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The number of $S


arXiv:math/0411484v3 [math.NT] 27 Jul 2005

¨ ¨ JURGEN KLUNERS Abstract. We prove that the number of quartic S4 –extensions of the rationals of given discriminant d is O? (d1/2+? ) for all ? > 0. For a prime number p we derive that the dimension of the space of octahedral modular forms of weight 1 and conductor p or p2 is bounded above by O(p1/2 log(p)2 ).

1. Introduction For a number ?eld k we denote by dk ∈ N the absolute value of the ?eld discriminant of k. The class group will be denoted by Clk and the p-rank rkp (A) of an abelian group A is de?ned to be the minimal number of generators of A/Ap . We denote by N the absolute norm. The symbol O? denotes the usual Landau symbol O, where the implied constant is depending on ?. In this note we answer a question of Akshay Venkatesh about the number of S4 – extensions of degree 4 with given discriminant d. It is conjectured that this number is O? (d? ) for all ? > 0. In average we have the stronger result (see [Bha02, Bel04]): 1 1 = c(S4 ), lim x→∞ x
K:dK ≤x

where K runs through all quartic S4 –extensions and c(S4 ) > 0 is explicitly given. We prove the bound O? (d1/2+? ) for all ? > 0 which improves the bound O? (d4/5+? ) given in [MV02]. As an application we give an upper bound for the dimension of the space of octahedral forms of weight 1 and given conductor N . In the general case the best known bound is O? (N 4/5+? ) for all ? > 0 given in [MV02]. For squarefree conductors this bound is improved to O? (N 2/3+? ) on average. In this note we are able to prove the upper bound O? (N 1/2+? ) in many cases, e.g. when N is prime or a square. The discrepancy between the expected bound O? (d? ) and the proven bound O? (d1/2+? ) for the number of S4 –extensions of discriminant d comes from the fact that we can only use weak bounds for the 3–rank of the classgroup of quadratic ?elds and the 2-rank of the classgroup of non-cyclic cubic ?elds. In order to understand the problems which arise we give the following easy example. Let us count the number of cubic S3 –extensions M/Q of discriminant d such that the normal closure contains a given quadratic extension k. Since every unrami?ed cyclic cubic extensions N/k corresponds to a cubic extension M we see that the number of elements h3 of order 3 in the classgroup Clk plays an important role. In the general case we can only use the estimate h3 ≤ # Clk and the latter one can be bounded by O(d1/2 log(d)) using Lemma 2. It is very di?cult to improve this trivial bound for elements of order p in the class group when p > 3. Just recently for p = 3 Helfgott and Venkatesh [HV04] (λ = 0, 44179) and independently Pierce
1991 Mathematics Subject Classi?cation. Primary 11R29; Secondary 11R16, 11R32.



[Pie05] (λ = 0.49108 or λ = 0.41667 in special cases) proved that for all ? > 0 we get: 3rk3 (Clk ) = O? (dλ+? ). k Using this improved bound it is straightforward to get the upper bound O? (dλ+? ) for the number of cubic S3 –extensions. In the following we would like to explain the idea of the proof of our main result. We will improve the following elementary approach given in [Duk95, p. 101]. In the worst case we cannot exclude the case that there exists a quadratic ?eld k/Q such 1/2 that 3rk3 (Clk ) = O(dk log(dk )). Using these unrami?ed C3 –extensions of k there rk3 (Clk ) are 3 3?1 ?1 non-cyclic cubic ?elds M of the same discriminant. In the worst case all these extensions have a large 2–rank, i.e. 2rk2 (ClM ) = O(dM log(dM )2 ). Every unrami?ed C2 -extension leads to an S4 -extension K of degree 4 of the same discriminant dK = dk = dM . Using this idea we get the upper bound O(dK log(dK )3 ) for the number of S4 –extensions of discriminant dK . As we see from the above example it is a problem for our upper estimates when rk2 (ClM ) and rk3 (Clk ) are big. We will use Theorem 1 proved by Frank Gerth III which says that rk3 (ClM ) has about the same size as rk3 (Clk ). This means that rk3 (ClM ) is big when rk3 (Clk ) is big. This implies that rk2 (ClM ) must be small. E.g. consider the special case that d is squarefree, i.e. the corresponding S3 – extension L is unrami?ed over k. Then the ?rst part of Theorem 1 and the above explained elementary approach already proves our wanted result, i.e. the number of S4 –extensions of disriminant d is bounded by O(d1/2+? ). 2. Parameterizing S4 –extensions Let K/Q be a quartic ?eld such that the normal closure N has Galois group S4 . Then there is a unique normal sub?eld L of degree 6 having Galois group S3 . We denote by M a sub?eld of L of degree 3 and by k the unique sub?eld of degree 2 of L (or N ). N
4 1/2


2 ? ? ?

e e

e e


M e e


S3 e


k ? e e 2? e ? Q

? ? S4

e e K ? ?

For n ∈ N we de?ne Rad(n) :=

p, where the product is only taken over primes.

To each K/Q as above we associate a triple (a, b, c) = (Rad(dk ), Rad(N (dL/k )), Rad(N (dN/L ))) ∈ N3



of squarefree numbers. We de?ne (1) Ψ : K → N3 , K → (a, b, c),

where K is the set of quartic S4 –extensions of Q up to isomorphy. Ψ is a well de?ned mapping with bounded ?bers. In the rest of this section we want to give upper bounds for the size of the ?bers, i.e. to give an upper bound for the numbers of ?elds K which are associated to a given triple (a, b, c)? Assuming√ this situation k is one of the following quadratic ?elds. If 2 ? a we get that k = Q( ±a) where the sign is positive if a ≡ 1 mod 4. If 2 | a then k is one √ √ of the following three ?elds: Q( a), Q( ?a), and Q( ±a/2), where the sign is positive when a/2 ≡ 3 mod 4. Therefore at most 3 quadratic ?elds are associated to a given a. The number of b’s for a given ?eld k can be easily bounded by the following lemma. In the following we denote by ω(b) the number of prime factors of b. Lemma 1. Let b ∈ N as above. Then all ?elds M (up to isomorphism) such that L/K is only rami?ed in primes dividing b are contained in the ray class ?eld of a := 3bOk . The number of those extensions can be bounded by 3r ? 1 , where r = rk3 (Clk ) + ω(b) + 2. 3?1

Proof. We are looking for all ?elds which are at most rami?ed in primes dividing b. We need to choose a in such a way that all these ?elds are sub?elds of the ray class ?eld of a. For primes p not dividing 3 it is su?cient that p | a. For the wildly rami?ed primes there exists a maximal exponent such that all these ?elds occur as sub?elds [Ser95, p. 58] of the ray class ?eld of a. Using elementary properties of the ray class group Cla we get that For all prime ideals p not dividing 3 we get that the 3-rank of (Ok /p)? is at most 1 which shows that rk3 (Ok /pOk ) ≤ 2. Equality can only occur in the case p ≡ 1 mod 3, where p ∈ P ∩ p. In this case there exists a C3 –extension of Q only rami?ed in p. Denote by A the 3–part of the ray class group Cla . We can write A := A+ ⊕ A? , where the classes in A+ are invariant under Gal(k/Q). Because a prime p ≡ 1 mod 3 increases the 3–rank of A+ by one, we get that all odd primes increase the 3–rank of A? by at most one. The theory used in [KF03, Section 6] shows that S3 –extensions correspond to quotients of index 3 of A? . Finally we need to estimate the 3–rank for Ok /pw for primes dividing 3. In [HPP03] it is proved that the p-rank of (Ok /pw )? is at most [kp : Qp ] + 1. In all cases it is su?cient to add 2 since there is one C3 -extension of Q only rami?ed in 3. We use the trivial class group bound which can be found in [Nar89, Theorem 4.4]. Lemma 2. For all n ∈ N there exists a constant c(n) such that for all number ?elds F of degree n we have: | ClF | ≤ c(n)dF log(dF )n?1 . Trivially, we have 3rk3 (Clk ) ≤ | Clk |. For a given cubic S3 -?eld M we prove a similar lemma as Lemma 1.

rk3 (Cla ) ≤ rk3 (Clk ) + rk3 ((Ok /a)? ).



Lemma 3. Let c ∈ N be as above. Then the number of S4 -extensions N which contain a given S3 -?eld M such that N (dN/L ) is only divisible by primes dividing c is bounded by 2r ? 1, where r = rk2 (ClM ) + 3ω(c) + 6. √ Proof. In [Bai80, Lemmata 4,5] it is proven that the Galois closure of M ( α) for α ∈ M has Galois group S4 if and only if N (α) is a square. If N (α) is a square this certainly implies that the norm of the principal ideal (α) is a square. Therefore we get an upper bound if we count all extensions such that the conductor is a square. For a prime p = 3 we have at most three possibilities to produce squarefree ideals of norm p2 . The 6 is computed in a similar way as in Lemma 1 and gives an upper bound for the contribution of primes above 3. Altogether we get the following upper bound for the number of S4 -?elds associated to a given triple (a, b, c): 3 r1 ? 1 r2 )(2 ? 1) ≤ 3/2 · 9 · 26 3rk3 (Clk ) 2rk3 (ClM ) 3ω(b) 8ω(c) , (2) 3( 3?1 where r1 = rk3 (Clk ) + ω(b) + 2, r2 = rk3 (ClM ) + 3ω(c) + 6. The following theorem relates the 3–parts of the classgroups of k and M . Theorem 1. (Gerth III) Let M/Q be a non-cyclic cubic extension and denote by L the normal closure of M and by k the unique quadratic sub?eld of L. Then the following holds. (i) If L/k is unrami?ed, then rk3 (ClM ) = rk3 (Clk ) ? 1. (ii) rk3 (ClM ) = rk3 (Clk ) + t ? 1 ? z ? y, where y ≤ t ? 1 and t is the number of prime ideals of Ok which ramify in L. Furthermore we have 0 ≤ z ≤ u where u is the number of primes which are totally rami?ed in M but split in k. (iii) rk3 (ClM ) ≥ rk3 (Clk ) ? u Proof. The ?rst part is Theorem 3.4 in [Ger76]. The second part is Theorem 3.5. The last part is an immediate consequence. Since we are only interested in the asymptotic behaviour we can ignore rami?cation in 2 and 3. Therefore we de?ne S := {2, 3} and aS to be the largest number dividing a which is coprime to S. Using this we easily see that dS = aS (bS )2 , M where M is one of the cubic extensions constructed above. Using Theorem 1 we get the following estimate for 3rk3 (Clk ) 2rk3 (ClM ) . Lemma 4. Let M, k be the ?elds de?ned before. Then there exists a constant C > 0 such that 3rk3 (Clk ) 2rk2 (ClM ) ≤ Ca1/2 b log(ab2 )2 3ω(b) . Proof. Theorem 1 shows rk3 (ClM ) ≥ rk3 (Clk ) ? ω(b). Therefore we get: Using Lemma 2 and the fact that dS = (ab2 )S di?ers from dM by something which M can be bounded by a constant we get the desired bound. Combining Lemma 4 and (2) we deduce the following corollary. Corollary 1. The number of elements of the ?ber Ψ?1 (a, b, c) is bounded by 33 25 Ca1/2 b log(ab2 )2 9ω(b) 8ω(c) . 3rk3 (Clk ) 2rk2 (ClM ) ≤ 3rk3 (ClM ) 3ω(b) 2rk2 (ClM ) ≤ 3ω(b) | ClM |.



3. Upper bounds for quartic S4 –extensions with given discriminant In this section we prove an upper bound for the number of quartic S4 –extensions with given discriminant. In order to do this we need to compute the discriminant dK using the triple (a, b, c). In a second step we determine how many triples may lead to the same discriminant. Let us assume that we have given a ?eld K ∈ K with Ψ(K) = (a, b, c) rami?ed in p. Assuming p = 2, 3 we can compute the cycle shape of a generator of the cyclic inertia group at p in the degree 4 representation of S4 . Here we denote by the cycle shape the length of the cycles if we decompose a group element into disjoint cycles. Using local theory we get for primes p > 3 the following identities, where vp denotes the ordinary p-valuation. The results are given in the following table: cycle shape vp (dK ) p | a, p ? bc 12 2 1 p | a, p | c, p ? b 4 3 p | b, p ? ac 13 2 p | c, p ? ab 22 2 The other cases cannot occur since in these cases the inertia group would not be cyclic. The cases p = 2 or p = 3 can be handled by analyzing the local Galois groups. We still use the de?nition aS for S := {2, 3} from the preceding section and get: dS = aS (bS )2 (cS )2 . K The contribution of the primes 2 and 3 is bounded by a constant factor. Therefore we ignore these primes in the following. Using the results of the preceding section it remains to count the number of triples (a, b, c) which may lead to the same discriminant. In the following let d be a discriminant of a quartic S4 –extension, Theorem 2. Let d = 2e2 3e3 d1 d2 d3 such that 6d1 d2 d3 is squarefree. Then the 2 3 number of S4 -?elds with discriminant d is bounded above by ? ? (i) C(d1 d3 )1/2 d2 log(d1 d3 d2 )2 18ω(d2) 8ω(d3 ) for a suitable C > 0. 2 1/2+? (ii) O? (d ) for all ? > 0. Proof. Using the above discussion all ?elds K/Q with Ψ(K) = (a, b, c) have the property: aS = d1 d3 , d3 | cS and (bc)S = d2 d3 .

Therefore we have 2ω(d2 ) possibilities for choosing bS . The number of possibilities for the 2 and the 3–part can be bounded by a constant. Using Corollary 1 the worst ? case is when bS = d2 and therefore we get for some computable constant C > 0 ? C2ω(d2 ) (d1 d3 )1/2 d2 log(d1 d3 d2 )2 9ω(d2 ) 8ω(d3 ) 2

as an upper bound. For the second statement we write xω(d) = O(d? ) for a given number x and get the desired result. Remark 1. For squarefree discriminants d we can derive the better upper bound O(d1/2 log(d)2 ). We can combine this result with well known results to get bounds for degree 4 ?elds.



Theorem 3. The number of degree 4 ?elds of given discriminant d is bounded above by O? (d1/2+? ) for all ? > 0. Proof. Using the theorem of Kronecker-Weber we easily get that the number of ?elds with Abelian Galois group is bounded by O? (d? ) for all ? > 0. Since D4 ?elds can be constructed by quadratic extensions over quadratic extensions and the 2-torsion part of the class group can be easily controlled, we get the same result for D4 -extensions. For A4 -extensions we use the same approach as in the S4 case. The main di?erence is that we have only one step where we have to consider class groups. This gives O? (d1/2+? ) for the number of such extensions with given discriminant d. Using more advanced methods [MV02] this number can be reduced to O? (d1/3+? ).

4. Upper bounds for the dimension of the space of octahedral modular forms of given conductor In this section we give upper bounds for the dimension of the space of octahedral ? modular forms of weight 1. Denote by GQ the absolute galois group Gal(Q/Q). Suppose we have given a quartic S4 –extension K/Q which gives rise to a projective representation ρ : GQ → PGL2 (C). The conductor of this projective representation ? is de?ned to be the product of the local conductors of ρ|GQp : GQp → PGL2 (C) ? ? which is the minimal p–power of a so–called local lift, see e.g. [Ser77, §6] or [Won99] for more details. In this section we count S4 –extensions using the above de?ned conductor. A prime p divides the conductor if and only if p divides the discriminant. To simplify all computations we ignore the contribution of the 2– and 3–part of the conductor. For all other (tamely) rami?ed primes we have the property that p exactly divides the conductor when the local Galois group is cyclic. Otherwise the local Galois group is dihedral and we get that p2 exactly divides the conductor [Won99, Prop. 1, p. 144]. To each projective representation with image S4 we can associate an octahedral modular form of the same conductor. This means that we get the corresponding bounds for the modular forms when we compute the bounds for the number of projective representations (see e.g. [Duk95, Won99] for more details). In order to use the results of Section 2 we need to compute the conductor of the associated modular form only using the triple (a, b, c). Similar to the discriminant case we can do all computations locally. In the discriminant case it was only important to know the inertia group. Now it is important to know the decomposition group. Let p > 3 be a divisor of abc. Then p exactly divides the conductor if the decomposition group is cyclic. In the following table we collect the information we r get (for p > 3) using the prime ideal factorization pOK = i=1 pei . We remark i that some cases can be distinguished by congruence conditions. In the last column we denote the letters which are divisible by p. The information vp (d) on the discriminant is not needed in this section.



p2 p2 p3 1 p2 p2 1 p2 1 p2 p2 1 2 p4 1 p4 1 p3 p2 1 p3 p2 1

Dp C2 C2 × C2 C2 × C2 or C4 C2 × C2 or C2 D4 C4 C3 D3

Ip C2 C2 C2 C2 C4 C4 C3 C3

vp (N ) 1 2 2 or 1 2 or 1 2 1 1 2

vp (d) 1 1 2 2 3 3 2 2

Let K/Q be a quartic S4 –extension with associated triple (a, b, c) and conductor N . Then we write a = a1 a2 , b = b1 b2 , c = c0 c1 c2 , where c0 := gcd(a, c) such that N S = (a1 a2 b1 b2 c1 c2 )S . 2 2 2 Since gcd(b, ac)S = 1 we easily see that aS , aS , bS , bS , cS , cS are pairwise coprime. 1 2 1 2 1 2 Using the above table we know that bi is (up to the 3–part) exactly divisible by the primes dividing b which are congruent to i mod 3 (i = 1, 2).
2 Theorem 4. Let N = 2n2 3n N1,1 N1,2 N2 such that 6N1,1 N1,2 N2 is squarefree. Fur3 thermore we assume that p | N1,i if and only if p ≡ i mod 3 (i = 1, 2). Then the number of S4 –?elds of given conductor N is bounded above by

p ≡ 3 mod 4 p ≡ 1 mod 4 p ≡ 1 mod 3 p ≡ 2 mod 3

p| a a c c a, c a, c b b

C54ω(N ) N1,1 N1,2 N2 log(N )2 for a suitable C > 0. Proof. We have 3ω(N ) possibilities to partition the primes into three sets corresponding to a, b, c. Furthermore we have at most 2ω(N ) possibilities for c0 . Using Corollary 1 we have the worst case when b is big. Primes dividing N1,2 cannot divide b. Here we get the worst case when these primes divide a. Therefore we get as an upper bound:
1/2 2 2 ? C3ω(N ) 2ω(N ) N1,1 N1,2 N2 log(N1,2 N1,1 N2 )2 9ω(N ) .


We easily get the desired result. To get good estimates for the dimension of the space of octahedral forms with given conductor we have to avoid that b1 is big. Using this we can derive the following corollaries. Corollary 2. Let p be a prime. Then the dimension of the space of octahedral modular forms of weight 1 and conductor p or p2 is bounded above by O(p1/2 log(p)2 ). Proof. The quadratic subextension must be rami?ed in at least one prime. Therefore p | a for all possible triples. Corollary 3. Assume that all primes which exactly divide N are congruent to 2 mod 3. Then the dimension of the space of octahedral forms of weight 1 and conductor N is bounded above by O(N 1/2+? ) for all ? > 0. Proof. We have N1,1 = 1 and the assertion follows. This improves the bound O(N 4/5+? ) given in [MV02]. We remark that in the case that b1 resp. N1,1 is big we only get the trivial linear bound using our method.



Acknowledgments I thank Karim Belabas and Gunter Malle for fruitful discussions and reading a preliminary version of the paper. References
Andrew Marc Baily, On the density of discriminants of quartic ?elds, J. Reine Angew. Math. 315 (1980), 190–210. [Bel04] Karim Belabas, Param?trisation de structures alg?briques et densit? de discriminants e e e [d’apr`s bhargava]., Seminaire Bourbaki 56eme annee (2004), no. 935. e [Bha02] Manjul Bhargava, Gauss composition and generalizations, Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 1–8. MR MR2041069 [Duk95] William Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices 2 (1995), 99–109. [Ger76] Frank Gerth, III, Ranks of 3-class groups of non-Galois cubic ?elds, Acta Arith. 30 (1976), no. 4, 307–322. [HPP03] Florian Hess, Sebastian Pauli, and Michael E. Pohst, Computing the multiplicative group of residue class rings, Math. Comput. 72 (2003), no. 243, 1531–1548. [HV04] Harald Helfgott and Akshay Venkatesh, Integral points on elliptic curves and 3-torsion in class groups, arXiv:math.NT/0405180, 2004. [KF03] J¨rgen Kl¨ners and Claus Fieker, Minimal discriminants for small ?elds with Frobenius u u groups as Galois groups, J. Numb. Theory 99 (2003), 318–337. [MV02] Philippe Michel and Akshay Venkatesh, On the dimension of the space of cusp forms associated to 2-dimensional complex Galois representations, Internat. Math. Res. Notices 38 (2002), 2021–2027. [Nar89] Wladislaw Narkiewicz, Elementary and analytic theory of algebraic numbers, Springer, 1989. [Pie05] Lillian Pierce, The 3-part of class numbers of quadratic ?elds, J. London Math. Soc. (2) 71 (2005), no. 3, 579–598. [Ser77] Jean-Pierre Serre, Modular forms of weight one and Galois representations, Algebraic number ?elds: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, pp. 193–268. , Local ?elds, Springer, New York, 1995. [Ser95] [Won99] Siman Wong, Automorphic forms on GL(2) and the rank of class groups, J. Reine Angew. Math. 515 (1999), 125–153. E-mail address: klueners@mathematik.uni-kassel.de ¨ Universitat Kassel, Fachbereich Mathematik/Informatik, Heinrich-Plett-Str. 40, 34132 Kassel, Germany. [Bai80]


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