Genus theory and the factorization of class equations over F p

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1 arxiv: v2 [math.nt] 10 Dec 2017 Genus theory and the factorization of class euations over F Patrick Morton March 30, 2015 As is well-known, the Hilbert class euation is the olynomial H D (X) whose roots are the distinct j-invariants of ellitic curves with comlex multilication by the maximal order R K in the imaginary uadratic field K = Q( D). A root of H D (X) generates the Hilbert class field Σ of K over K. The olynomial H D (X) always has a real root ξ, and over Q this root generates the real subfield Σ 0 = Q(ξ) of Σ. Recently, Stankewicz [st] found a criterion for the olynomial H D (X) to have a root (mod ), for a given odd rime for which the Legendre symbol (D/) = 1. This criterion can be stated as follows. Theorem (Stankewicz). If is an odd rime for which (D/) = 1, and does not divide the discriminant of H D (X), then H D (X) has a linear factor over F if and only if ( ) = 1, D, an odd rime. Stankewicz derives this criterion from his analysis of rational -adic oints on twists of Shimura curves. In this note I give a more direct roof of the criterion using genus theory and basic roerties of the Hilbert class field. The roof shows that the above theorem fits naturally into a discussion of genus theory. (See [co], [has2], and [ish].) 1

2 1 Necessity. Inthisandthenext sectiontheinteger N will denotethesuare-freeartof the fundamental uadratic discriminant D, and K is the imaginary uadratic field K = Q( N) with discriminant D. Theorem 1. Let Σ 0 denote the real subfield of the Hilbert class field Σ of the uadratic field K = Q( ( ) N). Assume is an odd rime for which N = 1. If has a rime divisor of degree 1 in Σ 0, then ( ) = 1, N, an odd rime. To rove this we use the decomosition D = D, = ( 1) ( 1)/2, odd, 2 = 4,8, 8, where the roduct is over all the rime divisors of D. The genus field of K is the field Ω, which is obtained by adjoining all the suare-roots to K, as varies over the rime divisors of D. It is the largest unramified extension of K which is abelian over Q, so that Ω Σ. Assume( that ) the odd rime has a first degree rime divisor in Σ 0. The N conditions = 1 and odd imly that does not divide D, and has a first degree rime divisor in every subfield of Σ 0. If is a rime 1 (mod 4), then =, so Q( ) Σ 0. Hence, has a first degree rime divisor in Q( ), which imlies that ( ) = 1, 1 (mod 4), N. (1.1) This imlies then that ( ) = 1, 1 (mod 4), N. (1.2) 2

3 If 2 D and 2 = 8, the same argument also gives 8). ( ) 2 = 1, so ±1 (mod On the other hand, if there are several rimes i 3 (mod 4), i = 1,2, then i Σ imlies that 1 2 = 1 2 Σ 0. Then has a first degree rime divisor in Q( 1 2 ), so we have ( ) 1 2 = 1, i 3 (mod 4), i N. It follows that ( ) 1 = ( ) 2, (mod 4), i N. (1.3) We get a similar conclusion when 2 D and 2 = 4, 8, namely ( ) ( ) 2 =, 2 = 4, 8, 3 (mod 4), N. (1.4) Now we use the fact that ( ) ( ) D N = = ( ) = 1. (1.5) D From (1.1), (1.3), (1.4) the terms with 1 (mod 4) or = 8 dro out, and we are left with ( ) r = 1, where r is the number of rime divisors of D with 3 (mod 4) or = ( ) 4, 8. But this imlies that r is odd and = 1 for all these rime divisors. Hence, ( ) = 1, if 3 (mod 4), N. Together with (1.2), this roves Theorem 1. 3

4 ( ) N Corollary 1. If does not divide the discriminantof H D (X), = 1, and H D (X) (mod ) has a root in F, then ( ) = 1, N, an odd rime. Proof. Since does not divide the discriminant of H D (X) and a real root of H D (X) generates Σ 0, it is clear that the factors of H D (X) mod corresond 1-1 to the rime divisors of in Σ 0. The corollary is now immediate from Theorem 1. 2 Sufficiency. Now we rove the converse of Theorem 1: Theorem 2. Let Σ 0 denote the real subfield of the Hilbert class field Σ of the uadratic field K = Q( ( ) N). Assume is an odd rime for which N = 1. If satisfies the condition ( ) = 1, N, an odd rime, then has a rime divisor of degree 1 in Σ 0. To rove this we consider the ( decomosition ) grou of a rime divisor P of in Σ. First we note that if = 1 for all odd rime divisors of N, then (1.1) holds, as does ( ) = 1, 3 (mod 4), N, rime. (2.1) Now (1.5) imlies that ( ) 2 = ( 1) r 1 ( 1) = ( 1) r, if 2 D, 4

5 where r 1 is the number of rimes 3 (mod 4) dividing N. But if 2 D, then either: N 1 (mod 4), ( in ) which( case ) 2 = 4 and r 1 is even, so that r is 2 4 odd, imlying that = = 1; ( 2 (mod 8), in which case 2 2 N) ( ) = 8 and r is again odd, giving 8 = = 1; ( N 6 (mod 8), in which case 2 2 or ) ( ) = 8 and r 1 is odd, giving 8 = = 1. Thus, if 2 D, we have (1.4) and the assertion in the sentence following (1.2). This shows that slits comletely in the real subfield Ω 0 of the genus field Ω. (Note that [Ω : K] = 2 t 1, where t is the number of distinct rime factors of D. Thus [Ω 0 : Q] = 2 t 1, as well.) Hence, the decomosition field of any rime divisor P of in Σ contains the field Ω 0, and therefore the decomosition grou G P is contained in H = Gal(Σ/Ω 0 ). It suffices to show G P = {1,τ} for some P, where τ is comlex conjugation. If this holds, then Σ 0, which is the fixed field of τ, is the largest field in which the rime below P has degree 1, i.e. = PP τ is a first degree rime divisor of in Σ 0. Let J = H Gal(Σ/K) be the subgrou of Gal(Σ/K) corresonding to Ω in the Galois corresondence, so that [H : J] = 2 and H = J Jτ. By the genus theory [has2], J corresonds to the subgrou of suares in Pic(R K ), in the Artin correondence between ideal classes in R K and elements of the Galois grou Gal(Σ/K). We now have what we need to comlete the roof. The decomosition grou G P is a subgrou of H of order 2. This is because is inert in K, so that () = R K is a rincial ideal and therefore slits comletely in the extension Σ/K. Furthermore, we know that K is not contained in the decomosition field of any P, and therefore G P J Gal(Σ/K). Hence, G P H is generated by some στ, with σ J. But σ = ψ 2 for some ψ Gal(Σ/K), by the characterization of the grou J, and ψ 1 G P ψ = G P ψ = {1,ψ 1 στψ}, with ψ 1 στψ = ψ 2 στ = τ. This shows that G P ψ = {1,τ} and comletes the roof. 5

6 ( ) ( ) N Corollary 2. If satisfies = 1 and = 1, for all odd rimes such that N, then H D (X) (mod ) has a root in F. Proof. Theorem 2 imlies that H D (X) has a linear factor over Q and therefore H D (X) (mod ) has a root in F. ( ) N Note that the hyothesis = 1 of Corollary 2 holds for all the rime divisors of the discriminant of H D (X) which do not divide D, by a result of Deuring [d]. 3 A rime decomosition law for Σ 0. The roof of Theorem 2 shows that when has a first degree rime divisor in Σ 0, then it has as many rime divisors of degree 1 as there are distinct elements ψ in Gal(Σ/K) for which G P ψ = ψ 1 G P ψ = {1,τ}, where P. (Note that the ψ are in distinct cosets of G P = {1,τ}, so the rime divisors P ψ are distinct.) This holds if and only if ψ 1 τψ = τ, i.e., if and only if ψ 2 = 1. The number of such elements ψ is exactly 2 t 1, since this is the order of the 2-Sylow subgrou of the class grou Pic(R K ). Thus we have: ( ) N Theorem 3. If is an odd rime for which = 1, and has a rime divisor of degree 1 in Σ 0, then it has exactly 2 t 1 such rime divisors, where t is the number of distinct rime factors of D. Taken together, Theorems 1-3 yield the following decomosition law for the real subfield Σ 0 of Σ. Prime Decomosition Law in Σ 0. Let be an odd rime that does not divide D. ( ) D (a) If = 1, then in Σ 0, slits into h(d)/f rimes of degree f over Q, where f is the order of a rime ideal divisor of in Pic(R K ). ( ) ( ) D (b) If = 1 and = 1 for all odd rime divisors of D, then in Σ 0, slits into r 1 = 2 t 1 rimes of degree 1 and r 2 = (h(d) 2 t 1 )/2 rimes of degree 2 over Q. 6

7 ( ) ( ) D (c) If = 1 and = 1 for some odd rime divisor of D, then in Σ 0, slits into h(d)/2 rimes of degree 2 over Q. This law immediately imlies the following density result. Theorem 4. The density of rimes Z + for which H D (X) has a linear factor (mod ) is d(p(σ 0 )) = 1 2h(D) + 1 2t, where t is the number of distinct rime factors of D. 4 Discriminant divisors. The Prime Decomosition Law roved in the last section raises the uestion: how do the rime divisors of the discriminant D slit in the real subfield Σ 0? While the Prime Decomosition Law in Section 2 refers to the uadratic character of with resect to the rime factors of the discriminant, a similar law for the rime factors of D cannot simly involve these same rime factors. For examle, we have the following congruence for the class euation of discriminant D = 8, for > 13, from [mor2, Theorem 1.1]: H 8 (t) (t 1728) 2ǫ 1 (t 8000) 2ǫ 2 (t+3375) 4ǫ 3 (t t ) 4ǫ 4 i (t 2 +a i t+b i ) 2 (mod ). The roduct is over the distinct, irreducible uadratic factors, different from H 15 (t) = t t , of the gcd of H 8 (t) and J (t) over F ; where J (t) n k=0 ( 2n+s 2k +s )( 2n 2k n k ) ( 432) n k (t 1728) k (mod ), with n = [/12] = ( e)/12, e {1,5,7,11}, and s = 1 2 ( 1 ( )) 4. 7

8 (See [mor1].) The exonents in the congruence for H 8 (t) are given by ǫ 1 = 1 ( ( )) 4 1, ǫ 2 = 1 ( ( )) 8 1, 2 2 ǫ 3 = 1 ( ( )) 7 1, ǫ 4 = 1 ( ( ))( ( )) Thus, H 8 (t) has a linear factor modulo if and only if one of the exonents ǫ 1,ǫ 2,ǫ 3 is 1, which is the case (for > 13) exactly when ( ) 3,5,7 (mod 8) or = +1, i.e., 3,5,6 (mod 7). 7 Thus, the uadratic character of with resect to the rime = 7 is also relevant in this case! Similar congruences are roven in [mor2] for the discriminants D = 3 (when 1 (mod 4)) and D = 12 (when 3 (mod 4)). References. [co] D. A. Cox, Primes of the form x 2 + ny 2, John Wiley and Sons, New York, [d] M. Deuring, Teilbarkeitseigenschaften der singulären Moduln der ellitischen Funktionen und die Diskriminante der Klassengleichung, Commentarii Mathematici Helvetici 19 (1946), [hak1] F. Halter-Koch, Arithmetische Theorie der Normalkörer von 2-Potenzgrad mit Diedergrue, J. Number Theory 3 (1971), [hak2] F. Halter-Koch, Geschlechtertheorie der Ringklassenkörer, J. reine angew. Math. 250 (1971), [hak3] F. Halter-Koch, Eine allgemeine Geschlechtertheorie und ihre Anwendung auf Teilbarkeitsaussagen für Klassenzahlen algebraischer Zahlkörer, Math. Annalen 233 (1978), [has1] H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörer, I, Ia, II, Physica Verlag, Würzburg- Vienna, [has2] H. Hasse, Zur Geschlechtertheorie in uadratischen Zahlkörern, J. Math. Soc. Jaan 3 (1951),

9 [ish] M. Ishida, The Genus Fields of Algebraic Number Fields, Lecture Notes in Mathematics 555, Sringer, Berlin, [lym] R. Lynch and P. Morton, The uartic Fermat euation in Hilbert class fields of imaginary uadratic fields, International J. of Number Theory 11 (2015), [mor1] P. Morton, Exlicit identities for invariants of ellitic curves, J. Number Theory 120 (2006), [mor2] P. Morton, Exlicit congruences for class euations, Functiones et Aroximatio Commentarii Math. 51 (2014), [st] J. Stankewicz, Twists of Shimura curves, Canad. J. Math. 66 (2014), ; and at htt://arxiv.org/abs/ Det. of Mathematical Sciences Indiana University - Purdue University at Indianaolis (IUPUI) 402 N. Blackford St., Indianaolis, IN, USA morton@math.iuui.edu 9