I = R = F q [T ]. Thus the ideals (n + D) and (n ? D) are relatively prime. Therefore p p (m)g = (n + D)(n ? D) = ag a0 g
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
p
p
3
where a and a0 are ideals such that (n + D) = ag and (n ? D) = a0 g . Taking norms we nd that N (m) = q2 deg(m) = N (a)2 so that N (a) = qdeg(m) . Now suppose that ar is principal for some r < g: p ar = (u + v D ): Then p N (a)r = qr deg(m) = N (u + v D) = qdeg(u ?v D) 2
2
and because the leading coecient of v2 D is not a square this is
qdeg(D) = qdeg(n ?am ) = qdeg(m ) g
2
g
= N (a)g : This is a contradiction unless r = g.
4. How often is D = n2 ? amg squarefree? In light of Lemma 1 we would like to construct a lower bound on the number of squarefree expressions D = n2 ? amg as n and m vary such that deg(n) = j and deg(m) = k and deg(mg ) > deg(n2 ). Regarding k as the independent parameter we will maximize the number of possible values of D by choosing j to be the optimally large value j = bgk=2c if gk is odd or j = bgk=2c ? 1 if gk is even. Let s(h) be 1 or 0 according as h is squarefree or not. Also let( 2 sz (h) = 1 if d does not divide h whenever 1 deg(d) z 0 otherwise. We would like estimate the sum X s(n2 ? amg ): deg(m)=k deg(n)=j
Lemma 2. By counting expressions n2 ? amg that are squarefree in the small factors we obtain the following sieving inequality: X 2 X X X sz (n ? amg ) s(n2 ? amg ) sz (n2 ? amg ) ? 1: m;n
m;n
m;n
m;n;p deg(p)>z n ?amg 0(p2 ) 2
With an appropriate choice of z (depending on k) we will show that for large k #fdistinct squarefree values of n2 ? amg g X X s(n2 ? amg ) sz (n2 ? amg ) m;n qj+k :
m;n
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
4
Several auxiliary functions will be useful for estimating the terms in Lemma 2. De ne the Mobius function on the nonzero elements of R. If h 2 R has factorization ap 1 p t t where a 2 F q and the pi are irreducible monic polynomials in R then 8 > if h 2 F q ; (?1)t if i = 1 for all i; :0 otherwise. For z 1 let Y P (z) = p 1
irreducible p deg(p)z
and let
Nm;z (j ) =
X deg(n)=j
sz (n2 ? amg ):
For xed m; h 2 R let m (h) = #fn 2 R=hR : n2 ? amg 0(h)g: Thus m (h) is the number of n 2 R=hR satisfying the congruence n2 ?amg 0 (mod h). We will use the following elementary estimate several times: Lemma 3. If (u) represents the number of irreducible polynomials in Fq [T ] of degree u > 0, then (u) qu =u. P Proof. Since qu = dju d (d), the upper bound is clear.
Lemma 4.
m (d1 d2 ) = m (d1 )m (d2 ) if d1 and d1 are coprime. m (p2 ) = qdeg(p) if p is irreducible and p divides m. m (p2 ) 2 if p is irreducible and p does not divide m. m (d2 ) 2(d) qdeg(m) for squarefree d where (d) is the number of distinct monic irreducible polynomials of degree 1 that divide d. Proof. The multiplicativity of m is an immediate consequence of the Chi1. 2. 3. 4.
nese remainder theorem. Suppose that n satis es n2 ? amg 0(p2 ) with p dividing m. Then p divides n. There are exactly qdeg(p) multiples of p modulo p2 . Suppose that n satis es n2 ? amg 0(p2 ) but that p does not divide m. The solution n must be a `lift' of a solution modulo p. That is n = n1 + pt where n21 ? amg 0(p). We know there are at most two solutions of the congruence modulo p. Then 0 (n1 + pt)2 ? amg (n21 ? amg ) + 2n1 pt (mod p2 )
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
implies
5
0 n1 ?pam + 2n1 t (mod p): g
2
When p does not divide n and (2; q) = 1 there is a unique t (mod p) satisfying the last congruence. Thus the solution n1 (mod p) gives rise to a unique solution n (mod p2 ). Therefore, in this case, n (p) 2. Now let (d) represent the number of distinct nonconstant monic polynomials dividing d where d is squarefree. Then Y Y Y Y m (d) = m(p2 ) m (p2) qdeg(p) 2 2 (d) qdeg(m) : pjd pjm
pjd (p;m)=1
pjd pjm
pjd (p;m)=1
The following lemma tells us a choice of z that allows the sieve in Lemma 2 to yield interesting information. Lemma 5. Given any > 0 we can choose (independently of m) so that if z = log(k) then
Nm;z (j ) = qj
Y
(1 ? m (p2 )q? deg(p ) ) + O(q(1+)k ): 2
deg(p)z
P Proof. Let Nm;z (j ) = deg(n)=j sz (n2 ? amg ). Then X X X Nm;z (j ) =
n
d monic d2 j(n2 ?amg ;P (z))
If j deg(d2 ) then
X
Nm;z (j ) =
X
X n n2 ?amg 0(d2 )
1:
1 = m (d2 )qj ?deg(d ) ;
X deg(n)=j n2 ?amg 0(d2 )
Thus
d d2 jP (z)
(d)
2
deg(n)=j n2 ?amg 0(d2 )
while if j deg(d2 ) then
(d) =
1 m (d2 ):
(d)fm (d2 )qj?deg(d ) + O(m (d2 ))g 2
djP (z) Y X (1 ? m (p2 )q? deg(p2 ) ) + O(m (d2 )): = qj deg(p)z djP (z)
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
Now
X djP (z)
n (d2 ) qdeg(m) = qdeg(m)
X djP (z)
6
2 (d)
Y
deg(p)z deg( m ) q 3qz :
(1 + 2)
Given any > 0 we can choose such that if z = log(k) then the last expression is bounded by qk for suciently large k. Therefore for suciently large k we have Y (1 ? m (p2 )q? deg(p ) ) + O(q(1+)k ): Nm;z (j ) = qj 2
deg(p)z
Lemma 6. We have X the 2lower bound X sz (n ? amg ) = Nm;z (j ) qj+k : m;n
m
Proof. We notice that Y Y Y (1 ? m (p2 )q? deg(p ) ) = (1 ? q? deg(p) ) (1 ? m (p2 )q? deg(p ) ) 2
2
deg(p)z
Then we sum m X overY
X X
Y
pjm deg(p)z
Y
pjm
(1 ? q? deg(p) )
(1 ? q? deg(p) ) =
(p;m)=1 deg(p)z
Y
(1 ? 2q? deg(p ) ) 2
all p
X djm
(d)q? deg(d) :
(1 ? m (p2 )q? deg(p ) ) 2
deg(m)=k deg(p)z
pjm deg(p)z
(d)q? deg(d) =
deg(m)=k djm X (d)q?2 deg(d) = qk deg(d)k
X
(d)q? deg(d) qk?deg(d)
deg(d)k = qk f(1 ? q?1 ) + O(q?k )g qk :
Summing the expression in Lemma 5 as m varies P such that deg(m) = k and apply the last inequality gives the lemma: m Nm;z (j ) qj +k . Lemma 7. Pm (m) log(k)qk Proof. X X k?deg(p) k X ?u qu (m) q q q u log(k)qk : m
p deg(p)k
uk
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
Lemma 8.
X
Proof. We may write
m;n;p deg(p)>z n2 ?amg 0(p2 )
X
m;n;p deg(p)>z 2 n ?amg 0(p2 )
where
1=
Mm;p(j ) =
7
1 = o(qi+j ):
X X m
p deg(p)>z
Mm;p (j )
X n n2 ?amg 0(p2 )
1:
Because Mm;p (j ) = m (p2 )qj ?deg(p ) if j deg(p2 ) and Mm;p (j ) m (p2 ) if j < deg(p2 ) we obtain an upper bound on Mm;p(j ) ( j?deg(p ) Mm;p(j ) 2(j?q deg(p) + 1) if (p; m) = 1, q if pjm. Summing over irreducible p results in X X X j?deg(p) Mm;p (j ) 2(qj ?deg(p ) + 1) + q 2
2
2
z<deg(p)j
z<deg(p)j (p;m)=1 qj?z qj
z<deg(p)j pjm
z + j + (m)qj?z :
Then summing the last expression over m gives
XX m p
j +k?z
j +k
X
Mm;p (j ) q z + q j + qj?z (m) m j + k j + k ? z q z + q j + log(k)qj+k?z k) qj+k ( zq1z + 1j + log( qz ) = o(qj +k ):
We have now shown (Lemmas 2, 6, and 8) that the number of squarefree values of n2 ? amg as m and n vary is X 2 s(n ? amg ) qj+k : m;n
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
8
It remains to be shown that there is not too much duplication among the expressions n2 ? amg . Lemma 9. The number of squarefree elements of the form n2 ? amg with deg(n) = j and deg(m) = k that are representable in more than one way is o(qj+k ). Proof. Let m1 and m2 be xed unequal polynomials such that D = n21 ? amg1 = n22 ? amg2 for some m1 and m2 . Then a(mg1 ? mg2 ) = n21 ? n22 = (n1 ? n2 )(n1 + n2 ): For xed m1 and m2 the choices for n1 and n2 are derived from the divisors of a(mg1 ? mg2 ) which cannot exceed deg(mg1 ? mg2 ) gk. The number of possible values of m1 and m2 is O(q2k ) and therefore the total number of squarefree elements of the form n2 ? amg that are duplicated is O(gkq2k ) which is o(qj +k ). We have now shown that there are qj +k distinct values of D = n2 ?amg . Since j = bgk=2c or j = bgk=2c ? 1 there are qgk( + g ) distinct p values of + ) ` ( g quadratic extensions F q (T; D) of F q (T ) D. Therefore there are q such that deg(D) `. This completes the proof of the theorem stated at the beginning. Acknowledgment. We wish to thank Yu-Ru Liu for her willingness to go over the details of this paper with the rst author. 1 2
1 2
1
1
References [1] E. Artin, Quadratische Korper im Gebiet der hoheren Kngruenzen I, II, Math. Zeitschrift 19 (1924) 153-246. [2] N. Ankeny and S. Chowla, On the divisibility of the class numbers of quadratic elds, Paci c Journal of Math., 5 (1955) p. 321-324. [3] H. Cohen and H.W. Lenstra Jr., Heuristics on class groups of number elds, Springer Lecture Notes, 1068 in Number Theory Noordwijkerhout 1983 Proceedings. [4] H. Davenport and H. Heilbronn, On the density of discriminants of cubic elds, II, Proc. Royal Soc., A 322 (1971), p. 405-420. [5] R. Gupta and M. Ram Murty, Class groups of quadratic functions elds, in preparation. [6] Eduardo Friedman and Lawrence C. Washington, On the distribution of divisor class groups of curves over nite elds, in Theorie des nombres (Quebec, PQ 1987), p. 227-239, de Gruyter, Berlin, 1989. [7] Christian Friesen, Class number divisibility in real quadratic function elds, Canad. Math. Bull., Vol. 35(3), 1992, p. 361-370. [8] P. Hartung, Proof of the existence of in nitely many imaginary quadratic elds whose class number is not divisible by 3, J. Number Theory, 6 (1974), 276-278. [9] T. Honda, A few remarks on class numbers of imaginary quadratic elds, Osaka J. Math., 12 (1975), 19-21. [10] M. Ram Murty, The ABC conjecture and exponents of class groups of quadratic elds, Contemporary Mathematics, Volume 210, 1998, pages 85-95. [11] M. Ram Murty, Exponents of class groups of quadratic elds, (to appear).
EXPONENTS OF CLASS GROUPS OF QUADRATIC FUNCTION FIELDS
9
die Klassenzahl imaginar quadratischer Zahlkorper, Abh. Math. [12] T. Nagell, Uber Seminar Univ. Hamburg, 1 (1992), p. 140-150. [13] P. Weinberger, Real Quadratic Fields with Class Numbers Divisible by , Journal of Number Theory, 5 (1973) p. 237-241. [14] Y. Yamaomoto, On unrami ed Galois extensions of quadratic number elds, Osaka J. Math., 7 (1970) 57-76. [15] Jiu-Kang Yu, Toward the Cohen-Lenstra conjecture in the function eld case, preprint. n
David A. Cardon, Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA
E-mail address :
[email protected]
M. Ram Murty, Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6, CANADA
E-mail address :
[email protected]
