ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI

Transcription

1 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI Abstract. In this paper, we evaluate the altings height of a CM abelian surface in terms of the Barnes double Gamma function at algebraic points. We illustrate this for Jacobians of genus hyperelliptic curves with complex multiplication by non-abelian CM fields. To perform these evaluations, we carry out an extensive study of the combinatorial geometry of certain finite sets of algebraic numbers which we call Shintani sets. In particular, we develop an algorithm to compute these sets. 1. Introduction and statement of results The classical Chowla-Selberg formula [CS67] is a remarkable identity which evaluates the Dedekind eta function at CM points in terms of Euler s Gamma function at rational numbers. The Chowla- Selberg formula can be reformulated geometrically as an identity for the altings height of a CM elliptic curve. In this paper, we evaluate altings heights of CM abelian surfaces in terms of the Barnes double Gamma function at algebraic points see Theorem 1.1. These identities can be viewed as explicit, two-dimensional analogs of the classical Chowla-Selberg formula. We illustrate this for Jacobians of genus hyperelliptic curves with complex multiplication by non-abelian CM fields see e.g. Theorem 1.. The analysis needed to perform these evaluations occupies the majority of this paper; among other things, we carry out an extensive study of the combinatorial geometry of certain finite sets of algebraic numbers which we call Shintani sets see Section 4, and develop an algorithm to compute the Shintani sets see Appendix B The Chowla-Selberg formula. In this section, we review the classical Chowla-Selberg formula and explain its geometric reformulation see e.g. Gross [Gro80] and Deligne [Del85]. or a very nice discussion of the Chowla-Selberg formula and its applications, we refer the reader to Zagier s article [Zag08]. Let K be an imaginary quadratic field of discriminant D < 0. Let ζ K s be the Dedekind zeta function, h D be the class number, w D be the number of units, χ D be the Kronecker symbol, and Lχ D, s be the Dirichlet L function associated to χ D. The Dedekind eta function is the weight 1/ modular form for SL Z defined by the infinite product C ηz := q 1/4 n=1 1 q n, q := e πiz. Using the factorization ζ K s = ζslχ D, s and Kronecker s first limit formula, one can establish the identity log Imτ C 1 ητ C 1 = h D log D/ 1 logπ + L χ D, 0, 1.1 Lχ D, 0 where the sum is over a complete set of CM points τ C 1 of discriminant D on the modular surface X = SL Z\H. There are h D such points, corresponding to the ideal classes C of K. 1

2 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI Now, recall that the Hurwitz zeta function is defined by 1 ζs, z =, Res > 1, Rez > 0. n + z s n=0 Lerch [Ler97] evaluated the second term in the Taylor expansion at s = 0 of the Hurwitz zeta function in terms of Euler s Gamma function Γz as d ζs, z ds = log Γ z, 1. s=0 where Γ z := Γz/ π. functions By expressing Lχ D, s as a linear combination of Hurwitz zeta Lχ D, s = D s D k=1 χ D kζs, k/d, 1.3 one can employ Lerch s identity 1. and Dirichlet s class number formula Lχ D, 0 = h D/w D to relate the logarithmic derivative of Lχ D, s at s = 0 to Euler s Gamma function at rational arguments, L χ D, 0 Lχ D, 0 = logd + w D h D D χ D k log Γk/D. 1.4 inally, by substituting 1.4 into 1.1 and exponentiating, one arrives at the Chowla-Selberg formula: 1 h D/ ImτC 1 ητ C 1 = 4π D Γk/D w Dχ Dk/ D C To explain the geometric reformulation of 1.5, we require the stable altings height of a CM abelian variety. Let be a totally real number field of degree n. Let E/ be a CM extension of and Φ be a CM type for E. Let X Φ be an abelian variety defined over Q with complex multiplication by O E and CM type Φ. We call X Φ a CM abelian variety of type O E, Φ. Let L Q be a number field over which X Φ has everywhere good reduction and choose a differential ω H 0 X Φ, Ω n X Φ. Then the altings height of X Φ is defined by h al X Φ := 1 [L : Q] σ:l C k=1 k=1 log ω σ ω σ XΦ σc. The altings height does not depend on the choice of L, ω, or X Φ. In particular, the altings height depends only on the choice of CM type Φ, and hence is often denoted by h al Φ. Assume now that E = Q D is an imaginary quadratic field and X = X Φ is a CM elliptic curve of type O E, Φ. Then the altings height of X can be calculated directly in terms of the CM values of ηz see e.g. [Gro80, Sil86, BSM15, BS + 17], h al X = log 3/ π 1 h D log Imτ C 1 ητ C C

3 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 3 The identity 1.6 allows us to express the pre Chowla-Selberg formula 1.1 in the equivalent form h al X = 1 L χ D, 0 Lχ D, log D 1 logπ. 1.7 Then by substituting 1.4 into 1.7, we can express the exact Chowla-Selberg formula 1.5 in the equivalent geometric form 1/ D D exp[h al X] = Γk/D w Dχ Dk/4h D. 1.8 π k=1 gcdk,d=1 1.. Evaluating the altings height of a CM abelian surface. In this section, we state our identities which evaluate altings heights of CM abelian surfaces in terms of the Barnes double Gamma function at algebraic points. We will need the following notation and definitions. Let be a real quadratic field. Let d be the discriminant, O be the ring of integers, and O be the group of units of, respectively. Let E be a CM extension of, and let E c be the Galois closure of E/Q. Then E/Q is either biquadratic with GalE/Q = Z/Z Z/Z, cyclic with GalE/Q = Z/4Z, or non-galois with GalE c /Q = D 4. Let d E be the discriminant, µ E/ = [O E : O ] be the index of the unit groups, h E be the class number, and D E/ be the relative discriminant, respectively. Let χ E/ be the quadratic narrow ray class character modulo D E/ associated to the extension E/. Next, we define the Barnes double Gamma function. Let z be a complex number and ω = ω 1, ω be a pair of complex numbers. Assume that z, ω 1 and ω have positive real part. or a complex number w C \, 0], let w s = exps log w, where log w = log w + i arg w with arg w < π. Define the double Hurwitz zeta function ζ H s, z, ω := m,n=0 1, Res >. z + mω 1 + nω s The function ζ H s, z, ω has a meromorphic continuation to C with simple poles at s = 1,. In analogy with Lerch s identity 1., the normalized Barnes double Gamma function is defined by log Γ z, ω := d ds ζ Hs, z, ω. s=0 Now, the function Γ z = Γz/ π has a simple pole at z = 0 with residue 1/ π. Hence, if we define ρ 1 := 1/ π, then we can write Γz = ργ z. Analogously, the function Γ z, ω has a simple pole at z = 0. Then if ρ ω 1 denotes the residue of Γ z, ω at z = 0, the Barnes double Gamma function is defined by Γ z, ω := ρ ωγ z, ω. More concretely, the Barnes double Gamma function can also be defined by Γ z, ω := z, ω 1, where z, ω is given by the Weierstrass product expansion z, ω := z exp γ ωz + z γ 1ω m,n 1 + z mω 1 + nω exp z + mω 1 + nω z mω 1 + nω, the product being over all pairs of integers m, n Z 0 with m, n 0, 0. The function z, ω is entire, and the constants γ ω, γ 1 ω are explicit higher analogs of Euler s constant γ.

4 4 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI inally, let ε > 1 be a generator of the group of totally positive units O,+ of, and let ε denote the conjugate of ε by the nontrivial automorphism in Gal/Q. Define the Shintani set by Rε, D E/ := { z = x + yε D 1 E/ x, y Q, 0 < x 1, 0 y < 1, D E/ z coprime to D E/ }, where z is the ideal generated by z. The Shintani set is finite, and embeds via the map α α, α as a subset of the Shintani cone Cε generated by the vectors 1, 1 and ε, ε, defined by Cε := { t 1 1, 1 + t ε, ε : t 1 > 0, t 0 } R +. Let B x := x x + 1/6 be the second Bernoulli polynomial. We will establish the following identity for the altings height of a CM abelian surface, which can be viewed as an explicit, two-dimensional analog of the classical Chowla-Selberg formula 1.8. Theorem 1.1. Let be a real quadratic field with narrow class number 1. Let E be a CM extension of with E/Q non-biquadratic. Let X = X Φ be a CM abelian surface of type O E, Φ. Then exp[h al X] = 1 1/4 de ε ce/ / Γ z σ, 1, ε σ µ E/ χ E/ D E/ z /4h E, π where d 3 ce/ := ε ε 4 z Rε,D E/ σ Gal/Q µ E/ h E z Rε,D E/ z=x+yε χ E/ D E/ z B x A non-abelian example. The first evaluation of the altings height of a CM abelian surface in terms of special Gamma values is due to Bost, Mestre, and Moret-Bailly [BMM-B], who evaluated the altings height of the Jacobian J C of the genus CM curve C : y = x as 1 h al J C = logπ 1 log Γ1/5 5 Γ/5 3 Γ3/5Γ4/ The Jacobian J C is a CM abelian surface of type O E, Φ where E is the abelian cyclic quartic CM field E = Qζ 5 whose real quadratic subfield Q 5 has narrow class number 1. Hence, we can use Theorem 1.1 to evaluate this altings height as see Section exp[h al J C ] = 1 π /10 Γ 5± 5 10, 1, 3± 5 5/4 Γ 3 5± 5 10, 1, 3± 5 5/4 ± Γ 1 5± 5 10, 1, 3± 5 5/4 By approximating the values of the Barnes double Gamma function appearing in this evaluation see Remark 5.1, we find that h al J C Γ 4 5± 5 10, 1, 3± 5 5/4. This serves as a numerical verification of Theorem 1.1. On the other hand, our primary objective is to demonstrate how Theorem 1.1 can be used to evaluate the altings heights of Jacobians of genus curves with complex multiplication by nonabelian CM fields, providing the first examples of this type. To illustrate this, consider the quartic polynomial px = X X The value given here differs from [BMM-B, Proposition 1] by addition of the number logπ, due to a difference in the normalization of the altings height.

5 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 5 whose roots are given by α = ± /. ix a root α and define a := α + 5. Then the hyperelliptic curve C a : y = a + 3x 6 + 4a 8x x 4 + a + 0x 3 + 4a + 5x + a + 4x is a genus curve defined over the real quadratic field Q 41. Its Jacobian J Ca is a CM abelian surface of type O E, Φ where E is the non-abelian quartic CM field E = Q / whose real quadratic subfield = Q 5 has narrow class number 1. We will establish the following identity for the altings height of the Jacobian J Ca see Example.3. Theorem 1.. The altings height of the Jacobian J Ca of the hyperelliptic curve C a defined by 1.9 is given by exp[h al J Ca ] = Γ m + 8m + n 13± 5 π 5 8, 1, 3± 5 cm,n 4, 0 m 4 ± 1 n 8 where the numbers cm, n {±1} are defined in Table 1. Values of cm, n m n Table The combinatorial geometry of Shintani sets. The algebraic numbers in the Shintani set Rε, D E/ at which the Barnes double gamma function Γ is evaluated in Theorem 1.1 are analogous to the rational numbers { k R D := D 1 } D Z 1 k D, gcdk, D = 1 at which Euler s Gamma function Γ is evaluated in the Chowla-Selberg formula 1.8. This analogy can be understood from the following group-theoretic perspective. The set of rational numbers defined by { k R D := D 1 } D Z 1 k D is a complete set of coset representatives for the quotient group / 1 D Z Z

6 6 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI inside the standard fundamental domain 0, 1] for the group R/Z. Similarly, in Proposition 4.3 we will show that the set of algebraic numbers defined by Rε, D E/ := {z = x + yε D 1 E/ is a complete set of coset representatives for the quotient group / GD E/ := D 1 E/ Z + Zε x, y Q, 0 < x 1, 0 y < 1} whose images under the embedding ι : R given by α α, α lie inside the standard fundamental parallelogram for the group P := {t 1 1, 1 + t ε, ε t 1, t R, 0 < t 1 1, 0 t < 1} ιgd E/ = R /ιz + Zε. Hence, after removing from the Shintani set Rε, D E/ the numbers z for which χ E/ D E/ z = 0 by enforcing the condition that D E/ z be coprime to D E/, we see that Rε, D E/ is analogous to R D. Now, observe that the size of the set R D can be expressed as #R D = volr/z vol0, 1] ϕd = ϕd, 1.10 volz dq where ϕ is the Euler ϕ-function. By a combinatorial analysis which involves a remarkable theorem of Pick which expresses the area of a lattice polygon in R in terms of the number of lattice points inside and on the boundary of the polygon, we will prove the following analog of 1.10 for the Shintani set Rε, D E/. Theorem 1.3. Let be a real quadratic field with narrow class number 1. Then the size of the Shintani set Rε, D E/ is given by where #Rε, D E/ = volr /ιz + Zε ϕd volιo E/ = volp ϕd E/ = ε ε ϕd E/, d d ϕd E/ := N /Q D E/ is the generalized Euler ϕ-function for number fields. p D E/ 1 1 N /Q p Remark 1.4. The quantity ε ε / d is always a positive integer. In fact, if one writes ε in terms of the integral basis {1, d + d /} for O as ε = a + bd + d / for some a, b Z with b 1, then ε ε / d = b. In Table, we display a list of the non-biquadratic quartic CM fields E of discriminant d E 4000 such that the corresponding totally real subfield has narrow class number 1. In the table, we also list all of the quantities needed to use the formula from Theorem 1.3 to give the size of the Shintani set Rε, D E/. We have placed a in each entry where the Galois group is GalE c /Q = Z/4Z. All other entries correspond to non-abelian quartic CM fields with Galois group GalE c /Q = D 4. The first three entries of the table correspond to the CM fields that we chose for the examples that we give in Section. Moreover, as the reader can observe from the table, these were chosen so that we have the least number of elements in the Shintani sets.

7 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 7 E d E d ε D E/ ε ε d ϕd E/ #Rε, D E/ Qζ 5 Q Q 5 Q Q 5 Q Q Q + Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Table. The number of elements in the Shintani set Rε, D E/ for d E Discussion. The periods of CM abelian varieties have been studied extensively by many authors; see for example the works of Gross [Gro78], Shimura [Shi79], Anderson [And8], Deligne [Del8], and Yoshida [Yos99]. Colmez [Col93] gave a conjectural generalization of the identity 1.7 which relates the altings height of a CM abelian variety X = X Φ of type O E, Φ to logarithmic derivatives of Artin L functions at s = 0. In [Yan10a, Yan10b, Yan13], Tonghai Yang proved the Colmez conjecture for a large class of non-biquadratic quartic CM fields, including the first non-abelian cases. This result is now known for all non-biquadratic quartic CM fields. In particular, if E is a non-biquadratic quartic CM field and X is a CM abelian surface of type O E, Φ, then the altings height of X is given by h al X = 1 L χ E/, 0 Lχ E/, log de d logπ, 1.11 where Lχ E/, s is the incomplete L function of the Hecke character χ E/ associated to the quadratic extension E/. To evaluate the altings height using 1.11, we need an analog of the identity 1.4 for the logarithmic derivative of Lχ E/, s at s = 0. In [Shi77], Shintani gave formulas for special values of ray class L functions of real quadratic fields in terms of double Gamma values. By building on Shintani s work, we will establish the required analog of 1.4. We stress that the primary

8 8 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI difficulty is to produce an identity in which all quantities involved are explicit and can be effectively computed. or example, we undertake an extensive study of the combinatorial geometry of certain finite sets of algebraic numbers appearing in Shintani s formulas which we call Shintani sets. One of our discoveries is an exact formula for the number of elements in a Shintani set; see Theorems 1.3 and 4.1. This formula plays a crucial role in our examples which evaluate altings heights of Jacobians of genus CM curves see Section. To explain, for a non-abelian CM field, the number of elements in the corresponding Shintani set will usually be very large. Using our formula, we can choose the CM fields so as to control the number of elements in the corresponding Shintani sets, and therefore give more manageable evaluations see Table. Roughly speaking, to prove this formula we introduce a group structure on Shintani sets, then use this structure to give a combinatorial analysis which reduces the problem to an application of Pick s theorem, which gives a formula for the area of a lattice polygon in R in terms of the number of lattice points inside and on the boundary of the polygon. As another consequence of our analysis, we discover certain orthogonality relations satisfied by ray class characters when evaluated on Shintani sets see Theorem 4.. This allows us to further simplify our identities for the altings height. inally, in Appendix B we develop an algorithm to compute the Shintani sets.. altings heights of Jacobians of CM curves of genus In this section, we use Theorem 1.1 to give explicit evaluations of altings heights of CM abelian surfaces X = X Φ of type O E, Φ where E is a non-biquadratic quartic CM field. It is known that such a CM abelian surface is the Jacobian J C of a nonsingular CM curve C of genus. In Appendix A, we use work of Bouyer and Streng [BS15] to give models for genus curves whose Jacobians have complex multiplication by a prescribed CM field E. In Appendix B, we give an algorithm to compute Shintani sets. All computer calculations in our examples were performed using SageMath [S + 09]. Example.1. Consider the cyclotomic field E = Qζ 5, where ζ 5 := e πi/5 is a primitive 5th root of unity. Then E is a CM extension of the real quadratic field = Q 5 with narrow class number 1 and E/Q is cyclic. The Jacobian J C of the genus curve C : y = x is a CM abelian surface of type O E, Φ. Note that E has class number h E = 1. Also, we have d E = 15 and d = 5. The relative discriminant is D E/ = 5O = 5. The ring of integers of is O = Z + Z and the group of units is O = {±εn 5 n Z}, where ε 5 := 1 + 5/ is the fundamental unit of. Since ε 5 < 0, the subgroup of totally positive units is given by O,+ = {ε n 5 n Z}. Hence, we let ε := ε 5 = 3 + 5/ be the generator of O,+. Since the roots of unity in E are the 10-th roots of unity µ 10 = {±ζ5 k k = 0, 1,, 3, 4}, we have O E = Z/10Z Z and O = Z/Z Z. Hence, µ E/ = [O E : O ] = 5. We computed the Shintani set { { m Rε, D E/ = 5 + m } 5 ε m = 1,, 3, 4 =, m 10 } m = 1,, 3, 4, and the corresponding values of the quadratic ray class character χ E/ modulo D E/, { χ E/ D , m = 1, 4 E/ 10 m = χ 5 m = 1, m =, 3.

9 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 9 rom the character values, we compute z Rε,D E/ z=x+yε χ E/ D E/ z B x = 4 m χ 5 mb = Hence, c = ce/ = 5/5. By substituting the preceding calculations in Theorem 1.1, we get exp[h al J C ] = / Γ 5± 5 π 10 m, 1, 3± χ 5m. m=1 ± m=1 Then expanding this expression yields the following result. Theorem.. The altings height of the Jacobian J C of the genus curve C : y = x is given by exp[h al J C ] = /10 Γ 5± 5 10, 1, 3± 5 5/4 Γ 3 5± 5 10, 1, 3± 5 5/4 π ± Γ 1 5± 5 10, 1, 3± 5 5/4 Γ 4 5± 5 10, 1, 3± 5. 5/4 Numerically, the value of this altings height is h al J C We now turn to our primary objective, which is to explicitly evaluate altings heights of Jacobians of genus curves with complex multiplication by non-abelian quartic CM fields. According to Table, the non-abelian quartic CM fields whose corresponding Shintani sets have the least number of elements are Q / Therefore, we work with these fields. and Q 5, Example.3. Consider the field E = Q where = / 0. Then E is a CM extension of the real quadratic field = Q 5 with narrow class number 1 and E/Q is non-galois with GalE c /Q = D 4. Define the genus curve C a : y = a + 3x 6 + 4a 8x x 4 + a + 0x 3 + 4a + 5x + a + 4x where a := α + 5 and α = ± / is a root of X X + 0. Then C a is defined over the real quadratic field Q 41 and the Jacobian J Ca is a CM abelian surface of type O E, Φ see Appendix A, Example A.1. Note that E has class number h E = 1. Also, we have d E = 105 and d = 5. The relative discriminant is D E/ = O =. Recall that ε 5 := 1 + 5/ is the fundamental unit of and ε := 3 + 5/ is the generator of the group of totally positive units O,+. Since the roots of unity in E are {±1}, we have O E = Z/Z Z and O = Z/Z Z. Hence, µ E/ = [O E : O ] = 1. We computed the Shintani set Rε, D E/. This set can be visualized geometrically in R + via the embedding α α, α as a finite subset of the Shintani cone as shown in igure 1. Cε := { t 1 1, 1 + t ε, ε : t 1 > 0, t 0 } R +, The shaded parallelogram in igure 1 is the subset of the Shintani cone Cε determined by the inequalities 0 < t 1 1 and 0 t < 1, which correspond to the inequalities appearing in the definition of Rε, D E/.

10 10 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI y 4 3 Cε 1 1, 1 ε, ε x igure 1. The embedding of Rε, D E/ into Cε In order to give a uniform description of the points in Rε, D E/, it is convenient to express them in terms of a Z-basis for D 1 E/. In particular, for the Z-basis given by D 1 E/ = Z 1 + Z , 8 we find that Rε, D E/ = { z m,n := m + 8m + n } 0 m 4, 1 n 8. We computed the corresponding values of the quadratic ray class character χ E/ modulo D E/, which are given in Table 3. cm, n := χ E/ DE/ z m,n {±1}, Values of cm, n m n Table 3 rom the character values, we compute χ E/ D E/ z B x = z Rε,D E/ z=x+yε 0 m 4 1 n 8 m + 5n cm, nb = ,

11 where we used ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 11 z m,n = x + yε = m + 5n m + n ε. 41 Hence, ce/ = 8 5/41. By substituting the preceding calculations in Theorem 1.1, we get exp[h al J Ca ] = π 5 0 m 4 1 n 8 ± Γ m + 8m + n 13± 5 8, 1, 3± 5 cm,n 4. We have now proved the following result, which is a restatement of Theorem 1. from the introduction. Theorem.4. The altings height of the Jacobian J Ca of the genus curve C a over Q 41 defined by.1 is given by exp[h al J Ca ] = Γ m + 8m + n 13± 5 π 5 8, 1, 3± 5 cm,n 4, 0 m 4 ± 1 n 8 where the numbers cm, n {±1} are given in Table 3. Numerically, the value of this altings height is approximately h al J Ca Example.5. Consider the field E = Q where = 5 0. Then E is a CM extension of the real quadratic field = Q with narrow class number 1 and E/Q is non-galois with GalE c /Q = D 4. Define the genus curve C a : y = x 6 + a + 4x 5 + 3a + 14x a + 8x 3 + 9a + 3x + 16a 16x 4a + 1. where a := α + and α = ± 5 ± 17/ is a root of X 4 + 5X +. Then C a is defined over the real quadratic field Q 17 and the Jacobian J Ca is a CM abelian surface of type O E, Φ see Appendix A, Example A.. Note that E has class number h E = 1. Also, we have d E = 1088 and d = 8. The relative discriminant is D E/ = O =. The fundamental unit of is ε 5 := 1 + and ε := 3 + is the generator of the group of totally positive units O,+. Since the roots of unity in E are {±1}, we have O E = Z/Z Z and O = Z/Z Z. Hence, µ E/ = [O E : O ] = 1. We computed the Shintani set Rε, D E/. This set can be visualized geometrically in R + via the embedding α α, α as a finite subset of the Shintani cone Cε := { t 1 1, 1 + t ε, ε : t 1 > 0, t 0 } R +, as shown in igure. In order to give a uniform description of the points in Rε, D E/, it is convenient to express them in terms of a Z-basis for D 1 E/. In particular, for the Z-basis given by D 1 E/ = Z 1 + Z 6 +, 17 we find that Rε, D E/ = { z m,n := m + 4m + n } 0 m 8, n Sm,

12 1 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI 6 y 4 Cε 1, 1 4 ε, ε x igure. The embedding of Rε, D E/ into Cε. where {, 3, 4} if m = 0 {1,, 3, 4} if m = 1,, 3 Sm := {1, 3} if m = 4 {0, 1,, 3} if m = 5, 6, 7 {0, 1, } if m = 8. We computed the corresponding values of the quadratic ray class character χ E/ modulo D E/, which are given in Table 4. cm, n := χ E/ DE/ z m,n {±1}, Values of cm, n n m rom the character values, we compute χ E/ D E/ z B x = where we used z Rε,D E/ z=x+yε z m,n = x + yε = Table 4 0 m 8 n Sm m + 9n 9 34 m + 9n 9 cm, nb = , + 4m n + 1 ε. 34

13 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 13 Hence, ce/ = 4 /17. Substituting the preceding calculations in Theorem 1.1 yields exp[h al J Ca ] = π 8 We have now proved the following result. 0 m 4 n Sm ± Γ m + 4m + n 1 6± 17, 1, 3 ± cm,n 4. Corollary.6. The altings height of the Jacobian J Ca of the genus curve C a over Q 17 defined by. is given by exp[h al J Ca ] = π 8 0 m 4 n Sm ± Γ m + 4m + n 1 6± 17, 1, 3 ± cm,n 4, where the numbers cm, n {±1} are given in Table 4. Numerically, the value of this altings height is approximately h al J Ca Ray class characters and L functions In this section, we review some facts we will need regarding Hecke Grössencharacters and narrow ray class characters following [Neu99, Section VII.6]. Let K be a number field of degree n = r 1 +r, where r 1 is the number of real embeddings of K and r is the number of pairs of complex conjugate embeddings of K. Let O K be the ring of integers of K. Let σ t : K R, t = 1,..., r 1 be the real embeddings, and let σ t : K C, t = r 1 + 1,..., r 1 + r be a fixed choice of complex embeddings such that the set of all complex embeddings is given by the σ t and their conjugates σ t. We define the Minkowski space K R := R r 1 C r = K Q R. Using the fixed choice of embeddings, we define a map j : K K R by jα := σ 1 α,..., σ r1 α, σ r1 +1α,..., σ r1 +r α. Let m be an integral ideal of K and I K m be the group of fractional ideals of K that are relatively prime to m. A Grössencharacter modulo m is a character χ : I K m S 1 for which there exists a pair of characters χ f : O K /m S 1, χ : K R S1, such that for every α O K with α, m = 1 we have χ α = χ f αχ α. Here when we write χ α, we identify α with its image jα K R to ease notation. The characters χ f and χ are called the finite and infinite parts of χ, respectively. The conductor of a Grössencharakter χ modulo m is the smallest integral ideal f χ such that f χ m and χ is the restriction of a Grössencharakter χ modulo f χ. A Grössencharakter χ modulo m is primitive if it is not the restriction of a Grössencharakter χ modulo m for any proper divisor m m in other words, the conductor is f χ = m. A character λ : K R S1 can be described explicitly as follows. Let x = x t K R and e = e t C r 1+r. We define x e := x et t, x := x t, and Nx := t x t. An admissible vector is a vector p = p t Z r 1+r such that p t {0, 1} for t = 1,..., r 1. Given a character λ : K R S1, there exist uniquely determined vectors p, q with p Z r 1+r admissible and q R r 1+r such that for every x K R, we have λx = Nx p x p+iq

14 14 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI where x p x p+iq := x pt t x t pt+iqt = signx t pt x t iqt. If a Grössencharacter χ modulo m has infinite part χ : K R S1 given by χ x = Nx p x p+iq, we say that χ is of type p, q. We now define the narrow ray class characters modulo m. Let P + K m < I Km be the subgroup of principal fractional ideals P + K m = { α I Km α K, α 0, α 1 mod m}, where the congruence α 1 mod m means that if α = β/γ with β, γ O K relatively prime to m, then β γ mod m. The narrow ray class group modulo m is the quotient group R + K m := I Km/P + K m. A narrow ray class character modulo m is a character χ : R + K m S1 of the narrow ray class group, or equivalently, a character χ : I K m S 1 such that χp + K m = {1}. The conductor of χ is the smallest integral ideal f χ m such that χ factors through R + K f χ. Given a narrow ray class character χ modulo m, there exists a Grössencharacter χ modulo m of type p, 0 for some admissible vector p = p t Z r 1+r with p t = 0 for t = r 1 + 1,..., r 1 + r, such that χa = χa for every fractional ideal a I K m see [Neu99, Chapter VII, Proposition 6.9] In particular, for α O K relatively prime to m, we have α p χ α = χ f αn α where χ f : O K /m S 1 is the finite part of χ. Assume now that K is a real quadratic field, so that r 1 =, r = 0, σ 1 = id, and σ is the nontrivial embedding. The four possibilities for the admissible vector p Z are 0, 0, 1, 0, 0, 1, and 1, 1. In particular, on principal integral ideals α coprime to m, the narrow ray class characters modulo m take one of the following forms: χ α = χ f α p = 0, 0, χ α = χ f α signα p = 1, 0, χ α = χ f α signα p = 0, 1, χ α = χ f α signn K/Q α p = 1, 1. The L function of a narrow ray class character χ modulo m is defined by Lχ, s := χa N K/Q a s, Res > 1 a O K where N K/Q a denotes the norm and the sum is over non-zero integral ideals here we define χa = 0 when a is not relatively prime to m. If the admissible vector corresponding to χ is p = p 1, p, then the completed L function is given by see [Neu99, pp ] Λχ, s := d K N K/Q m s/ π s+p 1+p s + / p1 s + p Γ Γ Lχ, s, where d K is the absolute value of the discriminant of K. If χ is primitive, the completed L function satisfies the functional equation Λχ, s = W χλχ, 1 s, 3.1

15 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 15 where W χ is an explicit complex number of absolute value 1 see [Neu99, Chapter VII, Theorem 8.5]. Remark 3.1. A calculation with the functional equation shows that if χ has admissible vector p = p 1, p, then the order of vanishing of Lχ, s at s = 0 is p 1 + p. 4. The combinatorial geometry of Shintani sets In this section, we undertake an extensive study of the combinatorial geometry of certain finite sets of algebraic numbers which arise in Shintani s decomposition of ray class L functions for real quadratic fields as finite linear combinations of Shintani zeta functions. We call these sets Shintani sets. We will prove an explicit formula for the size of Shintani sets. Moreover, we will use this formula to prove certain orthogonality relations for ray class characters with respect to Shintani sets Notation. The following notation and assumptions will remain fixed throughout this section. Let be a real quadratic field with narrow class number 1. Then = Q p for a prime p with p = or p 1 mod 4. Let O be the ring of integers, O be the group of units, and ε p > 1 be the fundamental unit of. or a subset S, let S + denote the corresponding subset of totally positive elements of S. Since O = {±εn p n Z}, we have O,+ = {ε n p n Z} if ε p 0 and O,+ = {ε n p n Z} if ε p < 0, where ε p denotes the conjugate of ε p. In particular, if we define ε := ε p if ε p 0 and ε := ε p if ε p < 0, then ε generates O,+. Let f O be an integral ideal of. Then the Shintani set associated to f is defined by Rε, f := { z = x + yε f 1 } x, y Q, 0 < x 1, 0 y < 1. Similarly, the restricted Shintani set associated to f is defined by Rε, f := { z = x + yε f 1 x, y Q, 0 < x 1, 0 y < 1, f z coprime to f }. or brevity, we will simply refer to both of these sets as Shintani sets. The set Rε, f is finite. To see this, let ιx := x, x denote the embedding of x into R. Then ιf 1 is a lattice in R, and ι Rε, f ιf 1 is the subset consisting of points u, v of the form u, v = x + yε, x + yε R where 0 < u < 1 + ε, 0 < v < 1 + ε. Since ιf 1 is discrete, it follows that Rε, f is finite. 4.. Main results. The main result of this section is the following theorem, which gives an explicit formula for the size of the Shintani sets. Theorem 4.1. Let f be a non-zero integral ideal of. Then the sizes of the associated Shintani sets are given by and where # Rε, f = volr /ιz + Zε volo #Rε, f = volr /ιz + Zε volo ϕf := N /Q f is the generalized Euler ϕ-function for number fields. p f N /Q f = ε ε d ϕf = ε ε d 1 1 N /Q p N /Q f ϕf,

16 16 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI We will also prove the following orthogonality relations for any narrow ray class character χ modulo f. Theorem 4.. Let f be an integral ideal of and χ be a narrow ray class character modulo f. Then we have 0, χ 1 χf z = ε ε ϕf, χ = 1. d z Rε,f 4.3. A group structure on Rε, f. In order to prove Theorems 4.1 and 4., we will introduce a binary operation on the Shintani set Rε, f which gives it the structure of a finite abelian group. We will need the following modified fractional part functions. or x R, define {x} 0,1] and {x} [0,1 by { {x}, if x Z {x} 0,1] := 1, if x Z and {x} [0,1 := {x}, where {x} is the usual fractional part of a real number x. Now, since f is a nonzero integral ideal of, then f 1 contains the ring of integers O, so that we have the chain of abelian groups under addition Z + Zε O f 1. In the following proposition, we relate the Shintani set Rε, f to the quotient group f 1 /Z + Zε. Proposition 4.3. or each w f 1 there is a unique element z w Rε, f such that w z w Z + Zε. Hence, the Shintani set Rε, f is a complete set of coset representatives for the quotient group f 1 /Z + Zε. Proof. Let w f 1. Since {1, ε} is a Q-basis for, then we can write w = X + Y ε for some unique X, Y Q. Now, define z w = x + yε := {X} 0,1] + {Y } [0,1 ε. Then { z w + X + Y ε if X Z w = z w + X 1 + Y ε if X Z. Since w f 1 and both X + Y ε f 1 and X 1 + Y ε f 1, we see that z w f 1. By construction, 0 < x 1 and 0 y < 1, so we conclude that z w = x + yε Rε, f. To see that this element is unique, suppose that z w = x + ỹε is another such element. Then z w z w = x x + y ỹε Z + Zε. By the uniqueness of the representation of an element of as a Q-linear combination of the basis elements {1, ε}, we see that x x Z and y ỹ Z. Then, since 1 < x x < 1 and 1 < y ỹ < 1, we conclude that x x = y ỹ = 0. Thus z w = z w, and this proves uniqueness. Let : Rε, f Rε, f Rε, f

17 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 17 be the binary operation defined as follows. or z 1, z Rε, f, by Proposition 4.3 we let z 1 z be the unique element of Rε, f such that z 1 + z z 1 z Z + Zε. Hence z 1 z is the unique element of Rε, f such that z 1 z + Z + Zε = z 1 + Z + Zε + z + Z + Zε. Remark 4.4. Observe that if z 1 = x 1 + y 1 ε Rε, f and z = x + y ε Rε, f, then z 1 z = {x 1 + x } 0,1] + {y 1 + y } [0,1 ε. Proposition 4.5. The Shintani set Rε, f is a finite abelian group with respect to the binary operation. Proof. irst, observe that Rε, f since 1 Rε, f. Now, the binary operation is clearly commutative, so we proceed to prove associativity, existence of a neutral element, and existence of inverses. Associativity: Let z 1, z, z 3 Rε, f. Then as an immediate consequence of the definition of, there are integers m, n, m, ñ Z such that and z 1 + z + z 3 z 1 z z 3 = m + nε Z + Zε z 1 + z + z 3 z 1 z z 3 = m + ñε Z + Zε. Therefore, by the uniqueness part of Proposition 4.3, it follows that z 1 z z 3 = z 1 z z 3. Neutral element: We show that 1 Rε, f is the neutral element for the operation. z Rε, f. Then since z + 1 z = 1 Z + Zε, we see by uniqueness that z 1 = z. Let Existence of inverses: Let z Rε, f. By Proposition 4.3, we define z to be the unique element of Rε, f such that This immediately implies that 1 z z Z + Zε. z + z 1 Z + Zε, so again by uniqueness we have z z = 1. This completes the proof that Rε, f, is a finite abelian group. Now, since has narrow class number 1, we can write f = α for some α O with α 0, and hence { Rε, f = z = x + yε 1 } α O x, y Q, 0 < x 1, 0 y < 1. Note that if z Rε, f, then αz O. We then define the projection map by π α z := αz + f. π α : Rε, f O /f

18 18 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI Lemma 4.6. The projection map π α : Rε, f O /f is a surjective group homomorphism. Moreover, we have for all w O /f. #π 1 α w = # kerπ α Proof. We first show that π α is a group homomorphism. Let z 1, z Rε, f. Then there are integers m, n Z such that z 1 + z = z 1 z + m + nε. Hence z 1 z = z 1 + z m + nε, and we have π α z 1 z = π α z 1 + z m + nε = αz 1 + z m + nε + f = αz 1 + αz + f = π α z 1 + π α z. Next, we show that π α is surjective. Let w = β +f O /f, where β O. Then since β/α f 1, by Proposition 4.3 there is a unique element z β/α Rε, f such that β/α z β/α Z + Zε. It follows that z β/α = β/α + m + nε for some integers m, n Z, and thus π α z β/α = β + f = w. inally, we note that by the first isomorphism theorem for groups, we have Rε, f/ kerπ α = O /f. Hence if w O /f, then the size of the fiber πα 1 w is given by #πα 1 w = # kerπ α Pick s theorem and the size of kerπ α. To prove Theorem 4.1, we will need an explicit formula for # kerπ α. Here we prove such a formula by translating the problem of determining # kerπ α to a lattice point counting problem and employing a remarkable theorem of Pick see e.g. [SS09, Section 3.5] which gives a formula for the area of a lattice polygon in R in terms of the number of lattice points inside and on the boundary of the polygon. Recall that f = αo. Hence for a point z Rε, f, we have Thus z kerπ α αz αo z α 1 αo = O. kerπ α = Rε, f O = {z = x + yε O x, y Q, 0 < x 1, 0 y < 1}. Using the embedding ι : R of into the Minkowski space Q R = R given by ιx := x, x, we see that geometrically, the kernel kerπ α is the set of lattice points of ιo that lie in the half-open parallelogram In particular, this means that P := {t 1 1, 1 + t ε, ε t 1, t R, 0 < t 1 1, 0 t < 1}. # kerπ α = #P ιo. Note this already shows that # kerπ α only depends on the base field, and not on the ideal f = αo. In the next proposition, we will make use of Pick s theorem to prove an explicit formula for #P ιo.

19 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 19 Proposition 4.7. We have # kerπ α = #P ιo = volr /ιz + Zε volo = ε ε d. Proof. Let L : R R be a linear transformation mapping ιo onto Z. or example, if we fix the integral basis {1, d + d } for O, such a linear transformation is given by Lx, y = 1 d d d d + d 1 1 [ x y]. Since interior resp. boundary points of P LP, it follows that are mapped to interior resp. boundary points of #P ιo = #LP Z. 4.1 Now, recall that a polygon P in R whose vertices have integer coordinates is called a lattice polygon. If P is a lattice polygon in R, then Pick s theorem asserts that AreaP = IP + BP where IP denotes the number of lattice points in the interior of P and BP denotes the number of lattice points on the boundary of P. With this notation, we have 1, #LP Z = ILP + BLP. Moreover, since LP is a lattice parallelogram, Pick s Theorem implies that and thus By basic linear algebra, we know that AreaLP = ILP + BLP 1, #LP Z = AreaLP + BLP AreaLP = detl AreaP = 1 AreaP d = 1 [ ] 1 ε det d 1 ε = ε ε d Thus, it only remains to compute the value BLP, which we will now show is equal to 4. In fact, the 4 lattice points on the boundary are precisely the 4 vertices of the parallelogram LP. d + d If we write ε = a + b for some a, b Z with b 1 note that b 1 since b = ε ε / d > 0, then from the explicit description of the linear transformation L given

20 0 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI above, we see that the four vertices of P get mapped to the following vertices of LP, as shown in igure 3 and igure 4. 0, 0 0, 0 1, 1 1, 0 ε, ε a, b ε + 1, ε + 1 a + 1, b y y 1, 1 ε + 1, ε + 1 a, b a + 1, b LP d + d ε = a + b a, b Z 0, 0 P ε, ε x b 1 0, 0 1, 0 x igure 3. The parallelogram P and the lattice points in P ιo. igure 4. The lattice parallelogram LP and the corresponding lattice points in LP Z. It is clear that there are no lattice points on the line segments 0, 01, 0 and a, ba + 1, b other than the vertices. We now show that the same is true for the other two line segments. If there were a lattice point m, n Z lying on the line segment 0, 0a, b other than 0, 0 or a, b, there would be a rational number p/q Q with p, q Z, gcd p, q = 1, and 0 < p/q < 1, such that m, n = p a, b. q Therefore qm = pa and qn = pb, and since gcd p, q = 1, this implies that q a and q b. Thus, writing a = a 0 q and b = b 0 q for some a 0, b 0 Z, we see that d + 1 = N /Q ε = N /Q a d d + d 0 q + b 0 q = N /Q q N /Q a 0 + b 0, and this is a contradiction because N /Q q = q > 1. Hence the only lattice points on the line segment 0, 0a, b are 0, 0 and a, b. Since the remaining line segment 1, 0a + 1, b is just obtained from the previous one after translation by the vector 1, 0, we see that the only lattice points on 1, 0a + 1, b are 1, 0 and a + 1, b. This completes the proof that Now, by equations , we see that BLP = #P ιo = ε ε d + 3.

21 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 1 Moreover, since the only vertex contained in the half-open parallelogram P is 1, 1, we have Hence the previous identity implies that #P ιo = #P ιo 3. # kerπ α = #P ιo = ε ε d. inally, observe that volr /ιz + Zε = AreaP = ε ε and volo = d. This completes the proof of the proposition Proof of Theorem 4.1. The Shintani sets can be written as the disjoint unions Rε, f = πα 1 w and Rε, f = πα 1 w. 4.4 w O /f w O /f Then using Lemma 4.6, we see that # Rε, f = w O /f #πα 1 w = # kerπ α N /Q f and #Rε, f = #πα 1 w = # kerπ α ϕf. w O /f inally, it follows from Proposition 4.7 that # Rε, f = ε ε N /Q f and #Rε, f = ε ε ϕf. d d Note that the equality involving the volumes follows from the last line in the proof of Proposition Proof of Theorem 4.. Recall from Section 3 that χ takes one of the following forms on principal integral ideals γ coprime to f, χ γ = χ f γ, χ γ = χ f γ signγ, χ γ = χ f γ signγ, χ γ = χ f γ signn /Q γ, where χ f : O /f S 1. Let f = α for some α O with α 0. Since every element of the Shintani set Rε, f is totally positive, we see that for z Rε, f we have αz 0, and it follows that In particular, for z Rε, f we have signαz = signα z = signn /Q αz = 1. χf z = χ αz = χ f αz.

22 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI Using the disjoint union 4.4, Lemma 4.6, and Proposition 4.7, we get χf z = χ f αz z Rε,f = w O /f z πα 1 w χ f w #πα 1 w O /f w = # kerπ α χ f w w O /f = ε ε χ f w. d w O /f inally, the orthogonality relations for characters of the finite group O /f yield { 0, χ 1 χ f w = # O w O /f /f = ϕf, χ = Shintani cycles in Rε, f. In this section, we continue our combinatorial study of Shintani sets and their relation to ray class characters. In particular, using the vantage point provided by the group law defined on the Shintani set Rε, f, we revisit some results already observed by Shintani in [Shi77]. This allows us to clarify and unify the theory, and moreover to set the stage for a potential generalization to higher dimensions currently being investigated by the authors. We will need the following important facts. Lemma 4.8. or w f 1 and M, N Z, we have Proof. Since f = α for some α O, we have gcdf w + M + Nε, f = 1 gcdf w, f = 1. gcdf w + M + Nε, f = 1 f w + M + Nε + f = O αw + M + Nε + α = O αw + α = O Proposition 4.9. If u O,+ f w + f = O gcdf w, f = 1. and z Rε, f, there exists a unique δu, z Rε, f such that δu, z uz Z + Zε. Proof. By Proposition 4.3, we know there is a unique element δ := δu, z Rε, f such that δu, z uz Z + Zε. 4.5 Thus, to prove that δ Rε, f, we need only show that gcdf δ, f = 1. By 4.5, we have δ = uz + M + Nε for some integers M, N Z. Then by Lemma 4.8, we have gcdf δ, f = 1 gcdf z, f = 1. However, since z Rε, f, it follows that gcdf z, f = 1.

23 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 3 Using Proposition 4.9, we define a map : O,+ Rε, f Rε, f u, z u z := δu, z. Proposition The map defines a group action of O,+ on Rε, f. Proof. irst, we prove that 1 z = z for all z Rε, f. Since z z = 0 Z + Zε, by uniqueness, we must have 1 z := δ1, z = z. Next, we prove that u v z = uv z for all u, v O,+ and z Rε, f. This is equivalent to δu, δv, z = δuv, z. We know that are the unique elements such that respectively. Hence we can write and δv, z, δuv, z, δu, δv, z Rε, f δv, z vz, δuv, z uvz, δu, δv, z uδv, z Z + Zε, for some integers m 1, n 1, m, n Z. equation into 4.7, we get δv, z vz = m 1 + n 1 ε 4.6 δu, δv, z uδv, z = m + n ε 4.7 Solving for δv, z in 4.6 and substituting the resulting δu, δv, z uvz = um 1 + n 1 ε + m + n ε. 4.8 We claim that the right hand side of 4.8 is in Z + Zε, and hence by uniqueness, δu, δv, z = δuv, z. Since u = ε n for some n Z, it suffices to prove that m 1 ε n + n 1 ε n+1 Z + Zε. To do this, we will show that ε N Z + Zε for every N Z. Now, since ε is an algebraic integer of degree, it satisfies ε = aε + b for some integers a, b Z, and hence ε Z + Zε. Then using an inductive argument, we can conclude that ε N Z + Zε for any N 0. On the other hand, for negative powers, we use the fact that ε 1 = ε. Writing ε in terms of the integral basis {1, d + d /}, we see that d + d ε = r + s for some integers r, s Z. Hence ε 1 = ε d d = r + s = r + sd ε Z + Zε. Moreover, since ε is also an algebraic integer of degree, we see that ε 1 satisfies ε = ãε + b for some integers ã, b Z. Therefore, as before, we conclude by an inductive argument that ε N Z + Zε for any N 1. Thus the claim is proved, which completes the proof of the proposition. An O,+ -orbit in Rε, f is called a Shintani cycle and denoted by C. Given z Rε, f, we define ẑ := z, where z Rε, f is the inverse of z under the group law defined in Proposition 4.5. Lemma The map z ẑ is an involution on Rε, f.

24 4 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI Proof. Clearly, we have ẑ = z = z. Hence, it remains to prove that ẑ Rε, f, i.e., that gcdf ẑ, f = 1. Now, by definition of z, we have z z Z + Zε, and therefore there exist integers m, n Z such that ẑ = z = m + nε z. It follows from Lemma 4.8 that Thus, we conclude that ẑ Rε, f. gcdf ẑ, f = gcdf m + nε z, f = gcdf z, f = 1. Given a cycle C in Rε, f, define the set Lemma 4.1. The set Ĉ is a cycle in Rε, f. Ĉ := {ẑ : z C}. Proof. Let C z0 denote the cycle containing an element z 0 Rε, f. Then it suffices to prove that Ĉ z0 = Cẑ0. irst, we prove that Ĉz 0 Cẑ0. Let w Ĉz 0. Then w = ẑ for some z C z0. Since z C z0, there exists u O,+ such that u z 0 = z, or equivalently, z Rε, f is the unique element such that z uz 0 Z + Zε. 4.9 We claim that u ẑ 0 = ẑ, or equivalently, that ẑ uẑ 0 Z + Zε, which implies that w = ẑ Cẑ0. Write ẑ 0 = m 1 + n 1 ε z 0, ẑ = m + n ε z for some integers m 1, m, n 1, n Z. Substituting these equations yields ẑ uẑ 0 = z uz 0 + m + n ε um 1 + n 1 ε. By 4.9, we have z uz 0 Z + Zε, and by the argument in the last paragraph of the proof of Proposition 4.10, we have um 1 + n 1 ε Z + Zε. Hence ẑ uẑ 0 Z + Zε, which proves the claim. This completes the proof that Ĉz 0 Cẑ0. A symmetric argument can be used to prove that Cẑ0 Ĉz 0. The cycle Ĉ is called the opposite cycle. It follows from Lemma 4.11 that the map C Ĉ is an involution on the set of cycles in Rε, f. Lemma Let χ be a narrow ray class character modulo f. Then for z Rε, f and integers M, N Z such that z + M + Nε 0, we have χf z + M + Nε = χf z. Proof. Suppose that gcdf z, f 1. Then by Lemma 4.8, we have trivially that χf z + M + Nε = χf z = 0. Next, suppose that gcdf z, f = 1. Then by Lemma 4.8, we have gcdf z + M + Nε, f = 1. Now, recall that if γ O with γ 0 and relatively prime to f, then χ γ = χ f γχ γ = χ f γ. Since has narrow class number 1, f = α for some α O with α 0. Because α 0, z 0, and z + M + Nε 0, it follows that χf z + M + Nε = χ αz + M + Nε = χ f αz + M + Nε = χ f αz = χf z, where we used the fact that αz + M + Nε αz mod f and that χ f : O /f S 1.

25 ALTINGS HEIGHTS O CM ABELIAN SURACES AND SPECIAL GAMMA VALUES 5 Lemma Let χ be a narrow ray class character modulo f and C be a cycle in Rε, f. Then the value χf z is independent of the choice of z C. Proof. Let z 1, z C. Then there exists u O,+ such that u z 1 = z. This implies that there exist integers M, N Z such that z uz 1 = M + Nε. Since uz 1 0, we have z M + Nε 0. It follows from Lemma 4.13 that χf z 1 = χf uz 1 = χf z M + Nε = χf z. By Lemma 4.14, given z C we may define χc := χf z. Proposition Let χ be a narrow ray class character modulo f with admissible vector p. Then { χĉ = χc, if p = 0, 0 or 1, 1 χc, if p = 1, 0 or 0, 1. Proof. Let γ be a principal integral ideal coprime to f. Then the infinite part of χ is given by see Section 3 1, if p = 0, 0 signγ, if p = 1, 0 χ γ = signγ, if p = 0, 1 signn /Q γ, if p = 1, 1. It follows that Now, because we have { χ γ χ γ = 1, if p = 0, 0 or 1, 1 1, if p = 1, 0 or 0, 1. χ f γ = χ γ χ γ = χ f γ = χ γ χ γ = χ γ χ γ χ fγ, { χ f γ, if p = 0, 0 or 1, 1 χ f γ, if p = 1, 0 or 0, Let f = α for some α O with α 0. Given z C, write ẑ = m + nε z for some integers m, n Z. Then χĉ = χf ẑ = χ αm + nε z = χ f αm + nε zχ αm + nε z = χ f αm + nε z = χ f αz, 4.11 where for the fourth equality we used αm + nε z 0, and for the last equality we used αm + nε z αz mod f and χ f : O /f S 1. A similar calculation shows that The proposition now follows from 4.10, 4.11 and 4.1. χc = χ f αz. 4.1 Corollary If there exists a narrow ray class character χ modulo f with admissible vector p = 1, 0 or 0, 1, then the involution C Ĉ on the set of cycles in Rε, f is fixed point free.

26 6 ADRIAN BARQUERO-SANCHEZ AND RIAD MASRI Proof. Suppose that C Ĉ has a fixed point C 0. Then Ĉ0 = C 0, so that χĉ0 = χc 0. On the other hand, by Proposition 4.15 we have χĉ0 = χc 0. It follows that χc 0 = χc 0, or χc 0 = 0. But this is impossible, since χc 0 takes values in the unit circle S 1 recall that χc 0 := χf z for z C 0 and gcdf z, f = 1 for z C 0. Thus C Ĉ is fixed point free. If a narrow ray class character χ modulo f has admissible vector 1, 0 or 0, 1, then one can use the symmetry relations satisfied by χ with respect to Shintani cycles to give an alternative proof of the orthogonality relations in Theorem 4.. Recall that B x = x x 1/6 is the second Bernoulli polynomial. Proposition If χ is a narrow ray class character modulo f with admissible vector p = 1, 0 or 0, 1, then χf z = 0 and z Rε,f z Rε,f z=x+yε χf z B x = 0. Proof. By Corollary 4.16, the involution C Ĉ is fixed point free. Hence, there is an even number of cycles C 1,..., C n in Rε, f, and a choice of cycles C i1, C i,..., C in giving a disjoint union Rε, f = C i1 Ĉi 1 C i Ĉi C in Ĉi n It follows from Proposition 4.15 and 4.13 that n χf z = #C ij χc ij + χĉi j = z Rε,f j=1 n #C ij χcij χc ij = 0. Next, given z Rε, f, write z = x z +y z ε and ẑ = xẑ +yẑε. By definition of the involution z ẑ, we have xẑ = x z or xẑ = 1 x z. It follows from Proposition 4.15, 4.13, and the relation that z Rε,f z=x z+y zε χf z B x z = = = = 0. B x = B 1 x n j=1 z C ij z=x z+y zε n j=1 z C ij z=x z+y zε n j=1 z C ij z=x z+y zε j=1 χc ij B x z + χĉi j B xẑ B x z χc ij + χĉi j B x z χc ij χc ij Remark The involution C Ĉ may not be fixed point free. or example, let = Q 5, so that ε = 3+ 5, and consider the integral ideal f = 5. Then the Shintani set decomposes into the union of two cycles Rε, f = C 1 C,