THE ANALYTIC CLASS NUMBER FORMULA FOR ORDERS IN PRODUCTS OF NUMBER FIELDS

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1 THE ANALYTIC CLASS NUMBER FORMULA FOR ORDERS IN PRODUCTS OF NUMBER FIELDS BRUCE W. JORDAN AND BJORN POONEN Abstract. We derive an analytic class number formula for an arbitrary order in a roduct of number fields. 1. Introduction Let O be an order in a roduct of m number fields for some nonnegative integer m. The 1-dimensional scheme Sec O has m irreducible comonents; in articular, it is irreducible if and only if m = 1. The scheme Sec O is regular if and only if O is the roduct of the full rings of integers of the m number fields. If Sec O fails to be regular at some oint, we call that oint a singularity and say that Sec O is singular. Let ζ O (s) be the zeta function of O as defined by Serre [Ser65,. 83] (see Section 3). In the case where O is the ring of integers of a number field, that is, the case in which Sec O is regular and irreducible, Dedekind [Dir94, Sulement XI, 184, IV], generalizing work of Dirichlet, roved an analytic class number formula for the leading term of the Laurent series of ζ O (s) at s = 1 (see also Hilbert s Zahlbericht [Hil97, Theorem 56]). The generalization to the regular and reducible case is immediate. In this aer we generalize further by roving an analytic class number formula for an arbitrary order in a roduct of number fields, thereby extending Dedekind s result to orders O for which Sec O is singular. We conclude by verifying our formula in an examle with a singularity: the fiber roduct of rings Z F Z. Remark 1. We exect a similar formula in the function field case, for a singular curve over a finite field, but we do not investigate this here. Various authors have defined other zeta functions attached to such a curve and have comuted their leading terms at s = 0 or s = 1 [Gal73, Gre89, ZG97a, ZG97b, Stö98], but these zeta functions are different from the zeta function in [Ser65] in general. 2. Orders in roducts of number fields If F is a number field with ring of integers O F, classical algebraic number theory defines the following invariants: the number of real embeddings r 1 (F ), the number of airs of comlex embeddings r 2 (F ), the discriminant Disc O F, the regulator R(O F ), the class number h(o F ) defined as the order of the Picard grou Pic O F, and the number w(o F ) of roots of unity in O F. All of these invariants occur in the analytic class number formula. In this section Date: August 21, Mathematics Subject Classification. Primary 11R54; Secondary 11R29. The second author was suorted in art by National Science Foundation grant DMS and DMS and grants from the Simons Foundation (# and # to Bjorn Poonen). 1

2 we extend each of these definitions to the case of an arbitrary order in a roduct of number fields Orders. Let K be a finite étale Q-algebra; in other words, K = K 1 K m for some number fields K i. Let O K be an order, i.e., a subring of K finitely generated as a Z-module such that QO = K. Equivalently, let O be a reduced ring that is free of finite rank as a Z-module, let K := O Q, and let m := # Sec K The invariants n, r 1, r 2, and r of K. Let n := [K : Q], so O has rank n over Z. Define r 1, r 2 Z 0 by K R R r 1 C r 2. Thus r 1 is the number of ring homomorhisms K R, and 2r 2 is the number of ring homomorhisms K C whose image is not contained in R. Let r = r(k) := r 1 + r 2 m Roots of unity, the Picard grou Pic O, the class number h(o), and the discriminant Disc O. Let µ(o) be the torsion subgrou of O, so µ(o) is the grou of roots of unity in O. Let w(o) := #µ(o). Let X := Sec O. Then Pic O := Pic X = H 1 (X, OX ) [Har77, Exercise III.4.5]. Let h(o) := # Pic O. Let Tr K/Q : K Q be the trace ma. Let e 1,..., e n be a Z-basis of O. As usual, the discriminant is defined by Disc O := det (Tr K/Q (e i e j )) 1 i,j n Z The normalization. Let Õi be the ring of integers in K i. Let Õ be the normalization of O. Since O is a finite Z-module, the normalization of O equals the normalization of Z in K; thus Õ = m i=1 Õi in K. Also, Õ is finite as an O-module, so Õ is another order in K. Thus #(Õ/O) <. The invariants n, r 1, r 2, and r deend only on K, so they are the same for O as for Õ. Let be a maximal ideal of O. Localizing the O-module Õ at yields a semilocal ring Õ. The quotient Õ/O (Õ/O) is finite, and is trivial for all excet the finitely many corresonding to singularities of Sec O. Each maximal ideal P of Õ lies above the maximal ideal O of O. Therefore, given a O, saying that a lies outside every P is the same as saying that a lies outside O ; since Õ is semilocal and O is local, this means that a Õ if and only if a O. In other words, the ma of sets Õ /O Õ/O is injective. Hence Õ /O is finite too, and trivial for all but finitely many. Injectivity of Õ/O Õ /O imlies that if a, a 1 Õ are such that their images in Õ land in O for every, then a, a 1 O. Thus Õ /O Õ /O is injective. Hence Õ /O is finite. 2

3 Remark 2. The finiteness of Õ /O follows from the more general fact, Theorem 3.8 in [BL17], that for any ring homomorhism f : R S having finite kernel and image of finite additive index, the induced ma R S has finite kernel and image of finite index. Proosition 3 (Dirichlet unit theorem for orders). The unit grou O is a finitely generated abelian grou of rank r. Proof. If O is the ring of integers in a number field, this is the Dirichlet unit theorem. In general, Õ is a roduct of such rings of integers, so the result holds for Õ. Since O is of finite index in Õ, the result holds for O too The logarithmic embedding and the regulator R(O). For x R, let λ R (x) = ln x. For x C, let λ C (x) = 2 ln x. Let (K R) = (R ) r 1 (C ) r 2 λ R r 1+r 2 be the homomorhism that alies λ R or λ C coordinate-wise, as aroriate. Let φ be the comosition O K (K R) λ R r 1+r 2. Since ker λ is bounded in K R, ker φ is finite; on the other hand, the codomain of φ is torsion-free; thus ker φ = µ(o). Suose that K is a field. The roof of the classical Dirichlet unit theorem shows that the image φ(õ ) is a full lattice in the hyerlane in R r 1+r 2 where the coordinates sum to 0. Under the rojection to R r = R r 1+r 2 1 defined by forgetting one coordinate, the hyerlane mas isomorhically to R r, and φ(õ ) mas to a full lattice in R r ; the covolume of this lattice is called the regulator, R(Õ). In the general case, φ(õ ) is a direct roduct of lattices in m i=1 Rr(Ki) = R r. As roved in Section 2.4, O is of finite index in Õ, so φ(o ) is again a full lattice L(O) in R r ; its covolume is denoted R(O). 3. The zeta function Retain the notation of the revious section. In what follows, ranges over rime ideals of O with finite residue field. Since O is finitely generated as a Z-algebra, these rime ideals are the same as the maximal ideals of O, which corresond to the closed oints of Sec O. Define N := #O/. Since Sec O is of finite tye over Z, it has a zeta function defined as an Euler roduct, as in [Ser65,. 83]: ζ O (s) = ( ) 1 N s 1. Work of Hecke imlies that ζ O (s) has a meromorhic continuation to the entire comlex lane, and that ζ O (s) has a ole at s = 1 of order m. The analytic class number formula roosed below gives the leading term of ζ O (s) at s = 1. Theorem 4 (Analytic class number formula for orders). Let O be an order in a roduct of number fields K = K 1 K m. Then (1) lim s 1 (s 1) m ζ O (s) = 2 r 1 (2π) r 2 w(o) Disc O h(o)r(o). 3

4 In the classical case when O is the ring of integers of a number field, ζ O is the Dedekind zeta function, and Theorem 4 was roved by Dedekind, as mentioned already in Section 1. Each factor in (1) is multilicative if O is a roduct of rings, so Theorem 4 holds for any roduct of rings of integers, and in articular for the normalization Õ of any O. To rove Theorem 4 for a general order O, we will relate the formulas for O and Õ. 4. Relating the invariants for O and Õ Let X = Sec O and X = Sec Õ. The inclusion O Õ induces a morhism π : X X that is an isomorhism above the comlement of a finite subset Z X. For maximal ideals O and P Õ, we write P when = P O, i.e., when π mas the closed oint P to the closed oint The zeta functions of O and Õ. Proosition 5. We have ζõ(s) lim s 1 ζ O (s) = P (1 NP 1 ) 1 (1 N 1 ) 1. Proof. By definition, ζõ(s) ζ O (s) = P (1 NP s ) 1, (1 N s ) 1 where, for all but finitely many, the fraction on the right is 1; cf. [Jen69, Theorem] The discriminants of O and Õ. Proosition 6. We have ( ) Disc Õ 2 Disc O = #Õ. O Proof. This is standard: Let A M 2 (Z) be the change-of-basis matrix exressing the Z-basis of O in terms of the Z-basis of Õ. Then #(Õ/O) = det A. On the other hand, the matrix whose determinant is Disc O is obtained from the matrix whose determinant is Disc Õ by multilying by A on the right and A T on the left, so Disc O = (det A) 2 Disc Õ The regulators of O and Õ. Proosition 7. R(Õ) #Õ R(O) O = w(õ) w(o). Proof. Let L = L(O) be as in Section 2.5, and let L = L(Õ); these are lattices in the same R r. Alying the snake lemma to 1 µ(o) O L 1 µ(õ) Õ L 0 4 0

5 yields an exact sequence 1 µ(õ) µ(o) Õ O L L 0 of finite grous, the last of which has order R(O)/R(Õ) Relating Pic O and Pic Õ via the Leray sectral sequence. View the abelian grou Õ /O as a skyscraer sheaf on X suorted at ; it is trivial for / Z. We have an exact sequence of sheaves on X 0 O X π O X Õ O 0. The corresonding long exact sequence in cohomology is (2) 0 O Õ Õ O Lemma 8. We have H 1 (X, π O X) Pic X. Proof. From the Leray sectral sequence ( ) Pic X H 1 X, π O X 0. H (X, R q π F ) = H +q ( X, F ) with F = O X we extract an exact sequence 0 H ( ) ( ) 1 X, π O X Pic X H 0 X, R 1 π O X. Lemma 9 below comletes the roof. Lemma 9. The sheaf R 1 π O X on X is 0. Proof. By [Har77, Proosition III.8.1], its stalk (R 1 π O X) x at a closed oint x of X is lim Pic U π 1 U, where U ranges over oen neighborhoods of x in X. Since π 1 (x) is finite, every line bundle on π 1 U becomes trivial on π 1 U for some smaller neighborhood U of x in X. Thus lim Pic π 1 U = 0. U Substituting the isomorhism of Lemma 8 into (2) yields an exact sequence of finite grous (3) 0 Õ O Õ O Pic X Pic X 0. Remark 10. For a more elementary derivation of (3), at least in the case where O is an integral domain; see [Neu99, Proosition I.12.9]. Next we comute the order of the second term in (3). Fix a nonzero ideal c of Õ such that c O; one ossibility is c = nõ, where n := (Õ : O). (In fact, there is a largest c the sum of all of them called the conductor of O.) 5

6 Lemma 11. The natural ma is an isomorhism. Õ O (Õ /c ) (O /c ) Proof. Case 1: c = O. Then 1 c, so c = Õ too; thus both sides are trivial. Case 2: c O. Then c O P for every maximal ideal P of Õ. If an element ā (Õ/c ) is lifted to an element a Õ, then a lies outside each P, so a Õ. Thus Õ (Õ/c ) is surjective. Similarly, O (O /c ) is surjective. Both surjections have the same kernel 1 + c, so the result follows. Lemma 12. If c O, then ) ) ( # (Õ /c = # (Õ /c ) 1 NP 1, P # (O /c ) = # (O /c ) ( 1 N 1). Proof. The maximal ideals of Õ/c are the ideals PÕ for P. An element of Õ/c is a unit if and only if it lies outside each maximal ideal. The robability that a random element of the finite grou Õ/c lies outside PÕ is 1 NP 1, and these events for different P are indeendent by the Chinese remainder theorem, so the first equation follows. The second equation is similar (but easier). Lemma 13. We have Proof. By Lemmas 11 and 12, Õ # O Õ # O = #Õ O P (1 NP 1 ). 1 N 1 = #Õ O P (1 NP 1 ) 1 N 1 ; this holds even if c = O since both sides are 1 in that case. Now take the roduct of both sides and use the isomorhism of finite grous Õ O Õ O. Proosition 14. #Õ O = h(õ) h(o) #Õ O P (1 NP 1 ) 1 N 1. Proof. Take the alternating roduct of the orders of the grous in (3) and use Lemma 13. 6

7 4.5. Conclusion of the roof. To comlete the roof of Theorem 4, we comare (1) for Õ to (1) for O. The ratio of the left side of (1) for Õ to the left side of (1) for O is The ratio of the right sides is Disc Õ Disc O 1/2 ( ζõ(s) lim s 1 ζ O (s). By Proositions 5, 6, 7, and 14, both ratios equal ) w(õ) 1 h(õ) w(o) h(o) R(Õ) R(O). P (1 NP 1 ) 1 (1 N 1 ) 1. Consider the ring 5. An examle with Sec O singular: a fiber roduct O := Z F Z = { (a, b) Z Z : a b (mod ) }. The normalization Õ of O is the ring Z Z; inverting all non-zerodivisors of O gives its ring of fractions K = Q Q. The scheme X := Sec O consists of two coies of the curve Sec Z crossing at the oint () Sec Z. The scheme X := Sec Õ is a disjoint union of two coies of Sec Z. The conductor of O is the Õ-ideal := { (a, b) Z Z : a b 0 (mod ) }. Above the rime of O there are two rimes of Õ, the two coies of (). Proosition 15. We have lim (s s 1 1)2 ζ O (s) = 1 1. Proof. The Riemann zeta function ζ Z (s) has a ole of order 1 at s = 1 with residue 1. Since the one ideal of norm in O is relaced by two ideals of norm in Õ, ζ O (s) = ( 1 s) ζõ(s) lim (s s 1 1)2 ζ O (s) = ( 1 1) lim(s 1) 2 ζõ(s) s 1 Proosition 16. We have ) 2 = ( 1 1) ( lim(s 1)ζ Z (s) s 1 = r 1 (2π) r 2 w(o) Disc O h(o)r(o) =

8 Proof. First, r 1 = 2, r 2 = 0, and r = = 0, so R(O) = 1. The trace ma on Õ or O sends (a, b) to a + b. The elements (1, 1) and (, 0) form a basis of O, so ( ) 2 Disc O = det 2 = 2. Inside Õ = Z Z = ±1 ±1 we have { O ±(1, 1) if is odd, = ±1 ±1 if = 2, so w(o) = By Lemma 11, the exact sequence (3) is { 2 if is odd, 4 if = 2. (4) 1 Õ O F F F Pic O Pic Õ. Õ The image of O in F F F F is ±1, even when = 2 in which case these grous are trivial. On the other hand, Pic Õ = Pic Z Pic Z = 0. Thus (4) yields Pic O F / ± 1, and { ( 1)/2 if is odd, h(o) = 1 if = 2. Combining the above calculations yields r 1 (2π) r 2 w(o) Disc O h(o)r(o) = if is odd if = 2 2 = 1 1. Proositions 15 and 16 verify Theorem 4 for O. Remark 17. Fiber roducts such as O arise as integral Hecke algebras of ellitic modular forms. For examle, the integral Hecke algebra T (cf. [Maz77,. 37]) for modular forms of weight 2 for the congruence subgrou Γ 0 (11) is Z F5 Z. Acknowledgments It is a leasure to thank Tony Scholl for helful discussions. We thank Carlos J. Moreno for bringing the article [Stö98] to our attention. 8

9 References [BL17] Alex Bartel and Hendrik W. Lenstra Jr., Commensurability of automorhism grous, Comositio Math. 153 (2017), no. 2, , DOI /S X X. [Dir94] P. G. Lejeune Dirichlet, Vorlesungen über Zahlentheorie, 4th ed., Braunschweig, Edited by and with sulements by R. Dedekind. [Gal73] V. M. Galkin, Zeta-functions of certain one-dimensional rings, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 3 19 (Russian). MR [Gre89] Barry Green, Functional equations for zeta functions of non-gorenstein orders in global fields, Manuscrita Math. 64 (1989), no. 4, , DOI /BF MR [Har77] Robin Hartshorne, Algebraic geometry, Sringer-Verlag, New York, Graduate Texts in Mathematics, No. 52. MR (57 #3116) [Hil97] David Hilbert, Die Theorie der algebraische Zahlkörer, Jahresbericht der Deutschen Mathematiker- Vereinigung 4 (1897), ; English transl., David Hilbert, The theory of algebraic number fields (1998), xxxvi+350. Translated from the German and with a reface by Iain T. Adamson; With an introduction by Franz Lemmermeyer and Norbert Schaacher. MR (99j:01027). [Jen69] W. E. Jenner, On zeta functions of number fields, Duke Math. J. 36 (1969), MR [Maz77] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), (1978). MR (80c:14015) [Neu99] Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Princiles of Mathematical Sciences], vol. 322, Sringer-Verlag, Berlin, Translated from the 1992 German original and with a note by Norbert Schaacher; With a foreword by G. Harder. MR (2000m:11104) [Ser65] Jean-Pierre Serre, Zeta and L functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harer & Row, New York, 1965, MR (33 #2606) [Stö98] Karl-Otto Stöhr, Local and global zeta-functions of singular algebraic curves, J. Number Theory 71 (1998), no. 2, , DOI /jnth MR [ZG97a] W. A. Zúñiga Galindo, Zeta functions and Cartier divisors on singular curves over finite fields, Manuscrita Math. 94 (1997), no. 1, 75 88, DOI /BF MR [ZG97b] W. A. Zúñiga-Galindo, Zeta functions of singular curves over finite fields, Rev. Colombiana Mat. 31 (1997), no. 2, MR Deartment of Mathematics, Baruch College, The City University of New York, One Bernard Baruch Way, New York, NY , USA address: bruce.jordan@baruch.cuny.edu Deartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA , USA address: oonen@math.mit.edu URL: htt://math.mit.edu/~oonen/ 9