On superspecial abelian surfaces over finite fields III
aa r X i v : . [ m a t h . N T ] F e b ON SUPERSPECIAL ABELIAN SURFACES OVER FINITEFIELDS III
JIANGWEI XUE, CHIA-FU YU, AND YUQIANG ZHENG
Abstract.
In the paper [On superspecial abelian surfaces over finite fields II.
J. Math. Soc. Japan , 72(1):303–331, 2020], Tse-Chung Yang and the first twocurrent authors computed explicitly the number | SSp ( F q ) | of isomorphismclasses of superspecial abelian surfaces over an arbitrary finite field F q of even degree over the prime field F p . There it was assumed that certain commutative Z p -orders satisfy an ´etale condition that excludes the primes p = 2 , ,
5. Wetreat these remaining primes in the present paper, where the computations aremore involved because of the ramifications. This completes the calculation of | SSp ( F q ) | in the even degree case. The odd degree case was previous treatedby Tse-Chung Yang and the first two current authors in [On superspecialabelian surfaces over finite fields. Doc. Math. , 21:1607–1643, 2016]. Alongthe proof of our main theorem, we give the classification of lattices over localquaternion Bass orders, which is a new input to our previous works. Introduction
Throughout this paper, p denotes a prime number, q = p a a power of p , and F q the finite field of q -elements. We reserve N for the set of strictly positive integers.Let k be a field of characteristic p , and ¯ k an algebraic closure of k . An abelian varietyover k is said to be supersingular if it is isogenous to a product of supersingularelliptic curves over ¯ k ; it is said to be superspecial if it is isomorphic to a productof supersingular elliptic curves over ¯ k . For any d ∈ N , denote by SSp d ( F q ) theset of F q -isomorphism classes of d -dimensional superspecial abelian varieties over F q . The classification of supersingular elliptic curves (namely, the d = 1 case) overfinite fields were carried out by Deuring [7, 8], Eichler [10], Igusa[11], Waterhouse[24] and many others since the 1930s.In a series of papers [25, 26, 27, 28], Tse-Chung Yang and the first two currentauthors attempt to calculate the cardinality | SSp d ( F q ) | explicitly in the case d = 2.More precisely, it is shown in [26] that for every fixed d > | SSp d ( F q ) | depends onlyon the parity of the degree a = [ F q : F p ], and an explicit formula of | SSp ( F q ) | isprovided for the odd degree case. The most involving part of this explicit calculationis carried out prior in [25, 27], which counts the number of isomorphism classes ofabelian surfaces over F p within the simple isogeny class corresponding to the Weil p -numbers ±√ p . For the even degree case, an explicit formula of | SSp ( F q ) | isobtained in [28] under a mild condition on p (see Remark 3.7 of loc. cit.), which Date : February 3, 2021.2020
Mathematics Subject Classification.
Key words and phrases. superspecial abelian surfaces, quaternion algebra, Bass order, conju-gacy classes of arithmetic subgroups. holds for all p ≥
7. We treat the remaining primes p ∈ { , , } in the presentpaper, thus completing the calculation of | SSp ( F q ) | in the even degree case.For the rest of the paper, we assume that q = p a is an even power of p . Allisogenies and isomorphisms are over the base field F q unless specified otherwise. Theset SSp ( F q ) naturally partitions into subsets by isogeny equivalence, which can beparametrized by (multiple) Weil numbers (see [26, § n ∈ N ,let ζ n be a primitive n -th root of unity, and π n be the Weil q -number ( − p ) a/ ζ n .By the Honda-Tate theorem, there is a unique simple abelian variety X n / F q up toisogeny corresponding to the Gal( ¯ Q / Q )-conjugacy class of π n . Moreover, the X n ’sare mutually non-isogenous for distinct n . Thanks to the Manin-Oort Theorem[29, Theorem 2.9], a simple abelian variety over F q is supersingular if and only if itis isogenous to X n for some n . Let d ( n ) be the dimension of X n . The formula for d ( n ) is given in [26, § p ∈ { , , } , we have • d ( n ) = 1 if and only if n ∈ { , , , } or ( n, p ) ∈ { (4 , , (4 , } ; • d ( n ) = 2 if and only if ( n, p ) = (4 ,
5) or n ∈ { , , , } .Given a superspecial abelian surface X/ F q , we have two cases to consider:(I) the isotypic case where X is isogenous to X /d ( n ) n for some n ∈ N with d ( n ) ≤ the non-isotypic case where X is isogenous to X n := X n × X n for apair n = ( n , n ) ∈ N with n < n and d ( n ) = d ( n ) = 1.Let o ( n ) (resp. o ( n )) denote the number of isomorphism classes of superspecialabelian surfaces over F q that are isogenous to X /d ( n ) n (resp. X n ). It was shown in[28, § o (1) = o (2) = 1 , o (3) = o (6) , o (5) = o (10);(1.1) o (1 ,
3) = o (2 , , o (1 ,
4) = o (2 , , o (1 ,
6) = o (2 , , o (3 ,
4) = o (4 , . (1.2)Thus we have | SSp ( F q ) | =2 + 2 o (3) + o (4) + 2 o (5) + o (8) + o (12)+ o (1 ,
2) + 2 o (2 ,
3) + 2 o (2 ,
4) + 2 o (2 ,
6) + 2 o (3 ,
4) + o (3 , . (1.3)As mentioned before, the value of each o ( n ) or o ( n ) in (1.3) has been workedout in [28] conditionally on p . To make explicit this condition, we uniformize thenotation. For each r ∈ N , let us denote˘ N r := { n = ( n , · · · , n r ) ∈ N r | < n < · · · < n r } . In particular, if r = 1, then ˘ N r = N and we drop the underline from n . For each n ∈ ˘ N r with r arbitrary, we define(1.4) A n := Z [ T ]( Q ri =1 Φ n i ( T )) , K n := Q [ T ]( Q ri =1 Φ n i ( T )) ≃ r Y i =1 Q ( ζ n i ) , where Φ n ( T ) ∈ Z [ T ] is the n -th cyclotomic polynomial. Clearly, A n is a Z -order in K n , so it is contained in the unique maximal order O K n := Q ri =1 Z [ T ] / (Φ n i ( T )).Let n ∈ ˘ N r with r ∈ { , } be an r -tuple appearing in (1.3). In the proof of[28, Theorem 3.3], it is assumed that(1.5) A n,p := A n ⊗ Z p is an ´etale Z p -order . This condition fails precisely in the following two situations:
UPERSPECIAL ABELIAN SURFACES 3 (C1) p is ramified in Q ( ζ n i ) for some 1 ≤ i ≤ r , or(C2) p divides the index [ O K n : A n ].If r = 1, then A n coincides with O K n , so (C2) is possible only if r = 2. For thereader’s convenience, we list the indices i ( n ) := [ O K n : A n ] from [28, Table 1]: n (1 ,
2) (2 ,
3) (2 ,
4) (2 ,
6) (3 ,
4) (3 , i ( n ) 2 1 2 3 1 4 Theorem 1.1.
Let n ∈ ˘ N r be an r -tuple appearing in (1.3), and p be a primesatisfying (C1) or (C2). Then the values of o ( n ) for each p are given by the followingtable n ,
2) (2 ,
3) (2 ,
4) (2 ,
6) (3 ,
4) (3 , p , , o ( n ) 2 2 1 1 3 3 1 2 3 2 8 2 Moreover, the number of isomorphism classes of superspecial abelian surfaces overa finite field F q of even degree over F p with p ∈ { , , } is given by (1.6) | SSp ( F q ) | = if p = 2 , if p = 3 , if p = 5 . Remark 1.2.
We provide an arithmetic interpretation of the values o ( n ). Let D = D p, ∞ be the unique quaternion Q -algebra up to isomorphism ramified precisely at p and ∞ , and Mat ( D ) be the algebra of 2 × D . Fix a maximal Z -order O in D . As explained in [28, §
1, p. 304], up to isomorphism, the arithmetic groupGL ( O ) depends only on p and not on the choice of O . An element x ∈ GL ( O ) offinite group order is semisimple, so its minimal polynomial over Q in Mat ( D ) isof the form P n ( T ) := Q ri =1 Φ n i ( T ) for some n = ( n , · · · , n r ) ∈ ˘ N r . It is not hardto show that r ≤ § o ( n ) counts the number of conjugacy classes of elements of GL ( O ) with minimalpolynomial P n ( T ), and | SSp ( F q ) | is equal to the total number of conjugacy classesof elements of finite group order in GL ( O ) (see [28, Proposition 1.1]). Actually,this arithmetic interpretation works for GL d ( O ) with any d ≥
2, not just for d = 2.The proof of Theorem 1.1 will occupy the remaining part of the paper. InSection 2, we recall from [28, § o ( n ). Theisotypic case (i.e. r = 1) will be treated in Section 3, and the non-isotypic case(i.e. r = 2) will be treated in Section 4.2. General strategy for computing o ( n )Keep the notation and the assumptions of the previous section. We recall from[28, § § o ( n ) with n ∈ ˘ N r for r ≤
2. Based on the arithmetic interpretation of o ( n ) in Remark 1.2, we furtherprovide a lattice description of o ( n ). Indeed, it is via this lattice description thatthe value of each o ( n ) is calculated. Unfortunately, the word “order” plays double duties in this paragraph: for the order insidean algebra and also for the order of a group element. To make a distinction, we always insert theword “group” when the second meaning applies.
JIANGWEI XUE, CHIA-FU YU, AND YUQIANG ZHENG
Let V = D be the unique simple left Mat ( D )-module, which is at the sametime a right D -vector space of dimension 2. Let M := O be the standard right O -lattice in V , whose endomorphism ring End O ( M ) is just Mat ( O ). For eachelement x ∈ GL ( O ) of finite group order with minimal polynomial P n ( T ) ∈ Z [ T ],there is a canonical embedding A n = Z [ T ] / ( P n ( T )) ֒ → Mat ( O ) sending T to x .This embedding equips M with an ( A n , O )-bimodule structure, or equivalently, afaithful left A n ⊗ Z O opp -module structure. Similarly, V is equipped with a faithfulleft K n ⊗ Q D opp -module structure. The canonical involution induces an isomor-phism between the opposite ring O opp and O itself (and similarly between D opp and D ), so we put(2.1) A n := A n ⊗ Z O , and K n := K n ⊗ Q D. Clearly, A n is a Z -order in the semisimple Q -algebra K n . It has been shown in[28, p. 309] that the K n -module structure on V is uniquely determined by the r -tuple n . From [26, Theorem 6.11] (see also [28, Lemma 3.1]), the above constructioninduces a bijection between the following two finite sets:(2.2) conjugacy classes of ele-ments of GL ( O ) with min-imal polynomial P n ( x ) ←→ isomorphism classesof A n -lattices in theleft K n -module V Therefore, we have o ( n ) = | L ( n ) | , where L ( n ) denote the set on the right.Now fix a pair ( n, p ) in Theorem 1.1 and in turn a left K n -module V . Givenan A n -lattices Λ ⊂ V , we write [Λ] for its isomorphism class, and O Λ for itsendomorphism ring End A n (Λ) ⊂ End K n ( V ). As a convention, the endomorphismalgebra E n := End K n ( V ) acts on V from the left, so it coincides with the centralizerof K n in Mat ( D ). Two A n -lattices Λ and Λ in V are isomorphic if and only ifthere exists g ∈ E × n such that Λ = g Λ .For each prime ℓ ∈ N , we use the subscript ℓ to indicate ℓ -adic completion. Forexample, A n,ℓ (the ℓ -adic completion of A n ) is a Z ℓ -order in the semisimple Q ℓ -algebra K n,ℓ , and Λ ℓ is an A n,ℓ -lattice in V ℓ . For each prime ℓ , let L ℓ ( n ) denote theset of isomorphism classes of A n,ℓ -lattices in the left K n,ℓ -module V ℓ . For almost allprimes ℓ , the Z ℓ -order A n,ℓ is maximal in K n,ℓ , in which case both of the followinghold by [6, Theorem 26.24]:(i) Λ ℓ is uniquely determined up to isomorphism (i.e. | L ℓ ( n ) | = 1), and(ii) O Λ ,ℓ is maximal in E n,ℓ .Let S ( n, p ) be the finite set of primes ℓ for which A n,ℓ is non-maximal. The profinitecompletion Λ b Λ := Q ℓ Λ ℓ induces a surjective map(2.3) Ψ : L ( n ) → Y ℓ L ℓ ( n ) ≃ Y ℓ ∈ S ( n,p ) L ℓ ( n ) . Two A n -lattices Λ and Λ in V are said to be in the same genus if Ψ([Λ ]) =Ψ([Λ ]), or equivalently, (Λ ) ℓ ≃ (Λ ) ℓ for every prime ℓ . The fibers of Ψ partition L ( n ) into a disjoint union of genera. Let G (Λ) := Ψ − (Ψ([Λ])) ⊆ L ( n ) be thefiber of Ψ over Ψ([Λ]), that is, the set of isomorphism classes of A n -lattices in thegenus of Λ. From [21, Proposition 1.4], we have(2.4) | G (Λ) | = h ( O Λ ) , where h ( O Λ ) denote the class number of O Λ . In other words, h ( O Λ ) is the numberof locally principal right (or equivalently, left) ideal classes of O Λ . UPERSPECIAL ABELIAN SURFACES 5
Therefore, the computation of o ( n ) can be carried out in the following two steps:(Step 1) Classify the genera of A n -lattices in the left K n -module V . Equivalently,classify the isomorphism classes of A n,ℓ -lattices in V ℓ for each ℓ ∈ S ( n, p ).(Step 2) Pick a lattice Λ in each genus and write down its endomorphism ring O Λ (at least locally at each prime ℓ ). The number o ( n ) is obtained by summingup the class numbers h ( O Λ ) over all genera. Remark 2.1.
The reason that condition (1.5) is assumed throughout the calcula-tions in [28] is to make sure that the Z p -order A n,p is a product of Eichler orders[28, Remark 3.7]. In our setting, p satisfies condition (C1) or (C2), so A n,p becomesmore complicated. This is precisely why the primes p ∈ { , , } are treated sepa-rately from the rest of the primes. Luckily for us, many A n,p turn out to be Bassorders (see Definition 3.1 below), which makes the classification of A n,p -latticesmore manageable. 3. The isotypic case
In this section, we calculate the values of o ( n ) for n ∈ { , , , , } and p | n .Keep the notation of previous sections. In particular, D = D p, ∞ is the uniquequaternion Q -algebra ramified precisely at p and ∞ , and O is a maximal Z -orderin D . Since A n is the maximal order in the n -th cyclotomic field K n , and O has reduced discriminant p , we have S ( n, p ) = { p } . In other words, the ℓ -adiccompletion A n,ℓ is non-maximal in K n,ℓ if and only if ℓ = p . It turns out that A n,p is always a Bass order in the quaternion K n,p -algebra K n,p . Therefore, theclassification of genera of the lattice set L ( n ) is then reduced to the classificationof lattices over local quaternion Bass orders.3.1. Classification of lattices over local quaternion Bass orders.
The mainreferences for this section are [2, 4] and [6, § F be a nonarchimedean localfield, and O F be its ring of integers. Fix a uniformizer ̟ of F and denote the thefinite residue field O F /̟O F by k . Let B be a finite dimensional separable F -algebra[6, Definition 7.1 and Corollary 7.6], and O be an O F -order (of full rank) in B . Wewrite Ov( O ) for the finite set of overorders of O , i.e. O F -orders in B containing O .A minimal overorder of O is a minimal member of Ov( O ) r { O } with respect toinclusion. Definition 3.1. An O F -order O in B is Gorenstein if its dual lattice O ∨ :=Hom O F ( O , O F ) is projective as a left (or right) O -module. It is called a Bass order if every member of Ov( O ) is Gorenstein. It is called a hereditary order if every leftideal of O is projective as a left O -module. If O is the intersection of two maximalorders, then it is called an Eichler order .We have the following inclusions of orders:(maximal) ⊂ (herediary) ⊂ (Eichler) ⊂ (Bass) ⊂ (Gorenstein) . If B is division, then Eichler orders are also maximal. Bass notes in [1] that Goren-stein orders are ubiquitous.Let I be a fractional left O -ideal (of full rank) in B . We say I is proper over O if its associated left order O l ( I ) := { x ∈ B | xI ⊆ I } coincides with O . From[3, Example 2.6 and Corollary 2.7], the following lemma provides an equivalentcharacterization of Gorenstein orders in certain types of F -algebras: JIANGWEI XUE, CHIA-FU YU, AND YUQIANG ZHENG
Lemma 3.2.
Suppose that B is either a commutative algebra or a quaternion F -algebra. Then O is Gorenstein if and only if every proper fractional left O -ideal I ⊂ B is principal (i.e. there exists x ∈ B × such that I = O x ). In the quaternion case, the above lemma can also be obtained by combining[9, Condition G4 or G4’, p. 1364] and [12, Theorem 1]. The lemma no longer holdsin general for orders in more complicated algebras. If every proper fractional left O -ideal I is principal, then O is Gorenstein, but the converse is not necessarilytrue. See [12, p. 220] and [3, p. 535] for some examples. Nevertheless, a properfractional left ideal over a Gorenstein order is always left projective according to[18, Theorem 5.3, pp. 253–255]. However, unlike the situation over commutativerings, a projective module over a non-commutative ring may not be locally free.Brzezinski [3, Proposition 2.3] gave a precise characterization of the orders O suchthat every proper fractional left O -ideal I ⊂ B is principal (Such orders are called strongly Gorenstein by him).For the rest of Section 3.1, we assume that char( F ) = 2 and B is a quaternion F -algebra. The reduced trace and reduced norm maps of B are denoted by Tr : B → F and Nr : B → F respectively. We write d ( O ) for the reduced discriminant of O ,which is a nonzero integral ideal of O F . From [2, Proposition 1.2], O is hereditaryif and only if d ( O ) is square-free. Thus if B is division, then O is hereditary if andonly if O is the unique maximal order of B ; if B ≃ Mat ( F ), then O is hereditaryif and only if O is isomorphic to Mat ( O F ) or (cid:2) O F O F ̟O F O F (cid:3) . Theorem 3.3.
The following are equivalent:(a) every left O -ideal is generated by at most 2 elements;(b) O is Bass;(c) every indecomposable O -lattice is isomorphic to an ideal of O ;(d) O ⊇ O L for some semisimple quadratic F -subalgebra L ⊆ B . Indeed, the implications ( a ) ⇒ ( b ) ⇒ ( c ) hold in much more general settingsaccording to [6, § c ) ⇒ ( a ) is proved by Drozd, Kirichenkoand Roiter [9] (see [6, p. 790]). Lastly, the equivalence ( b ) ⇔ ( d ) is proved byBrzezinski [4, Proposition 1.12]. See Chari et al. [5] for more characterization ofquaternion Bass orders.We recall the notion of Eichler invariant following [2, Definition 1.8].
Definition 3.4.
Let k ′ / k be the unique quadratic field extension. When O Mat ( O F ), the quotient of O by its Jacobson radical J ( O ) falls into the followingthree cases: O / J ( O ) ≃ k × k , k , or k ′ , and the Eichler invariant e ( O ) is defined to be 1 , , − O ≃ Mat ( O F ), then its Eichler invariant is defined to be 2.For example, if B is division and O is the unique maximal order, then e ( O ) = − e ( O ) = 1 if and only if O is a non-maximalEichler order. Note that e ( O ) = 1 can only occur when B ≃ Mat ( F ). Moreover, if e ( O ) = 0, then O is automatically Bass by [2, Corollary 2.4 and Propoisition 3.1].The classification of lattices over Eichler orders is well known (see [28, p. 315] forexample), which we recall as follows. UPERSPECIAL ABELIAN SURFACES 7
Lemma 3.5.
Suppose B ≃ Mat ( F ) and let O ≃ (cid:20) O F O F ̟ e O F O F (cid:21) be an Eichlerorder. Let M be an O -lattice in a finite left B -module W . Then (3.1) W ≃ (cid:20) FF (cid:21) ⊕ u , and M ≃ u M i =1 (cid:20) O F ̟ e i O F (cid:21) , where the e i ’s are integers such that ≤ e i ≤ e and e i ≤ e i +1 for all i . Moreover,the isomorphism class of M is uniquely determined by these e i ’s. Henceforth we assume that e ( O ) ∈ { , − } . Let n ( O ) be the unique non-negativeinteger such that d ( O ) = ( ̟ n ( O ) ). Suppose that O is Bass but non-hereditary.From [2, Proposition 1.12], O has a unique minimal overorder M ( O ), which is alsoBass by definition. According to [2, Propositions 3.1 and 4.1],(3.2) n ( M ( O )) = ( n ( O ) − e ( O ) = − ,n ( O ) − e ( O ) = 0 , and e ( M ( O )) = e ( O ) if M ( O ) is also non-hereditary. Thus starting from M ( O ) := O , we define M i ( O ) := M ( M i − ( O )) recursively to obtain a unique chain of Bassorders terminating at a hereditary order M m ( O ):(3.3) O = M ( O ) ⊂ M ( O ) ⊂ M ( O ) ⊂ · · · ⊂ M m − ( O ) ⊂ M m ( O ) , where m is given as follows • m = n ( O ) − e ( O ) = 0; and • m = ⌊ n ( O ) / ⌋ if e ( O ) = −
1, where x
7→ ⌊ x ⌋ is the floor function on R .The order M m ( O ) is called the hereditary closure of O and will henceforth bedenoted by H ( O ). If e ( O ) = −
1, then H ( O ) is always a maximal order by [2,Proposition 3.1]. Thus when e ( O ) = − n ( O ) is even if B ≃ Mat ( F ), and n ( O )is odd if B is division. If e ( O ) = 0, then • H ( O ) ≃ (cid:2) O F O F ̟O F O F (cid:3) if B ≃ Mat ( F ), and • H ( O ) is the unique maximal order if B is division.Note that O is hereditary (i.e. m = 0) if and only if e ( O ) = − B is division,so m is strictly positive in the remaining cases.From [4, Proposition 1.12], there exists a quadratic field extension L/F such that O L embeds into O , and O = O L + J ( H ( O )) c , where(3.4) c = ( n ( O ) / e ( O ) = − B ≃ Mat ( F ) ,n ( O ) − L/F is the unique unramified quadratic field extension if e ( O ) = −
1, and itis a ramified quadratic field extension if e ( O ) = 0. In the latter case, the ramifiedquadratic extension L/F can be arbitrary if n ( O ) = 2 according to [4, (3.14)];and it is uniquely determined by O if n ( O ) ≥ F is nondyadic according to[17, Lemma 3.5]. From the proof of [4, Theorem 3.3 and 3.10], any two embeddings of O L into O are conjugateby an element of the normalizer of O , thus expression (3.4) does not depend on the choice of theembedding O L ֒ → O . JIANGWEI XUE, CHIA-FU YU, AND YUQIANG ZHENG
Lemma 3.6.
Suppose that O ⊂ B is a Bass order with e ( O ) ∈ { , − } . Let N bean indecomposable left O -lattice.(1) If B is division, then (3.6) N ≃ M i ( O ) for some ≤ i ≤ m. (2) Suppose that B is split, i.e. B ≃ Mat ( F ) . Fix an identification of H ( O ) with (cid:2) O F O F ̟O F O F (cid:3) (resp. Mat ( O F ) ) if e ( O ) = 0 (resp. − ).(a) If e ( O ) = 0 , then N is isomorphic to one of the following O -lattices: (3.7) (cid:20) O F ̟O F (cid:21) , (cid:20) O F O F (cid:21) , or M i ( O ) with ≤ i ≤ m − . (b) If e ( O ) = − , then N is isomorphic to one of the following O -lattices: (3.8) (cid:20) O F O F (cid:21) or M i ( O ) with ≤ i ≤ m − . Proof.
According to the Drozd-Krichenko-Roiter Theorem [6, Theorem 37.16],(3.9) N ⊗ O F F ≃ ( B if B is division ,F or Mat ( F ) if B ≃ Mat ( F ) . First, suppose that B ≃ Mat ( F ) and N ⊗ O F F ≃ F . Then N ≃ O F asan O F -module, and End O F ( N ) is a maximal order in B containing O . It followsfrom (3.3) that End O F ( N ) contains H ( O ), which equips N with a canonical H ( O )-module structure. Therefore, if e ( O ) = −
1, then H ( O ) = Mat ( O F ), and hence N is homothetic to (cid:2) O F O F (cid:3) . Similarly, if e ( O ) = 0, then H ( O ) = (cid:2) O F O F ̟O F O F (cid:3) , and hence N is homothetic to (cid:2) O F O F (cid:3) or (cid:2) O F ̟O F (cid:3) .Next, suppose that N ⊗ O F F ≃ B . Then we regard N as a fractional left idealof O . Let O l ( N ) = { x ∈ B | xN ⊆ N } be the associated left order of N . Clearly, O l ( N ) contains O , so O l ( N ) = M i ( O ) for some 0 ≤ i ≤ m . In particular, O l ( N )is Gorenstein. It follows from Lemma 3.2 that N ≃ O l ( N ) as O -lattices.Clearly, if B is division, then M i ( O ) is indecomposable for every 0 ≤ i ≤ m . Onthe other hand, if B ≃ Mat ( F ), then the hereditary closure H ( O ) = M m ( O ) is decomposable as an O -lattice. Thus N
6≃ M m ( O ) in this case. It remains to showthat M i ( O ) is indecomposable for the remaining i ’s. Suppose otherwise so that M i ( O ) = N ⊕ N , where each N j is an O -lattice in N j ⊗ O F F ≃ F . Then M i ( O ) = O l ( M i ( O )) = O l ( N ⊕ N ) = O l ( N ) ∩ O l ( N ) . Since O l ( N i ) is a maximal order in Mat ( F ) for each i , this would imply that M i ( O )) is an Eichler order (i.e. e ( M i ( O )) ∈ { , } ), contradicting to the fact that e ( M i ( O )) = e ( O ) ∈ { , − } for 0 ≤ i ≤ m − (cid:3) Applying the Krull-Schmidt-Azumaya Theorem [6, Theorem 6.12], we immedi-ately obtain the following proposition.
Proposition 3.7.
Suppose that O ⊂ B is a Bass order with e ( O ) ∈ { , − } . Let M be an O -lattice in a finite left B -module W .(1) If B is division, then M ≃ L mi =0 M i ( O ) ⊕ t i with ( t , · · · , t m ) ∈ Z m +1 ≥ and P mi =0 t i = dim B W .(2) If B ≃ Mat ( F ) , then W ≃ Mat ,u ( F ) for some u ≥ . There are twocases to consider: UPERSPECIAL ABELIAN SURFACES 9 (2a) if e ( O ) = 0 , then (3.10) M ≃ (cid:20) O F ̟O F (cid:21) ⊕ r M (cid:20) O F O F (cid:21) ⊕ s M m − M i =0 M i ( O ) ⊕ t i with ( r, s, t , · · · , t m − ) ∈ Z m +2 ≥ and r + s + 2 P m − i =0 t i = u ;(2b) if e ( O ) = − , then (3.11) M ≃ (cid:20) O F O F (cid:21) ⊕ s M m − M i =0 M i ( O ) ⊕ t i , with ( s, t , · · · , t m − ) ∈ Z m +1 ≥ and s + 2 P m − i =0 t i = u .In all cases, the isomorphism class of M is uniquely determined by the numericalinvariants r, s (if applicable) and the t i ’s. Explicit computations.
Recall that our goal is to compute the value of o ( n )for n ∈ { , , , , } and p | n . As explained in Section 2, o ( n ) coincides with thenumber of isomorphism classes of A n -lattices in the left K n -module V = D (See(1.4) and (2.1) for the definition of A n and K n ). From [23, Theorem 11.1], the n -thcyclotomic field K n has class number 1 for each n ∈ { , , , , } .Since K n is totally imaginary and p does not split completely in K n , we have K n = K n ⊗ Q D = Mat ( K n ). Thus as a left Mat ( K n )-module,(3.12) V ≃ ( Mat ( K n ) if n ∈ { , } ,K n if n ∈ { , , } . From this, we can easily write down its endomorphism algebra(3.13) E n = End K n ( V ) = ( Mat ( K n ) if n ∈ { , } ,K n if n ∈ { , , } . In particular, we see that every arithmetic subgroup of E × n is infinite, and hencethe abelian surfaces in these isogeny classes have infinite automorphism groups.Given an A n -lattice Λ ⊂ V , its endomorphism ring O Λ = End A n (Λ) is an A n -order in E n . Therefore, if n ∈ { , , } , then O Λ = A n and h ( O Λ ) = 1. Nowsuppose that n ∈ { , } . We are going to show in (3.19) that det( O × Λ ,p ) = A × n,p .On the other hand, at each prime ℓ = p , we have O Λ ,ℓ ≃ Mat ( A n,ℓ ) since O Λ ,ℓ is maximal. Thus if we write b O Λ (resp. b A n ) for the profinite completion of O Λ (resp. A n ), then det( b O × Λ ) = b A × n . The same proof of [22, Corollaire III.5.7(1)] showsthat h ( O Λ ) = h ( A n ) = 1. In conclusion, for every pair ( n, p ) with n ∈ { , , , , } and p | n , we have(3.14) o ( n ) = | L p ( n ) | , which is consistent with [28, (4.3)]. Lemma 3.8.
Suppose that n ∈ { , , , } and p | n . Let U p := K n,p be the uniquesimple K n,p -module. Then up to isomorphism, there is a unique A n,p -lattice in U p .Proof. For each pair ( n, p ) under consideration, p is totally ramified in K n . Itfollows from [15, § e ( A n,p ) = e ( O p ) = − . From [2, Proposition 3.1], A n,p is a Bass order. Thus the lemma is a direct appli-cation of Corollary 3.7. (cid:3) Lemma 3.9.
Suppose that n ∈ { , , , } and p | n . Then (3.16) o ( n ) = ( if n ∈ { , } , if n ∈ { , } . Proof.
First, suppose that n ∈ { , } . Then V p ≃ K n,p is a simple Mat ( K n,p )-module. From Lemma 3.8, | L p ( n ) | = 1, and hence o ( n ) = 1 by (3.14).Next, suppose that n ∈ { , } . We have already seen in the proof of Lemma 3.8that A n,p is a Bass order with Eichler invariant −
1. Let ̟ n = 1 − ζ n be theuniformizer of the local field K n,p . The reduced discriminant d ( A n,p ) is given by(3.17) d ( A n,p ) = d ( A n,p ⊗ Z p O p ) = d ( O p ) A n,p = pA n,p = ̟ n A n,p . Thus the chain of Bass orders in (3.3) reduces to A n,p ⊂ M ( A n,p ), where M ( A n,p ) =Mat ( A n,p ) under a suitable identification K n,p = Mat ( K n,p ). In this case, V p isa free Mat ( K n,p )-module of rank 1. From (3.11), every A n,p -lattice Λ p in V p isisomorphic to either A n,p or Mat ( A n,p ). Correspondingly, the endomorphism ring O Λ ,p is given by(3.18) O Λ ,p = End A n,p (Λ p ) ≃ ( A n,p if Λ p ≃ A n,p , Mat ( A n,p ) if Λ p ≃ Mat ( A n,p ) . In both cases, we have(3.19) det( O × Λ ,p ) = A × n,p . Indeed, this is clear if O Λ ,p ≃ Mat ( A n,p ). In the case O Λ ,p ≃ A n,p , let L be theunique unramified quadratic field extension of K n,p . From (3.4), O Λ ,p contains acopy of O L , which implies that det( O × Λ ,p ) ⊇ N L/K n,p ( O × L ) = A × n,p . On the otherhand, det( O × Λ ,p ) is obviously contained in A × n,p , so equality (3.19) holds in this caseas well. We conclude that o ( n ) = | L p ( n ) | = 2 if n ∈ { , } and p | n . (cid:3) Lemma 3.10. o (12) = 3 if p ∈ { , } .Proof. Since 2 and 3 are ramified in K n , 2 is inert in Q ( ζ ) and 3 is inert in Q ( ζ ),the p -adic completion K n,p is a field extension of degree 4 over Q p with residuedegree 2, so e ( A n,p ) = e ( A n,p ⊗ Z p O p ) = 1 by [15, § d ( A n,p ) = ̟ p A n,p , where ̟ p denotes a uniformizer of K n,p . From[2, Proposition 2.1], we may identify A n,p with the Eichler order (cid:20) A n,p A n,p ̟ p A n,p A n,p (cid:21) .Since V p ≃ K n,p is a simple Mat ( K n,p )-module, every A n,p -lattice in V p is isomor-phic to one of the following lattices:(3.20) (cid:20) A n,p ̟ p A n,p (cid:21) , (cid:20) A n,p ̟ p A n,p (cid:21) , (cid:20) A n,p A n,p (cid:21) . Therefore, o (12) = | L p (12) | = 3 by (3.14). (cid:3) Remark 3.11.
We have proved (cf. (3.14)) that in the isotypic case, every genusin the set L ( n ) of lattice classes has class number one. This holds also in the case p ∤ n according to [28, (4.3)]. UPERSPECIAL ABELIAN SURFACES 11 The non-isotypic case
In this section, we compute the values of o ( n ) for n = ( n , n ) ∈ ˘ N and p satisfying condition (C1) or (C2) (or both). More explicitly, the pairs ( n, p ) arelisted in the following table: n = ( n , n ) (1 ,
2) (2 ,
3) (2 ,
4) (2 ,
6) (3 ,
4) (3 , p , , i ( n ) 2 1 2 3 1 4Here we have also included the index i ( n ) = [ O K n : A n ], where O K n = A n × A n is the unique maximal order of K n .Since K n = K n × K n , we have K n = K n × K n . Consequently, the left K n -module V = D decomposes into a product V n × V n , where each V n i is a simple left K n i -module with dim D V n i = 1. In turn, E n = E n × E n with E n i := End K ni ( V n i ).If n i ∈ { , } , then K n i = Q , so we have(4.1) K n i = D, V n i ≃ D, E n i ≃ D if n i ∈ { , } . If n i ∈ { , , } , then K n i is an imaginary quadratic extension of Q , and p does notsplit completely in K n i . Thus(4.2) K n i ≃ Mat ( K n i ) , V n i ≃ K n i , E n i = K n i if n i ∈ { , , } . To avoid conflict of notations between V n i and V ℓ , we will always write the fullexpression V ⊗ Q ℓ instead of V ℓ for the ℓ -adic completion of V . On the other hand,the subscript n will never be expanded out explicitly as ( n ,n ) nor n ,n , so thereshould be no ambiguity about A n i ,ℓ := A n i ⊗ Z Z ℓ .If ℓ ∈ N is a prime with ℓ ∤ i ( n ) and ℓ = p , then A n,ℓ = O K n ,ℓ , and O ℓ ≃ Mat ( Z ℓ ), which implies that A n,ℓ = A n,ℓ ⊗ Z ℓ O ℓ ≃ Mat ( O K n ,ℓ ) . Thus A n,ℓ is maximal in K n,ℓ for such an ℓ . From this, we can easily write the set S ( n, p ) of primes at which A n is non-maximal:(4.3) S ( n, p ) = ( { , } if n = (3 ,
6) and p = 3; { p } otherwise . Recall that the class number h ( O ) is given by the following formula [22, Proposi-tion V.3.2](4.4) h ( O ) = p −
112 + 13 (cid:18) − (cid:18) − p (cid:19)(cid:19) + 14 (cid:18) − (cid:18) − p (cid:19)(cid:19) , where (cid:0) · p (cid:1) denotes the Legendre symbol. In particular, h ( O ) = 1 if p = 2 ,
3. Wealso note that h ( A n i ) = 1 for every n i ∈ { , , } . Given d ∈ N , we write Q ℓ d forthe unique unramified extension of degree d over Q ℓ , and Z ℓ d for its ring of integers. Lemma 4.1. o (2 ,
3) = 1 if p = 3 , and o (3 ,
4) = 2 if p ∈ { , } .Proof. For n = (2 ,
3) or (3 , A n = A n × A n , and hence A n = A n × A n .Any A n -lattice Λ ⊂ V decomposes into a product Λ n × Λ n , where each Λ n i is an A n i -lattice in V n i . Thus the techniques developed in Section 3 applies here.First suppose that n = (2 ,
3) and p = 3. In this case, A n = Z ⊗ Z O = O ,and V n ≃ D by (4.1). Since h ( O ) = 1, there is a unique isomorphism class of O -lattices in V n . As S ( n, p ) = { } in this case, we consider the A n , -lattices in V n ⊗ Q ≃ K n , . From Lemma 3.8, there is a unique A n , -lattice up toisomorphism in V n ⊗ Q . This implies that there is only a single genus of A n -lattices in V n . Note that End A n (Λ n ) = A n ≃ Z [ ζ ], which has class number1. Thus there is a unique isomorphism class of A n -lattices in V n . As a result, o (2 ,
3) = 1 · p = 3.Next, suppose that n = (3 ,
4) and p = 2. Since 2 is ramified in K n = Q ( ζ ), thesame proof as above shows that there is a unique isomorphism class of A n -latticesin V n in this case. On the other hand, 2 is inert in K n = Q ( ζ ), so A n , = Z .It follows from [15, Lemma 2.10] that(4.5) A n , = A n , ⊗ Z O = Z ⊗ Z O ≃ (cid:20) Z Z Z Z (cid:21) . Hence every A n , -lattices in V n ⊗ Q Q is isomorphic to either (cid:2) Z Z (cid:3) or (cid:2) Z Z (cid:3) . Wefind that there are two genera of A n -lattices in V n , each consisting of a uniqueisomorphism class since h ( A n ) = h ( Z [ ζ ]) = 1. Therefore, o (3 ,
4) = (1 + 1) · p = 2.Lastly, the value of o (3 ,
4) for p = 3 can be computed in exactly the same wayas above, since 3 is ramified in K n and inert in K n . (cid:3) Lemma 4.2. o (1 ,
2) = 3 if p = 2 .Proof. Set n = (1 ,
2) throughout this proof. In this case, A n is non-maximal onlyat p = 2, and O is the unique maximal order in the division quaternion Q -algebra D . From [28, (5.4)], we have A n = { ( a, b ) ∈ Z × Z | a ≡ b (mod 2) } , which implies that(4.6) A n, = A n ⊗ Z O = { ( x, y ) ∈ O × O | x ≡ y (mod 2 O ) } . In particular, A n, is a subdirect sum of two copies of O , so it is a Bass order by[9, Proposition 12.3].Let P be the unique two-sided prime ideal of O above p = 2. We put(4.7) R := { ( x, y ) ∈ O × O | x ≡ y (mod P ) } , which has index 4 in the maximal order O := O × O in K n = D × D . Indeed, R / ( P × P ) ≃ O / P ≃ F while O / ( P × P ) ≃ F × F . Clearly, both R and O areoverorders of A n . We claim that there are no other overorders except A n itself. Itis enough to prove this locally at p = 2. From A n, / (2 O × O ) ≃ O / O , wefind that(4.8) A n, / J ( A n, ) ≃ O / J ( O ) ≃ F , where J ( · ) denotes the Jacboson radical. Hence A n, is completely primary inthe sense of [18, p. 262]. Now according to [18, Lemma 6.6], every non-maximalcompletely primary Gorenstein order has a unique minimal overorder. As A n, hasindex 4 in R , the left A n, -module R / A n, is isomorphic to the unique simple left A n, -module F . Thus R coincides with the unique minimal overorder of A n, .Similarly, R is also completely primary, and O is the unique minimal overorderof R by the same argument. This verifies the claim about the overorders of A n . Let { R i | ≤ i ≤ n } be a finite set of (unital) rings, and R := Q ni =1 R i be their directproduct. A ring T is called a subdirect sum of the R i ’s if there exists an embedding ρ : T → R such that every canonical projection pr i : R → R i maps ρ ( T ) surjectively onto R i . UPERSPECIAL ABELIAN SURFACES 13
In this case, V ⊗ Q is a free left module of rank 1 over K n, . For any A n, -lattice in V ⊗ Q , its associated left order necessarily coincides with one of the threeoverorders of A n, . Since A n, is completely primary, it is indecomposable as a leftmodule over itself. Taking into account that A n, is Gorenstein, it follows from[3, Proposition 2.3] that every proper A n, -lattice in the V ⊗ Q is principal. Thisholds for R as well by the same token. On the other hand, every proper O -latticein the V ⊗ Q is principal since O is maximal. Therefore, every A n, -lattice in V ⊗ Q is isomorphic to one of the following(4.9) A n, , R , O . We find that there are three genera of A n -lattices in V represented by A n , R and O respectively. Clearly,End A n ( A n ) = A opp n , End A n ( R ) = R opp , End A n ( O ) = O opp . In each case, the opposite ring can be canonically identified with the original ringitself, so we drop the superscript opp henceforth.It remains to show that h ( A n ) = h ( R ) = h ( O ) = 1. This clearly holds true forthe maximal order O = O × O since h ( O ) = h ( O ) · h ( O ) = 1 · h ( A n ) = 1, then h ( R ) = 1 since h ( A n ) ≥ h ( R ) by [28, (6.1)]. Let b O (resp. c A n ) bethe profinite completion of O (resp. A n ). Since h ( O ) = 1, it follows from [28, (6.3)]that(4.10) h ( A n ) = | O × \ b O × / ( c A n ) × | . Clearly, A × n, ⊇ O and A × n,ℓ = b O × ℓ for every prime ℓ = 2, so there is an b O × -equivariant projection( O / O ) × ≃ O × / (1 + 2 O ) ։ b O × / ( c A n ) × . Hence to prove h ( A n ) = 1, it is enough to show that the canonical projection O × → ( O / O ) × is surjective. Since O = O × O , this amounts to show that(4.11) ϕ : O × → ( O / O ) × is surjective . From [22, Proposition V.3.1], O × ≃ SL ( F ), which has order 24. On the otherhand, | ( O / O ) × | = | ( O / P ) × | = | F × | · | F | = 3 · . Thus to prove the surjectivity of ϕ , it suffices to show that ker( ϕ ) = {± } . Sinceevery α ∈ O × has finite group order, this follows from a well-known lemma of Serre[19, Theorem, p. 17–19] (See also [16, Lemma, p. 192], [20] and [13, Lemma 7.2] forsome variations and generalizations). Therefore, h ( A n ) = 1 as claimed.In conclusion, we have o (1 ,
2) = h ( A n ) + h ( R ) + h ( O ) = 1 + 1 + 1 = 3 . (cid:3) For n = (3 , o ( n ) for p ∈ { , } will be calculated in a few steps. Lemma 4.3. o (3 ,
6) = Q ℓ ∈ S ( n,p ) | L ℓ ( n ) | for both p ∈ { , } , where the set S ( n, p ) is given in (4.3) .Proof. We identify both A and A with Z [ ζ ] via the following maps: Z [ T ] / ( T + T +1) → Z [ ζ ] , T ζ , and Z [ T ] / ( T − T +1) → Z [ ζ ] , T
7→ − ζ . As a result, the maximal order O K n of K n is identified with Z [ ζ ] × Z [ ζ ]. From[28, (5.7)], we have(4.12) A n = { ( a, b ) ∈ O K n | a ≡ b (mod 2 Z [ ζ ]) } . In particular, A n / (2 O K n ) ≃ F . Hence [ O K n : A n ] = 4, and the overorders of A n are precisely O K n and A n itself. From (4.2), E n = K n , so the endomorphism ring O Λ of any A n -lattice Λ ⊂ V is an overorder of A n . Thus O Λ is equal to either A n or O K n . Since h ( A n ) = h ( O K n ) = 1 by [28, (5.8)], we conclude that (cid:3) (4.13) o (3 ,
6) = Y ℓ ∈ S ( n,p ) | L ℓ ( n ) | . Lemma 4.4. o (3 ,
6) = 2 if p = 3 .Proof. From (4.3), S ( n,
3) = { , } , so we need to classify A n,ℓ -lattices in V ⊗ Q ℓ for both ℓ = 2 ,
3. For ℓ = 2, we have O = Mat ( Z ). Hence(4.14) A n, = A n, ⊗ Z O = Mat ( A n, ) . Recall that V = K n by (4.2). Applying Morita’s equivalence, we reduce the clas-sification of A n, -lattices in V ⊗ Q to that of A n, -lattices in K n, . Now A n, isa commutative Bass order over Z , so it follows from Lemma 3.2 that each A n, -lattice in K n, is isomorphic to an overorder of A n, (namely, either A n, or O K n , ).Therefore, | L ( n ) | = 2.Next, we compute | L ( n ) | . Since 3 is coprime to [ O K n : A n ] = 4, we have A n, = O K n ⊗ Z Z = A n , × A n , , and hence(4.15) A n, = A n, ⊗ Z O = A n , × A n , . (4.16)Here A n i , = Z [ ζ ] ⊗ Z O for each i = 1 ,
2. On the other hand, V n i ⊗ Q ≃ K n i , by (4.2). It follows from Lemma 3.8 that | L ( n ) | = 1.We conclude from Lemma 4.3 that o (3 ,
6) = 2 · p = 3. (cid:3) Proposition 4.5. o (3 ,
6) = 8 if p = 2 .Proof. In this case, S ( n,
2) = { } by (4.3). Let Q be the unique unramifiedquadratic extension of Q , and Z be its ring of integers. Then A n i , = Z [ ζ ] = Z for both i = 1 ,
2. The same calculation as in (4.5) shows that(4.17) O K n , ⊗ Z O = ( Z × Z ) ⊗ Z O ≃ (cid:20) Z Z Z Z (cid:21) × (cid:20) Z Z Z Z (cid:21) . For simplicity, put E := (cid:2) Z Z Z Z (cid:3) and identify ( O K n , ⊗ Z O ) with B := E × E .Then it follows from (4.12) that(4.18) A n, = { ( x, y ) ∈ B | a ≡ b (mod 2 E ) } . In particular, 2 B is a two-sided ideal of A n, contained in J ( A n, ). We put(4.19) ¯ A n, := A n, / (2 B ) = E / E , ¯ B := B / B = ( E / E ) × ( E / E ) , where ¯ A n, embeds into ¯ B diagonally.From (4.2), V n i ⊗ Q = K n i , = h Q Q i for each i = 1 ,
2. For simplicity, we identify V ⊗ Q with Mat ( Q ) and regard the i -th column as a K n i , -module for each i . UPERSPECIAL ABELIAN SURFACES 15
Any B -lattice in V ⊗ Q is isomorphic to one of the following(4.20) (cid:20) Z Z Z Z (cid:21) , (cid:20) Z Z Z Z (cid:21) , (cid:20) Z Z Z Z (cid:21) , (cid:20) Z Z Z Z (cid:21) . From Lemma 4.3, we are only concerned with A n, -lattices in V ⊗ Q , so for ease ofnotation we drop the subscript from Λ and write Λ for an A n, -lattice in V ⊗ Q .Replacing Λ by g Λ for a suitable g ∈ E × n, if necessary, we assume that B Λ is equalto one of the B -lattices ∆ in (4.20). Clearly 2∆ ⊆ Λ ⊆ ∆ since 2 B ⊆ A n, . Thus¯Λ := Λ / (2∆) is an ¯ A n, -submodule of ¯∆ := ∆ / B -module, ¯∆is spanned by ¯Λ. Fix one ∆ in (4.20). Let S (∆) be the set of A n, -sublattices Λ of∆ satisfying B Λ = ∆, and S ( ¯∆) be the set of ¯ A n, -submodules ¯ M of ¯∆ satisfying¯ B ¯ M = ¯∆. For any ¯ M ∈ S ( ¯∆), there is a unique A n, -sublattice Λ of ∆ satisfyingΛ ⊇
2∆ and ¯Λ = ¯ M . Moreover, B Λ = ∆ by Nakayama’s lemma. Hence theassociation Λ ¯Λ induces a bijective map:(4.21) S (∆) ∼ −→ S ( ¯∆) , Λ ¯Λ . The group End B (∆) × acts on both S (∆) and S ( ¯∆), and the map in (4.21) isEnd B (∆) × -equivariant as well. Two members Λ , Λ ′ of S (∆) are A n, -isomorphicif and only if there exists an α ∈ End B (∆) × such that α Λ = Λ ′ . Therefore, wehave established the following bijection(4.22) ( Isomorphism classes of A n, -lattices Λ in V ⊗ Q with B Λ ≃ ∆ ) ←→ ( End B (∆) × -equivalent classesof ¯ A n, -submodules ¯ M in ¯∆such that ¯ B ¯ M = ¯∆ ) . Note that End B (∆) = Z × Z for every ∆ in (4.20), so the action of End B (∆) × on S ( ¯∆) factors through F × × F × .Recall from (4.19) that ¯ A n, = E / E , and ¯ B = ( E / E ) . We describe the4-dimensional F -algebra E / E more concretely. Let R be the commutative F -algebra F × F . Regard Q := R as an ( R, R )-bimodule such that for each ( a, d ) ∈ R and ( b, c ) ∈ Q , the multiplications are given by the following rules:(4.23) ( a, d ) · ( b, c ) = ( ab, dc ) , ( b, c ) · ( a, d ) = ( bd, ca ) . We form the trivial extension [14, Example 1.14] of R by Q and denote it as ¯ E := (cid:10) F F F F (cid:11) , where R is identified with the diagonal of (cid:10) F F F F (cid:11) and Q is identified withthe anti-diagonal. More explicitly, the product in ¯ E is defined by the following rule:(4.24) (cid:28) a bc d (cid:29) · (cid:28) a ′ b ′ c ′ d ′ (cid:29) := (cid:28) aa ′ ab ′ + bd ′ ca ′ + dc ′ dd ′ (cid:29) . For each x ∈ Z , let ¯ x be its canonical image in F = Z / Z . One easily checksthat the following map induces an isomorphism between E / E and ¯ E : (cid:20) Z Z Z Z (cid:21) → (cid:28) F F F F (cid:29) , (cid:20) x y z w (cid:21) (cid:28) ¯ x ¯ y ¯ z ¯ w (cid:29) . Henceforth E / E will be identified with ¯ E via this induced isomorphism. Eachcolumn of ¯ E admits a canonical ¯ E -module structure. These two ¯ E -modules willbe denoted by (cid:2) F F (cid:3) † and (cid:2) F F (cid:3) ‡ respectively. It is clear from (4.24) that (cid:2) F (cid:3) † (resp. (cid:2) F (cid:3) ‡ ) is the unique 1-dimensional (as an F -vector space) ¯ E -submodule of (cid:2) F F (cid:3) † (resp. (cid:2) F F (cid:3) ‡ ). We leave it as a simple exercise to check that (cid:2) F F (cid:3) † and (cid:2) F F (cid:3) ‡ are non-isomorphic ¯ E -modules. Now for each ∆ in (4.20), we classify the ( F × × F × )-orbits of S ( ¯∆). For i = 1 , i be the i -th column of ∆ so that ¯∆ = ¯∆ × ¯∆ . Let ¯ M ⊆ ¯∆ be an ¯ E -submodule, and pr i : ¯ M → ¯∆ i be the projection map to the factor ¯∆ i . Then(¯ E × ¯ E ) ¯ M = ¯∆ if and only if both pr i are surjective. Thus(4.25) dim F ¯ M ≥ M ∈ S ( ¯∆) . Suppose that ¯ M ∈ S ( ¯∆) from now on.If dim F ¯ M = 2, then both pr i are ¯ E -isomorphisms, and ¯ M is the graph of theisomorphism pr ◦ pr − : ¯∆ → ¯∆ . Necessarily, ∆ is equal to either (cid:2) Z Z Z Z (cid:3) or (cid:2) Z Z Z Z (cid:3) . In these two cases, any isomorphism ¯∆ → ¯∆ is a scalar multiplicationby F × . After a suitable multiplication by an element of F × × F × , we may identify¯ M with the diagonal of ¯∆ × ¯∆ .Next, suppose that dim F ¯ M = 3. Then ker(pr ) is a 1-dimensional submoduleof ¯ M ∩ ¯∆ , so it must coincide with the unique 1-dimensional submodule of ¯∆ . Asimilar result holds for ker(pr ). We claim that ¯∆ ≃ ¯∆ in this case as well. Ifnot, without lose of generality, we may assume that ¯∆ = (cid:2) F F (cid:3) † and ¯∆ = (cid:2) F F (cid:3) ‡ so that ¯∆ = ¯ E . By the above discussion, ¯ M ⊇ (cid:10) F F (cid:11) . Since dim F ¯ M = 3, thereexists u, v ∈ F × such that¯ M = F (cid:28) u v (cid:29) + F (cid:28) (cid:29) + F (cid:28) (cid:29) . Indeed, both u and v have to be nonzero since pr i is surjective for each i = 1 , h u v i ∈ ¯ E × , which implies that ¯ M = ¯ E = ¯∆. This contradicts dim F ¯ M =3 and verifies our claim. It remains to consider the cases ¯∆ = (cid:2) F F (cid:3) † × (cid:2) F F (cid:3) † or¯∆ = (cid:2) F F (cid:3) ‡ × (cid:2) F F (cid:3) ‡ . For simplicity, we write them as Mat ( F ) † and Mat ( F ) ‡ respectively. Suppose that ¯∆ = Mat ( F ) † . Then there exists u, v ∈ F × such that¯ M = F (cid:20) u v (cid:21) + F (cid:20) (cid:21) + F (cid:20) (cid:21) . Multiplication by ( u − , v − ) ∈ F × × F × sends ¯ M to¯ M := (cid:26)(cid:20) a bc d (cid:21) ∈ Mat ( F ) † (cid:12)(cid:12)(cid:12)(cid:12) a = b (cid:27) . Thus all 3-dimensional members of S (Mat ( F ) † ) are in the same ( F × × F × )-orbit.A similar result holds for ¯∆ = Mat ( F ) ‡ .Lastly, if dim F ¯ M = 4, then ¯ M = ¯∆.Combining (4.20), (4.22) and the above classifications, we find that | L ( n ) | = X ∆ | ( F × × F × ) \ S ( ¯∆) | = 3 + 3 + 1 + 1 = 8 . Thus o (3 ,
6) = 8 if p = 2 according to Lemma 4.3. (cid:3) Proposition 4.6.
Suppose that p ∈ { , } . Then (4.26) o (2 , p ) = ( if p = 2 , if p = 3 . UPERSPECIAL ABELIAN SURFACES 17
Proof.
Let n = ( n , n ) = (2 , p ) for p ∈ { , } . Then K n = Q × K n , and K n ≃ D × Mat ( K n ) by (4.1) and (4.2). Here K n is equal to Q ( √− Q ( √− p = 2 or p = 3. Let p be the unique ramified prime ideal of A n above p . From [28, (5.5) and (5.6)], we have(4.27) A n = { ( a, b ) ∈ Z × A n | a ≡ b (mod p ) } , which is a suborder of index p in O K n = Z × A n .From (4.3), S ( n, p ) = { p } . Clearly, O p is canonically a subring of A n ,p = A n ⊗ Z O p , and ( p A n ,p ) ∩ O p = p O p . It follows from (4.27) that(4.28) A n,p = { ( x, y ) ∈ O p × A n ,p | x ≡ y (mod p A n ,p ) } . For simplicity, let us put C := O p × A n ,p , and J := J ( O K n ,p ) = p Z p × p , where p denotes the maximal ideal of A n ,p by an abuse of notation. Then J C ⊆ A n,p , sowe further define three quotient rings¯ O p := O p /p O p , (4.29) ¯ C := C / ( J C ) = ( O p /p O p ) × ( A n ,p / p A n ,p ) = ¯ O p × ¯ O p , (4.30) ¯ A n,p := A n,p / ( J C ) = ¯ O p , (4.31)where ¯ A n,p embeds into ¯ C diagonally by (4.28). From [22, Corollary II.1.7], O p contains a copy of Z p , and there exists η ∈ O p such that(4.32) O p = Z p + Z p η, η = p, and xη = η ˜ x, ∀ x ∈ Z p . Here x ˜ x denotes the unique nontrivial Q p -automorphism of Q p . If we write ¯ η for the canonical image of η in ¯ O p , then(4.33) ¯ O p = F p + F p ¯ η. In particular, dim F p ¯ O p = 2, and the Jacobson radical J ( ¯ O p ) = F p ¯ η is the unique1-dimensional submodule of ¯ O p .From (4.1) and (4.2), V n is a free module of rank 1 over K n ≃ D , and V n = K n is a simple module over K n ≃ Mat ( K n ). Fix a suitable identification of K n ,p =Mat ( K n ,p ) so that the hereditary closure H ( A n ,p ) is equal to Mat ( A n ,p ). FromLemma 3.8, every C -lattice in V ⊗ Q p is isomorphic to(4.34) ∆ := ∆ × ∆ = O p × (cid:20) A n ,p A n ,p (cid:21) . Put ¯∆ := ∆ /J ∆ = ¯∆ × ¯∆ , where(4.35) ¯∆ = O p /p O p = ¯ O p , and ¯∆ = ∆ / p ∆ . The same proof as (4.22) shows that there is a bijection(4.36)
Isomorphism classesof A n,p -lattices in V ⊗ Q p ←→ ( End C (∆) × -equivalent classesof ¯ A n,p -submodules ¯ M in ¯∆such that ¯ C ¯ M = ¯∆ ) . Clearly, dim F p ¯∆ = 2. When regarded as an A n ,p -module, ¯∆ is isomorphic to theunique simple A n ,p -module F p by (4.30) and (4.33). We fix such an isomorphismand write ¯∆ = F p . Note that End C (∆) = O p × A n ,p , so the action of End C (∆) × on ¯∆ factors through ¯ O × p × F × p . Recall that ¯ C = ¯ O p × ¯ O p and ¯ A n,p = ¯ O p by (4.30)and (4.31). Let ¯ M ⊆ ¯∆ be an ¯ O p -submodule, and pr i : ¯ M → ∆ i be the canonical projectionsfor i = 1 ,
2. Then ( ¯ O p × ¯ O p ) ¯ M = ¯∆ if and only if both pr i are surjective. Supposethat this is the case. Then necessarilydim F p ¯ M ≥ dim F p ¯∆ = 2 . If dim F p ¯ M = 3, then ¯ M = ¯∆.Now suppose that dim F p ¯ M = 2. Then ker(pr ) is a 1-dimensional submoduleof ¯∆ , so ker(pr ) = F p ¯ η . Therefore, there exists a, c ∈ F × p such that(4.37) ¯ M = F p (¯ η,
0) + F p ( a, c ) ⊆ ¯∆ = ( F p + F p ¯ η ) × F p . Indeed, both a, c have to be nonzero since pr i is surjective for each i = 1 ,
2. Multi-plication by ( ca − , ∈ ¯ O × p × F × p sends ¯ M to the following submodule of ¯∆:(4.38) ¯Γ := { ( a + b ¯ η, c ) ∈ ¯∆ | a = c } . Let Γ be the unique A n,p -sublattice of ∆ such that Γ ⊇ J ∆ and Γ /J ∆ = ¯Γ. Thenevery A n,p -lattice in V ⊗ Q p is isomorphic to either ∆ or Γ. Let P be the uniquetwo-sided prime ideal of O p . We computeEnd A n,p (Γ) = { ( x, y ) ∈ O p × A n ,p | ( x, y )¯Γ ⊆ ¯Γ } = { ( x, y ) ∈ O p × A n ,p | x ≡ y (mod P ) } . (4.39)Here P ∩ A n ,p = p for any embedding of A n ,p into O p , so the congruence relationdoes not depend on the choice of such an embedding.We have seen from above that there are two genera of A n -lattices in V . Accordingto [22, Proposition V.3.1], h ( O ) = 1 for p ∈ { , } , so O is the unique maximalorder in D up to D × -conjugation. If e ∆ is an A n -lattices in V with e ∆ ⊗ Z p = ∆,then(4.40) End A n ( e ∆) ≃ O × A n . Similarly, if e Γ is the unique A n -sublattice of e ∆ such that e Γ ⊗ Z p = Γ, then End A n ( e Γ)is the unique suborder of End A n ( e ∆) satisfying(4.41) End A n ( e Γ) ⊗ Z ℓ = ( End A n ,p (Γ) if ℓ = p, End A n ( e ∆) ⊗ Z ℓ otherwise . For simplicity, let us put O := End A n ( e ∆) and R = End A n ,p (Γ). From the generalstrategy explained in Section 2, we have(4.42) o (2 , p ) = h ( O ) + h ( R ) for p = 2 , . Here h ( O ) = h ( O ) h ( A n ) = 1 · h ( R ).Replacing e ∆ by g e ∆ for a suitable g ∈ E × n if necessary, we assume that O = O × A n . Let b O (resp. b R ) be the profinite completion of O (resp. R ). We apply[28, (6.3)] to obtain(4.43) h ( R ) = | O × \ b O × / b R × | = | O × \ O × p / R × p | . From (4.39), P × p ⊆ R p , and O p / ( P × p ) = F p × F p , R p / ( P × p ) = F p . where F p embeds into F p × F p diagonally. It follows that O × p / R × p can be furthersimplified into ( F × p × F × p ) / diag( F × p ). On the other hand, O × = O × × A × n . Since UPERSPECIAL ABELIAN SURFACES 19 p ∈ { , } , the natural map O × → F × p × F × p sends {± } × A × n onto F × p × F × p , whichcontains diag( F × p ). Therefore, h ( R ) = | O × \ O × p / R × p | = | ( O × × A × n ) \ ( F × p × F × p ) / diag( F × p ) | = | ( O × × { } ) \ ( F × p × F × p ) / ( F × p × F × p ) | = |O × \ F × p / F × p | = |O × \ F × p | . (4.44)Here we have used freely the commutativity of F × p × F × p .If p = 2, we have already shown in the proof of Lemma 4.2 that the canonical map O × → ( O / O ) × is surjective (see (4.11)). Consequently, O × → ( O p / P ) × = F × issurjective as well. Thus h ( R ) = 1 if p = 2.Next, suppose that p = 3. According to [22, Exercise III.5.2], D = (cid:18) − , − Q (cid:19) , and O = Z + Z i + Z j Z i (1 + j )2 . Here { , i, j, ij } is the standard Q -basis of D . We have O × = (cid:26) ± , ± i, ± ± j , ± i (1 ± j )2 (cid:27) . Note that j ∈ P , so the image of O × in ( O p / P ) × = F × is equal to {± ¯1 , ± ¯ i } , whichhas index 2 in F × . Therefore, h ( R ) = 2 if p = 3.In conclusion, we find that o (2 ,
4) = 1 + 1 = 2 if p = 2, and o (2 ,
6) = 1 + 2 = 3if p = 3. (cid:3) Remark 4.7.
Assume that p ∈ { , } . We have shown that except for ( n, p ) =((2 , ,
3) (and ((1 , ,
3) by [28, Remark 3.2(i)]), every genus in the set L ( n ) oflattice classes has class number one. For the exceptional cases, L ( n ) consists oftwo genera; one genus has class number one while the other one has class numbertwo. Concluding the Proof of Theorem 1.1.
For p = 2 , ,
5, the value of o ( n ) is listed inthe following table (see [28] for the values not covered in the present paper): n ,
2) (2 ,
3) (2 ,
4) (2 ,
6) (3 ,
4) (3 , p = 2 3 2 2 1 3 3 2 2 4 2 8 p = 3 2 3 2 4 3 3 1 4 3 2 2 p = 5 3 1 1 4 4 4 2 0 6 0 8Formula (1.6) is obtained by plugging in the above values in (1.3). (cid:3) Acknowledgments
J. Xue is partially supported by Natural Science Foundation of China grant
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UPERSPECIAL ABELIAN SURFACES 21 (Xue) Hubei Key Laboratory of Computational Science (Wuhan University), Wuhan,Hubei, 430072, P.R. China.
Email address : xue [email protected] (Yu) Institute of Mathematics, Academia Sinica and NCTS, Astronomy-MathematicsBuilding, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, TAIWAN. Email address : [email protected] (Zheng) School of Mathematics and Statistics, Wuhan University, Luojiashan, 430072,Wuhan, Hubei, P.R. China Email address : [email protected] Current address : (Zheng) Academy of Mathematics and Systems Science, Chinese Academyof Science, No. 55, Zhongguancun East Road, Beijing 100190, China
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