Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants

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1 Bhargava, M. (2007) Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants, International Mathematics Research Notices, Vol. 2007, Article ID rnm052, 20 ages. doi:0.093/imrn/rnm052 Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants Manjul Bhargava Princeton University Corresondence to be sent to: Using Serre s mass formula [5] for totally ramified extensions, we derive a mass formula that counts all (isomorhism classes of) étale algebra extensions of a local field F having a given degree n. Alongtheway, we also rove a series of mass formulae for counting étale extensions of a local field F having certain roerties, such as a chosen rime slitting or ramification behavior. We then use these mass formulae to formulate a heuristic that redicts the asymtotic number of number fields, of fixed degree n and bounded discriminant, whose Galois closures have Galois grou the full symmetric grou S n over Q. Analogous redictions are made for the asymtotic density of those number fields having secified local behaviors at finitely many laces. All these redictions are in full agreement with the known results in degrees u to five. Introduction Let N n (X) denote the number of degree n number fields having absolute discriminant at most X and whose Galois closure has Galois grou S n. It is an old and well-known Received October 3, 2006; Acceted June 4, 2007 Communicated by Morris Weisfeld See htt:// journals/imrn/ for roer citation instructions. Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208 c The Author Published by Oxford University Press. All rights reserved. For ermissions, lease journals.ermissions@oxfordjournals.org.

2 2 ManjulBhargava conjecture that the limit N n (X) c n = lim X X (.) exists and is ositive for n >. The conjecture has recently been roven for n 5 (see [], [3], [4]). Forn > 5, however, numerical and theoretical evidence for the conjecture has been scant, and as far as I am aware no values for c n have been enunciated, even conjecturally, for general n. The urose of this note is to resent heuristics for the exected number of S n -number fields of given degree n and discriminant D. These heuristics allow us to derive exlicit values for the constants c n for all n. As evidence of their correctness, these values agree with the values of c n already established for n 5. In general, the values of c n are seen to take the form of certain Euler roducts whose comlexities increase with n. The first 0 values of c n are rovided in Table. At the heart of our heuristics is the derivation of a mass formula that counts all étale extensions of a local field. This counting formula generalizes, and indeed uses in Table Conjectured values of c n for n 0. n c n ζ(2) ζ(3) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

3 Mass Formulae for Extensions of Local Fields 3 an essential way, the remarkable mass formula for totally ramified extensions of a local field due to Serre [5]. Given a finite extension K of Q having nonzero discriminant, let us use Disc (K) to denote the highest ower v(disc(k)) of dividing the discriminant of K. Then Serre showed that [K:Q ]=n totally ramified Disc (K) #Aut(K) =. (.2) n In Section 2, we show how to use Serre s mass formula to rove analogous mass formulae for all étale extensions of Q, and for étale extensions of Q having any desired slitting/ramification behavior. For examle, our total mass formula for all étale extensions of Q is given by the following theorem. Theorem.. We have [K:Q ]=n étale Disc (K) #Aut(K) = n k=0 q(k, n k) k, (.3) where q(k, n k) denotes the number of artitions of k into at most n k arts. In Section 2, we also rove analogous mass formulae for secial clusters of étale extensions having certain tyes of fixed slitting or ramification behavior (see Proositions 2. and 2.2). These local mass formulae, together with an assumtion about indeendence of local laces, lead us to heuristics with a great deal of redictive ower regarding the discriminant distribution of global fields. To state the general formula for the limit (.), we require again the artition function. Let q(n, k) denote the standard restricted artition function, i.e., q(n, k) gives the number of artitions of n into at most k arts. Also, we use r 2 (S n ) to denote the number of 2-torsion elements in the symmetric grou S n. Then our heuristics imly: Conjecture.2. Let N n (X) denote the number of S n -number fields of degree n having absolute discriminant at most X.Then N n (X) lim X X = r 2(S n ) 2 n! ( n k=0 ) q(k, n k) q(k, n k + ) k. Our heuristics also have a number of other interesting consequences. Suose K is an S n -number field of degree n that is unramified at a given rime, and let K denote Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

4 4 ManjulBhargava the Galois closure of K. Then the Artin symbol ( K/) is defined as a conjugacy class in S n. The Chebotarev density theorem states that, for a fixed S n -number field K of degree n, the density of rimes for which the Artin symbol ( K/) equals a given conjugacy class is roortional to the size of the conjugacy class. Now suose instead of varying the rime and fixing the field K, we fix the rime and vary the field K. What then do we exect about the distribution of the Artin symbol ( K/)? Our heuristics redict the following comlement to Chebotarev density: Conjecture.3. Let be a fixed rime, and let K run through all S n -number fields of degree n in which does not ramify, the fields being ordered by the size of the discriminants. Then the Artin symbol ( K/) takes the value χ S n with a frequency roortional to the size of the conjugacy class χ. In fact, our heuristics also allow us to redict the density of discriminants of S n -fields K of degree n for which K has any of the various ossible ramification tyes at. Inarticular, we may describe the robability that a rime ramifies in a random S n -number field of degree n. Conjecture.4. When ordered by absolute discriminant, the roortion of degree n, S n -number fields that ramify at is given by n k= n q(k, n k) n k. q(k, n k) n k k=0 For n =, 2, 3, 4, and 5, this uts the robabilities of ramification at 0, , and , ,, resectively. As evidence for the correctness of these heuristics, we deduce the following from results of [], [3], [4]: Theorem.5. Conjectures.2.4 are trueforn 5. The heuristics have a number of further consequences (also now roven for n 5) which we will describe in Section 5. For examle, these heuristics redict the roortion of S n -fields of degree n that are totally real, or have ositive (or negative) discriminant, or have secified comletions at any of a finite number of laces. The organization of this article is as follows. In Section 2, we develo mass formulae for étale extensions of a local field, utilizing Serre s mass formula as the basic ingredient. In Section 3, we lay down the critical assumtions behind our heuristics, Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

5 Mass Formulae for Extensions of Local Fields 5 which reduces the evaluation of c n to certain sums which are conveniently related to the mass formulae of Section 2. These sums are evaluated in Section 4. In Section 5, Conjectures.2.4 and some other related conjectures are then derived under these hyotheses. In Section 6, we formulate more recise conjectures for individual discriminants. In Section 7, we study more closely the relationshi between the constants c n and the artition function q(n, k). Finally, in Section 8 we end by examining the situation for other Galois grous. Our methods may be reminiscent of the class grou heuristics of Cohen and Lenstra [8] and Cohen and Martinet [9]; indeed, it should now be ossible to formulate very general conjectures that naturally incororate both their conjectures and the conjectures laid out here. 2 Extensions of Serre s mass formula In 978, Serre [5] roved the remarkable mass formula (.2) for totally ramified extensions K of the local field Q. The urose of this section is to extend Serre s formula to other ossible slitting and ramification behaviors for K/Q, and in articular, to obtain such a counting formula for all étale extensions of Q. To state these slightly more general mass formulae, we need a bit of notation. Given an étale extension K of Q (res. Q ), let us define the slitting symbol (K, ) by (K, ) = (f e fe r r ) if the ideal () in O K factors into rime ideals as () = P e Pe r r with O K /P i = f i, (2.) where O K denotes the integral closure of Z (res. Z ) in K. LetT n denote the set of all ossible symbols (K, ) for a degree n étale extension of Q (res. Q ). For examle, we have T 3 = {(), (2), (3), ( 2 ), ( 3 )}. For σ = (f e fe r r ), define Disc (σ) in the natural way by Disc (σ) = f i (e i ). (Note that this agrees with the usual field discriminant at when is tamely ramified, but not when is wildly ramified.) Furthermore, define #Aut(σ) to be r i= f i times the number of ermutations of the factors f e i i that reserve σ. For examle, #Aut( ) = ( ) ( 2 6 ) = Then our analogue of Serre s mass formula for an arbitrary slitting symbol is as follows. Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

6 6 ManjulBhargava Proosition 2.. (Mass formula for extensions of a fixed slitting tye) Let σ T n be any slitting symbol of degree n.then [K:Q ]=n étale (K,) = σ Disc (K) #Aut(K) = Disc (σ) #Aut(σ). (2.2) Proof. Note that an étale Q -algebra K of degree n satisfies (K, ) = σ = (f e f e r r ) recisely when K = K K r, where K i is a field extension of Q satisfying (K i, ) = (f e i i ). Counting isomorhism classes of such algebras K, weighted by #Aut(K), is equivalent to counting such ordered direct sums K K r, weighted by #Aut(K ) #Aut(K r ) N, where N denotes the number of ermutations of the factors f e i i reserving σ. We claim that the total number of fields K i /Q having degree e i f i (counted with weights Disc (K i ) #Aut(K i ) ), with inertial degree f i and ramification index e i, is given by f i (e i ) f i. This is because any Q has a unique unramified extension F of degree f i, and the number of isomorhism classes of totally ramified extensions K i of F of degree e i, where each extension is weighted by Disc (K i ) #Aut F (K i ), is then by Serre s mass formula [5] f i (e i ) for totally ramified extensions of a local field F (see Aendix).However, nonisomorhic extensions of F may become isomorhic as extensions of Q (leading to an overcounting by some factor g i ), and it may be that #Aut F (K i ) = #Aut Q (K i ). Let h i denote the number of automorhisms of F over Q that extend to automorhisms of K i /Q. Then any maximal set of distinct isomorhism classes of extensions K i /F that become isomorhic as extensions of Q must have #Aut Q (F)/h i = f i /h i elements, so that the above overcounting factor g i is given by f i /h i. The observation that #Aut Q (K i ) = h i #Aut F (K i ) now roves the claim. We conclude that thesum ontheleft-handside of (2.2) is equal to ( r as desired. i= ) f i(e i ) f i N = Disc (σ) #Aut(σ), Equation (2.2) of Proosition 2. allows one to systematically relace certain sums over étale Q -algebras by sums of the same tye over slitting symbols. We now extend Proosition 2. to certain secial clusters of slitting tyes. Given a slitting symbol σ T n with Disc (σ) = k, we may naturally associate to σ a Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

7 Mass Formulae for Extensions of Local Fields 7 artition of k as follows. For σ = (f e fe r r ), let π(σ) be the artition ofk given by k = (e ) + + (e ) + + (e r ) + + (e r ), }{{}}{{} f times f r times where arts equal to zero are, as usual, ignored. We call π(σ) the ramification artition of σ. Ifσ T n and Disc (σ) = k, then by construction π(σ) is a artition of k into at most n k arts. Conversely, given a artition π of k into at most n k arts, there exist slitting symbols σ T n such that π(σ) = π. It is clear that many different slitting symbols σ could give rise to the same artition π.however, in this regard, we have the following roosition. Proosition 2.2. (Mass formula for extensions having a fixed ramification artition) Let k and n be integers with 0 k n, and let π 0 be any artition of k into at most n k arts. Then [K:Q ]=n étale π((k,))=π 0 Disc (K) Proof. By Proosition 2., it is equivalent to rove σ T n π(σ)=π 0 Disc (σ) which in turn is equivalent to σ T n π(σ)=π 0 #Aut(σ) =, #Aut(K) = k. (2.3) #Aut(σ) = k since Disc (σ) = k for any σ with π(σ) = π 0. Let us use n t to denote the number of arts in π 0 equal to t, so that k = k t= n t t. Given a slitting symbol σ T n with π(σ) = π 0, we attach to σ = ( r i= f e i i ) a conjugacy class χ = χ(σ) in G = S n S nk as follows. Namely, let χ = (χ,..., χ k ), where χ t S nt is the conjugacy class of any element x t S nt that is a roduct of disjoint cycles having lengths f i, f i2,..., where e i, e i2,... are the e i s for which e i = t. Under this corresondence, we have #Z(x) = #Aut(σ), where Z(x) denotes the centralizer of Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

8 8 ManjulBhargava x = (x,..., x k ) in G = S n S nk.inarticular, n! n k!/#aut(σ) = #χ. Summing over σ, we have σ T n π(σ)=π 0 n! n k! #Aut(σ) = χ G #χ = n! n k!, and this comletes the roof of the roosition. Note that if π 0 is the emty artition of zero, then Proosition 2.2 states [K:Q ]=n unramified =. (2.4) #Aut(K) Proosition 2.3. (Mass formula for general étale extensions of Q ) We have [K:Q ]=n étale n Disc (K) #Aut(K) = q(k, n k) k. k=0 Proof. Summing the mass formula (2.3) of Proosition 2.2 over all artitions π 0 yields the desired result. Thus we now have a mass formula for all étale extensions of Q for all finite rimes. All that remains now is the derivation of an analogous mass formula at the archimedean lace. Proosition 2.4. (Mass formula for general étale extensions of R) We have [K:R]=n étale #Aut(K) = r 2(S n ). n! Proof. As the only field extensions of R are R and C, the étale extensions of R of degree n are those given by E n,k = R n 2k C k, where k = 0,,..., n/2. We associate to the extension E n,k the conjugacy class χ n,k S n of any element x S n that is exressible as a roduct of k disjoint 2-cycles. Then we see that #Z(x) = #Aut(E n,k ) = (n k)! k! 2 k, where Z(x) is the centralizer of x in S n ; it follows that n!/#aut(e n,k ) = #χ n,k.notingthat n/2 k=0 χ n,k S n is the set of all 2-torsion elements in S n, we obtain n/2 k=0 as was desired. n/2 n! #Aut(E n,k ) = k=0 #χ n,k = r 2 (S n ) Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

9 Mass Formulae for Extensions of Local Fields 9 Proosition 2.4 can also be divided into cases of ositive and negative discriminant. Let r 2 + (S n) and r2 (S n) denote the number of 2-torsion elements in S n having signature +and resectively. Then [K:R]=n étale Disc(K)>0 #Aut(K) = r+ 2 (S n) n! and [K:R]=n étale Disc(K)<0 #Aut(K) = r 2 (S n). n! 3 How many number fields are exected of a given degree and discriminant? Let D be any integer, and n any ositive integer. We wish to address the question of the exected number of S n -number fields K of degree n such that the discriminant of K is D. The sign and divisibility of D by various rimes restrict the ossible comletions of K at the infinite and finite laces, resectively. If D is ositive (res. negative), then the comletion K = K R of K at infinity will be an étale R-algebra which contains C as a direct summand an even (res. odd) number of times. Similarly, if e D, then the ossibilities for the comletion K = K Q of K at are restricted to those for which e Disc(K /Q ). We may therefore ask how often we actually exect each of these local extensions to occur. It is a common hilosohy in number theory, going back at least to the Dirichlet class number formula, that isomorhism classes of algebraic objects tend to occur, and thus should be counted, by weights that are inversely roortional to the cardinalities of their resective automorhism grous. This indeed was also the fundamental assumtion behind the Cohen-Lenstra heuristics. In the case of (local or global) fields K of a fixed degree, this heuristic hilosohy can be understood simly via the observation that the number of subfields of K isomorhic to K is roortional to /#Aut(K). Our first assumtion is thus the understanding that all global and local extensions K are to be counted with weight #Aut(K).Inarticular, given only the knowledge that t Disc(K), where K is an S n -number field of degree n, and given an étale Q -algebra K of degree n with t Disc(K /Q ), the robability that K Q = K is roortional to #Aut(K ). Let us say that an étale extension K of Q is discriminant-comatible with D Z if v (Disc(K )) = v (D), where v denotes the valuation of Z at. Then the total (weighted) number of local extensions of Q of degree n that are discriminant- Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

10 0 Manjul Bhargava comatible with D is given by [K :Q ]=n v (Disc(K ))=v (D) #Aut(K ). (3.) In articular, given a number field K of degree n and discriminant D, and any fixed étale Q -algebra F discriminant-comatible with D, the robability that K Q = F is #Aut(F) divided by (3.). Similarly, we say an étale extension of R is discriminant-comatible with D Z if sgn(disc(k )) = sgn(d), where sgn denotes the sign function on R (taking values ±). The total (weighted) number of étale extensions of R of degree n that are discriminantcomatible with D is given by [K :R]=n sgn(disc(k ))=sgn(d) #Aut(K ). (3.2) Thus, given a number field K of degree n and discriminant D, and any fixed étale R- algebra F discriminant-comatible with D, the robability that K R = F is #Aut(F) divided by (3.2). Our second assumtion is that, for S n -fields, we exect the comletions K Q ν at (finite or infinite) laces to behave indeendently. For examle, suose we are given an S n -number field K of degree n and discriminant D, and local étale extensions K µ /Q µ and K ν /Q ν at two distinct laces µ and ν such that both K µ and K ν are discriminantcomatible with D. Then the robability that K Q µ = K µ and K Q ν = K µ is simly the roduct of the individual robabilities for the events K Q µ = K µ and K Q ν = K µ described above. (Incidentally, this would not be true for fields K having general Galois grous G, where local comletions K ν = K Q ν can have deendencies. But for G = S n, we exect such indeendence of rimes.) Our basic heuristic assumtion, then, which includes both assumtions discussed above, is that the exected (weighted) number of global S n -number fields of discriminant D is simly the roduct of the (weighted) number of local extensions of Q ν that are discriminant-comatible with D, where ν ranges over all laces of Q (finite and infinite). Thatis, if E n (D) denotes the exected number of S n -number fields of degree n Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

11 Mass Formulae for Extensions of Local Fields and discriminant D, then we have E n (D) = [K:R]=n étale sgn(d)=sgn(disc(k)) #Aut(K) [K:Q ]=n étale v (D)=v (Disc(K)) #Aut(K). (3.3) The latter infinite roduct makes sense, because the sum inside this roduct is seen to equal for sufficiently large (see (2.4)). More generally, for each rime let Σ be a set of isomorhism classes of étale extensions of Q of degree n, and let Σ be a set of isomorhism classes of étale extensions of R of degree n. Let us assume that, for all but finitely many rimes, the set Σ contains all the isomorhism classes of étale extensions of Q.LetE n,σ (D) denote the exected number of S n -number fields K having discriminant D such that K Q Σ for all rimes and K R Σ. Then our heuristics give that E n,σ (D) = K Σ sgn(d)=sgn(disc(k)) #Aut(K) K Σ v (D)=v (Disc(K)) #Aut(K). (3.4) The roduct again makes sense, because again all but finitely many factors in this roduct are seen to equal by (2.4). 4 Summing exectations over D For a sequence of local secifications Σ = (Σ, Σ 2, Σ 3,...), where Σ ν denotes some subset of étale extensions of Q ν as in Section 3, let N n,σ (X) denote the number of S n -number fields K of degree n having absolute discriminant at most X such that K Q ν Σ ν for all laces ν of Q. As a generalization of (.), we conjecture that the limit N n (X) c n,σ = lim X X (4.) exists and is greater than zero for n >. In accordance with our heuristics, in order to obtain the corresonding value of c n,σ, we wish to average (3.4) over all D ( X, X) subject to the global condition that D 0, (mod 4). Because E n,σ (D) is a multilicative function, it is an easy matter to Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

12 2 Manjul Bhargava carry out this average in the limit as X. We find c n,σ = X lim = X D ( X,X) [K:R]=n étale K Σ 2 E n,σ (D) #Aut(K) [K:Q ]=n étale Q Σ Disc (K) #Aut(K). (4.2) The 2 in the first (archimedean) factor corresonds to the robability that an integer in the range ( X, X) has any given sign (+ or ), while the constant Disc (K) in the Euler factor at corresonds to the robability that an integer is exactly divisible by Disc (K). 5 Conjectures.2.4, and other consequences Conjecture.2 now follows by substituting the results of Proosition 2.3 into (4.2). To obtain Conjecture.3, we firststate: Conjecture 5.. Let K be an étale degree n extension of Q, and let K run through all S n -number fields of degree n, the fields being ordered by the size of the discriminants. Then the relative frequency of those fields K for which K Q = K is roortional to Disc (K ) #Aut(K ). Conjecture 5. follows from (4.2). Combining Conjecture 5. together with Proosition 2., we then obtain: Conjecture 5.2. Let be a fixed rime, and let σ T n be a slitting symbol. Let K run through all S n -number fields of degree n, the fields being ordered by the size of the discriminants. Then the relative frequency of those fields K for which (K, ) = σ is roortional to Disc (σ) #Aut(σ). Conjecture.3 now follows from Conjecture 5.2, together with the observation that if an S n -number field K of degree n is unramified at with σ = (K, ), then n!/#aut(k) is equal to the size of the conjugacy class in S n reresented by the Artin symbol ( K/). Conjecture.4 is an immediate consequence of (4.2) and Proosition 2.2. Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

13 Mass Formulae for Extensions of Local Fields 3 Finally, as evidence for the truth of these conjectures, we have Theorem 5.3. Conjectures.2.4 and are true for n 5. Proof. See [], [3], and [4]. 6 More recise conjectures for individual discriminants We have seen that our heuristics give very accurate information on the asymtotic distribution of discriminants of S n -number fields of degree n for n 5 (and conjecturally for all n). It is interesting to ask how well the heuristics work for individual discriminants. In general, it is clear that E n (D) will not exactly equal the number of S n -number fields of degree n, since usually E n (D) is not an integer but only a rational number. But it is still interesting to ask for an accurate robabilistic estimate for the number of S n -number fields of degree n having a fixed discriminant D. As far as the asymtotic enumeration of number fields of absolute discriminant at most X was concerned, it was sufficient to consider simly the -divisibility of numbers D ( X, X) for various rimes. However, in the context of individual discriminants D, it is ossible to obtain finer robabilistic estimates by considering also the quadratic character of D modulo these rimes. This is because the discriminant of an extension of Q is not just a ower of, but is well-defined as a squareclass in Z (i.e., as an element of (Z \ {0})/Z 2 ). Let us say that an étale extension K of Q is strongly discriminant-comatible with D Z if Disc(K/Q ) is in the same squareclass as D Z. Now define E n(d) = [K:R]=n sgn(d)=sgn(disc(k)) #Aut(K) 2 [K:Q ]=n étale Disc(K)/D Z 2 #Aut(K). (6.) This roduct makes sense because one checks again that the Euler factors are eventually forsufficiently large. Asymtotically, the quantities X D= X E n(d) and X D= X E n(d) behave similarly as X, but for individual discriminants D, we susect that E n(d) will give the more accurate estimate for the number of S n -number fields of degree n and discriminant D. For examle, for n = 2, E n(d) gives exactly the right number of quadratic fields of discriminant D. It would be interesting to check the redictive accuracy of both E n(d) and E n (D) against known tables of cubic, quartic, and quintic fields. Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

14 4 Manjul Bhargava 7 Final remarks We end with a few remarks on the constants c n.first, since the time of the work of Davenort and Heilbronn [], it has been observed that the constants c, c 2, and c 3 are rational multiles of ζ(), ζ(2), and ζ(3) resectively. Table and Theorem. make it clear that this attern does not continue, and the values of c n in general do not aear to be related to zeta values. Instead, it aears that the olynomials in occurring in the Euler roducts in the exressions for c n are generally irreducible olynomials of degree n, with relatively small coefficients, times the factor ( ). There are excetions to this irreducibility, however, as one observes that c 7 and c 0 both factor into a rational multile of ζ(3) times the roduct of lower degree Euler factors! Even though c n is not related to ζ(n) in general, zeta values do make their aearance again in the limit. More recisely, one finds that the Euler roduct occurring in c n converges to the roduct of zeta values ζ(2)ζ(3)ζ(4)ζ(5) as n. Indeed, if P(n) denotes the number of artitions of n, then the Euler roduct aroaches ( ) P(k) P(k ) k = ( ) P 2 (k) k = ( ) n=2 kn = ζ(n), n=2 k=0 k=0 where we have used P 2 (k) to denote the number of artitions of k where each art is at least 2. Nowwenotethatthefactorr 2 (S n )/n! occurring in the exression for c n raidly aroaches zero as n tends to infinity. Since we have just shown that the Euler factors in c n converge to a finite constant, it follows that the sum of c n over all n also converges. This suggests that the total number N(X) of all S n -extensions of degree n (for any n), having absolute discriminant less than X, should also grow linearly with X; indeed, if c denotes the finite constant n= c n, then it is natural to conjecture that N(X) lim X X = c. I do not know if c has an alternate exression not involving an infinite sum. Numerically, c.4. 8 Further questions It is an interesting question to try and understand both the local and global analogues of the results/conjectures of this aer for general Galois grous. k=0 Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

15 Mass Formulae for Extensions of Local Fields 5 8. Local masses Suose G is a transitive subgrou of S n. Consider the set of G-isomorhism classes of mas φ : Gal(Q /Q ) G, where two such mas are said to be G-isomorhic if one can be obtained from the other by comosing with a conjugation by an element g G. Any such ma φ cuts out an étale extension K = K φ of Q having degree n. Define a function M Q (G, s) by M Q (G, s) = G-isomorhism classes of φ #Aut(φ) Disc (K φ ) s, (8.) where Aut(φ) G denotes the number of G-automorhisms of φ (equivalently, Aut(φ) is the centralizer in G of the image of φ). Then our mass formula in Theorem. imlies that M Q (G, s) is a olynomial in / s that is indeendent of, when G = S n and s =. For what other grous G and real numbers s is this the case? More generally, suose G is any finite grou and ρ : G GL(V) is a linear reresentation of G. Consider again the set of isomorhism classes of mas φ : Gal(Q /Q ) G. Toanysuchmaφ, we may associate the comosite ma φ : Gal(Q /Q ) GL(V) given by ρ φ. Define M Q (G, ρ, s) by M Q (G, ρ, s) = G-isomorhism classes of φ #Aut(φ) s Artin( φ), (8.2) whereweuseartin( φ) to denote the Artin conductor of φ. TheninthecaseofG = S n and ρ the standard reresentation, onechecksthatm Q (G, ρ, s) = M Q (S n, s). Again, we may ask for which reresentations ρ of a grou G and which values of s is M Q (G, ρ, s) a olynomial f in / s, where f is indeendent of?finally, what if the Artin conductor is relaced by some other invariant altogether? Some interesting results in these directions have recently been obtained by Kedlaya and Wood. For examle, Kedlaya [2] has shown there is such a olynomial indeendent of for M Q (G, ρ, s) when G is the Weyl grou W(B n ), ρ is its standard reresentation, and s = ; the same result also holds for W(D n ) and W(G 2 ) if = 2. Meanwhile, Wood has shown that there is such a olynomial indeendent of when G is taken to be any grou formed from symmetric grous via cross roducts and wreath roducts, and the Artin conductor is relaced by a slightly different tye of counting function (see [6] for details). Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

16 6 Manjul Bhargava 8.2 Global distribution of number fields Next, we turn to the global scenario. Let G again denote a transitive ermutation subgrou of S n.wewishtoconsider G-number fields, i.e., degree n field extensions of Q whose Galois closure has Galois grou isomorhic to G, and where an exlicit such isomorhism to G has been secified. We make this more recise. For a degree n field extension K of Q, let K denote the Galois closure of K. ThenGal( K/Q) naturally acts on the embeddings of K into K, giving Gal( K/Q) the structure of a transitive ermutation grou on n elements. We say that a air (K, ψ) is a G-number field if K is a degree n field extension of Q and ψ : Gal( K/Q) G is an isomorhism of ermutation grous (thus including a secific bijection between the underlying sets). Anisomorhism between two G-number fields (K, ψ ) and (K 2, ψ 2 ) is then simly an isomorhism η : K K 2 such that the induced ma η : Gal( K /Q) Gal( K 2 /Q) satisfies ψ = ψ 2 η.ag-number field (K, ψ) is usually denoted simly by K when the accomanying isomorhism ψ is understood. Note that an isomorhism class of G-number fields K is equivalent to a G-isomorhism class of surjective homomorhisms φ K : Gal(Q/Q) G. The latter leads to a G-isomorhism class of local homomorhisms φ K,ν : Gal(Q ν /Q ν ) G for any (finite or infinite) lace ν of Q, which in turn cuts out a local étale extension K Q ν of Q ν for any lace ν. For each lace ν of Q, let Σ ν be a set of G-isomorhism classes of mas φ ν : Gal(Q ν /Q ν ) G. We assume that for all but finitely many laces ν, the set Σ ν contains all such G-isomorhism classes of mas φ ν : Gal(Q ν /Q ν ) G. LetN G,Σ (X) denote the number of isomorhism classes of G-number fields K with absolute discriminant less than X and satisfying φ K,ν Σ ν for all ν, where each such isomorhism class of G-number field K is counted with a weight of /#Aut(K) = /#Aut(φ K ).Also, let n G,Σ (D) denote the weighted number of such G-number fields satisfying φ K,ν Σ ν for all ν but with absolute discriminant exactly D. Then the asymtotic behavior of N G,Σ (X) is determined by the behavior of the Dirichlet series ξ Q (G, Σ, s) = ξ(g, Σ, s) = at its rightmost ole. n G,Σ (D)D s (8.3) D= This Dirichlet series was first considered in the abelian case by Wright [8], and more generally by Cohen et. al. in [7]. Very little is known about this Dirichlet series in general, although the analytic continuation to the left of the rightmost ole is now known in many cases such as G = S n (n 5), abelian grous, and D 4 (see [], [3], [4], [8],and [7] for details). Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

17 Mass Formulae for Extensions of Local Fields 7 We may similarly count all Galois homomorhisms φ : Gal(Q/Q) G, not just the surjective ones corresonding to fields. These will in general corresond to certain étale G-extensions of Q having degree n. Counting such extensions by discriminant would lead to an analogous Dirichlet series ξ Q (G, Σ, s) = ξ(g, Σ, s) = ñ G,Σ (D)D s, (8.4) D= where ñ G,Σ (D) denotes the number of G-extensions étale over Q having absolute discriminant D and allowed local behavior dictated by Σ. We wish to comare these global counts with local ones. Following Section 8., consider the local weighted sums M Q (G, Σ, s) defined by M Q (G, Σ, s) = #Aut(φ) Disc (K φ ) s. (8.5) φ Σ Let M(G, Σ, s) = M Q (G, Σ, s) denote the roduct of the local factors M Q (G, Σ, s) over all, together with an archimedean factor and an as yet unsecified constant factor C(G): M(G, Σ, s) = C(G) #Aut(φ) φ Σ M Q (G, Σ, s). (8.6) We say that a Dirichlet series with nonnegative real coefficients is secial if it converges in some oen right half-lane Re(s) s 0, has a meromorhic continuation to the closed right-half lane Re(s) s 0, and is analytic in this region with the excetion of a single ole at s = s 0. We say that two such secial Dirichlet series f (s) = a n n s and g(s) = b n n s are asymtotically equivalent if their corresonding rightmost oles are located at the same oint s = s 0, and if the order of the ole of f g at s 0 is strictly lower than the order of the ole of f or g at s 0. By standard Tauberian theorems, if two secial Dirichlet series f and g are asymtotically equivalent then ( X n= a n)/( X n= b n) as X. Conjecture.2 in the introduction is thus imlied by our (slightly stronger) conjecture that, for C(G) = /2, the series ξ(s n, Σ, s) and M(S n, Σ, s) are secial and in fact are asymtotically equivalent for any choice of Σ. (Note that ξ(s n, Σ, s) has a comletely different order of growth than both ξ(s n, Σ, s) and M(S n, Σ, s) when n > 2.) Due to the derivation of sufficiently strong error terms in the works [], [3], and [4], this stronger version of Conjecture.2 is also now known for n 5. This naturally leads to the question as to when there exists a constant C(G) such that M(S n, Σ, s) is asymtotically equivalent to ξ(g, Σ, s), or to ξ(g, Σ, s), indeendent of Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

18 8 Manjul Bhargava the choice of Σ? What then is the value of C(G)? (The existence of such a C(G) means, heuristically seaking, that the robabilities of given local comletions are indeendent at any finite set of laces and are roortional to local sums of the form (8.5) at s =.) In the case of abelian grous, it follows from the works of Mäki [3 4] and Wright [8] that such a C(G) exists (for which the series M(G, Σ, s) and ξ(g, Σ, s) are asymtotically equivalent for all Σ) whenever the grou has rime exonent. It also follows from the work [5] (see also [2]) that such a constant C(S 3 ) exists when counting via the regular reresentation of S 3. More generally, if ρ is a faithful linear reresentation of G on a vector sace V, then we may count global fields (or étale extensions) by the Artin conductor of the reresentation ρ. One may also count global extensions by more general functions, such as the conductor in the abelian case, or by other suitable roducts of the ramified rimes. In the case of abelian grous G, Wood [7] has shown that there exist analogous C(G) (for which the analogous M and ξ are asymtotically equivalent for all Σ) for a number of different ways of counting abelian extensions, including the counting of G by conductor. Aendix All conjectures, theorems, and questions formulated in Sections 8 have natural analogues over an arbitrary local or global field. Theorem.andEquation(.2) remain true when Q is relaced by an arbitrary local field F with finite residue field of cardinality P; Disc (K) is relaced by Norm(Disc(K/F)) (viewed as a ower of P); and is also relaced by P on the right hand sides of (.2) and (.3). The roofs are identical. In order to obtain the analogue of Conjecture.2 over an arbitrary global number field, we need also a mass formula analogous to Proosition 2.4 over the archimedean local field C, the comlex numbers. This is trivial, as there is only one étale extension of C of degree n for any n, namely C n, and evidently /#Aut(C n ) = /n!. Conjecture.2 over an arbitrary base then becomes: Conjecture A. Let F be any number field having r real and r 2 comlex embeddings. Let N n (F, X) denote the number of S n -field extensions of F of degree n whose absolute discriminants have norm at most X.Then N n (F, X) lim = ( ) r ( ) ( r2 X X 2 Res ζ r2 (S n ) n ) q(k, n k) q(k, n k + ) F(s) s= n! n! N k, where the roduct runs over all rime ideals of F. k=0 Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

19 Mass Formulae for Extensions of Local Fields 9 Conjectures.3.4 and can be formulated similarly. Conjectures.2.4 and over an arbitrary base field are now known for n 3 thanks to the work of Datskovsky and Wright [0]. It is likely they will soon be theorems also for n = 4and n = 5. The work in Sections 2 8 can also be adated to an arbitrary base local or global field in an analogous way. In Section 8.2, the constant C(G) occurring in (8.6) must be relaced by C(G, F) to reflect its deendence on the base field F. For examle, Conjecture A may be rehrased to state that with C(S n, F) = 2 Res s= ζ F (s), the series ξ F (G, Σ, s) and M F (G, Σ, s) are asymtotically equivalent for all choices of local secifications Σ = (Σ ν ). References [] Belabas, K., M. Bhargava, and C. Pomerance. Error Terms for the Davenort-Heilbronn Theorems. Prerint. [2] Belabas, K., and E. Fouvry. Discriminants Cubiques et Progressions Arithmétiques. Prerint. [3] Bhargava, M. The Density of Discriminants of Quartic Rings and Fields. Annals of Mathematics 62 (2005): [4] Bhargava, M. The Density of Discriminants of Quintic Rings and Fields. Annals of Mathematics (forthcoming). [5] Bhargava, M., and M. M. Wood. The Density of Discriminants of S 3 -Sextic Number Fields. Proceedings of the American Mathematical Society (forthcoming). [6] Chebotarëv, N. G. Oredelenie Plotnosti Sovokunosti Prostykh Chisel, Prinadlezhashchikh Zadannomu Klassu Podstanovok (Determination of the Density of the Set of Prime Numbers, Belonging to a Given Substitution Class). Izvestiya Rossijskoj Akademii Nauk 7 (923): ; Sobranye Sochinenii I: [7] Cohen, H., F. Diaz y Diaz, and M. Olivier. Enumerating Quartic Dihedral Extensions of Q. Comositio Mathematica 33 (2002): [8] Cohen, H., and H. W. Lenstra. Heuristics on Class Grous of Number Fields. P in Number Theory, Noordwijkerhout, 983, Lecture Notes in Mathematics 068. Berlin: Sringer, 984. [9] Cohen, H., and J. Martinet. Étude Heuristique Des Groues De Classes Des Cors De Nombres. Journal Fur Die Reine Und Angewandte Mathematik 404 (990): [0] Datskovsky, B., and D. J. Wright. The Adelic Zeta Function Associated to the Sace of Binary Cubic Forms II. Local Theory. Journal Fur Die Reine Und Angewandte Mathematik 367 (986): [] Davenort, H., and H. Heilbronn. On the Density of Discriminants of Cubic Fields II. Proceedings of the Royal Society of London Series A-Mathematical and Physical Sciences 322, no. 55 (97): Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208

20 20 Manjul Bhargava [2] Kedlaya, K. S. Mass Formulas for Local Galois Reresentations (with an aendix by Daniel Gulotta). International Mathematics Research Notices (forthcoming). [3] Mäki, S. On the Density of Abelian Number Fields. Annales Academiae Scientiarum Fennicae Series A-Mathematica Dissertationes 54 (985). [4] Mäki, S. The Conductor Density of Abelian Number Fields. Journal of the London Mathematical Society (2) 47, no. (993): [5] Serre, J.-P. Une Formule De Masse Pour Les Extensions Totalement Ramifiés De DegréDonné d un Cors Local. Comtes Rendus Hebdomadaires des Séances de l Académie des Sciences. Séries A et B 286, no. 22 (978): A03 A036. [6] Wood, M. M. Mass Formulas for Local Galois Reresentations to Wreath Products and Cross Products. Prerint. [7] Wood, M. M. On the Probabilities of Local Behaviors in Abelian Field Extensions. (forthcoming). [8] Wright, D. J. Distribution of Discriminants of Abelian Extensions. Proceedings of the London Mathematical Society (3) 58 (989): Downloaded from htts://academic.ou.com/imrn/article-abstract/doi/0.093/imrn/rnm052/ by guest on 8 December 208