The a-numbers of Jacobians of Suzuki curves
Holley Friedlander, Derek Garton, Beth Malmskog, Rachel Pries, Colin Weir
aa r X i v : . [ m a t h . N T ] O c t The a -numbers of Jacobians of Suzuki curves Holley Friedlander Derek Garton Beth Malmskog Rachel PriesColin WeirOctober 20, 2018
Abstract
For m ∈ N , let S m be the Suzuki curve defined over F m +1 . It is well-known that S m is supersingular,but the p -torsion group scheme of its Jacobian is not known. The a -number is an invariant of theisomorphism class of the p -torsion group scheme. In this paper, we compute a closed formula for the a -number of S m using the action of the Cartier operator on H ( S m , Ω ).Keywords: Suzuki curve, maximal curve, Jacobian, p-torsion, a-number.MSC: 11G20, 14G50, 14H40. Let m ∈ N , q = 2 m +1 , and q = 2 m . The Suzuki curve S m ⊂ P is defined over F q by the homogeneousequation: W q ( Z q + ZW q − ) = Y q ( Y q + Y W q − ) . This curve is smooth and irreducible with genus g = q ( q −
1) and it has exactly one point at infinity [8,Proposition 1.1]. The number of points on the Suzuki curve over F q is S m ( F q ) = q + 1; this number isoptimal in that it reaches Serre’s improvement to the Hasse-Weil bound [8, Proposition 2.1].In fact, S m is the unique F q -optimal curve of genus g [2]. This shows that S m is the Deligne-Lusztigvariety of dimension 1 associated with the group Sz ( q ) = B ( q ) [7, Proposition 4.3]. The curve S m has theSuzuki group Sz ( q ) as its automorphism group; the order of Sz ( q ) is q ( q − q + 1) which is very largecompared with g . Because of the large number of rational points relative to their genus, the Suzuki curvesprovide good examples of Goppa codes [4, Section 4.3], [5], [8].The L -polynomial of S m is (1 + √ qt + qt ) g [7, Proposition 4.3]. It follows that S m is supersingularfor each m ∈ N . This fact implies that the Jacobian Jac( S m ) is isogenous to a product of supersingularelliptic curves and that Jac( S m ) has no 2-torsion points over F . However, there are still open questionsabout Jac( S m ). In this paper, we address one of these by computing a closed formula for the a -number ofJac( S m ).The a -number is an invariant of the 2-torsion group scheme Jac( S m )[2]. Specifically, if α denotes thekernel of Frobenius on the additive group G a , then the a -number of S m is a ( m ) = dim F Hom( α , Jac( S m )[2]).It equals the dimension of the intersection of Ker( F ) and Ker( V ) on the Dieudonn´e module of Jac( S m )[2].Having a supersingular Newton polygon places constraints upon the a -number but does not determine it. The a -number also gives partial information about the decomposition of Jac( S m ) into indecomposable principallypolarized abelian varieties, Lemma 4.3, and about the Ekedahl-Oort type of Jac( S m )[2], see Section 4.2.In Section 4, we prove that the a -number of S m is a ( m ) = q ( q + 1)(2 q + 1) /
6, see Theorem 4.1. Theproof uses the action of the Cartier operator on H ( S m , Ω ) as computed in Section 3.Author Pries was partially supported by NSF grant DMS-11-01712. We would like to thank the NSF forsponsoring the research workshop for graduate students at Colorado State University in June 2011 where thework on this project was initiated. We would like to thank Amy Ksir and the other workshop participantsfor their insights. 1 The a -number Suppose A is a principally polarized abelian variety of dimension g defined over an algebraically closedfield k of characteristic p >
0. For example, A could be the Jacobian of a k -curve of genus g . Considerthe multiplication-by- p morphism [ p ] : A → A which is a finite flat morphism of degree p g . It factors as[ p ] = V ◦ F . Here, F : A → A ( p ) is the relative Frobenius morphism coming from the p -power map on thestructure sheaf; it is purely inseparable of degree p g . The Verschiebung morphism V : A ( p ) → A is the dualof F .The kernel of [ p ] is A [ p ], the p -torsion of A , which is a quasi-polarized BT group scheme. In other words,it is a quasi-polarized finite commutative group scheme annihilated by p , again having morphisms F and V .The rank of A [ p ] is p g . These group schemes were classified independently by Kraft (unpublished) [10] andby Oort [13]. A complete description of this topic can be found in [12] or [13].Two invariants of (the p -torsion of) an abelian variety are the p -rank and a -number. The p -rank of A is r ( A ) = dim F p (Hom ( µ p , A [ p ])), where µ p is the kernel of Frobenius on the multiplicative group G m . Then p r ( A ) is the cardinality of A [ p ] (cid:0) F p (cid:1) . The a -number of A is a ( A ) = dim k (Hom ( α p , A [ p ])), where α p is thekernel of Frobenius on the additive group G a . It is well-known that 1 ≤ a ( A ) + r ( A ) ≤ g . Another definitionfor the a -number is a ( A ) = dim F p (Ker( F ) ∩ Ker( V )) . If X is a (smooth, projective, connected) k -curve, then the a -number of A = Jac( X ) equals the dimensionof the kernel of the Cartier operator C on H ( X, Ω ) [11, 5.2.8]. The reason for this is that the action of C on H ( X, Ω ) is the same as the action of V on V Jac( X )[ p ]. This is the property that we use to calculatethe a -number a ( m ) of the Jacobian of the Suzuki curve S m . In this section, we compute the action of the Cartier operator on the vector space of regular 1-forms for theSuzuki curves.
Let m ∈ N , q = 2 m +1 , and q = 2 m . Consider the Suzuki curve S m ⊂ P defined over F q by the homogeneousequation: W q ( Z q + ZW q − ) = Y q ( Y q + Y W q − ) . The curve S m is smooth and irreducible and has one point P ∞ at infinity (when W = Y = 0 and Z = 1).Consider the irreducible affine model of S m defined by the equation z q + z = y q ( y q + y ) (1)where y := Y /W and z := Z/W .The following result is well-known, see e.g., [8, Proposition 1.1]. We include an alternative proof thatillustrates the geometry of some of the quotient curves of S m and an important point about the a -number. Lemma 3.1.
The curve S m has genus g = q ( q − .Proof. The set F ∗ q = { µ , . . . , µ q − } can be viewed as a set of representatives for the q − F ∗ in F ∗ q .The Suzuki curve has affine equation z q − z = f ( y ) where f ( y ) = y q + q + y q +1 ∈ F ( y ). For 1 ≤ i ≤ q − Z i be the Artin-Schreier curve with equation z i − z i = µ i f ( y ). As seen in [3, Proposition 1.2], the set { Z i → P y | ≤ i ≤ q − } is exactly the set of degree 2 covers Z → P y which are quotients of S m → P y .By [6, Proposition 3], an application of [9, Theorem C], there is an isogenyJac ( S m ) ∼ ⊕ q − i =1 Jac ( Z i ) . By Artin-Schreier theory, µ i f ( y ) can be modified by any polynomial of the form T − T for T ∈ F [ y ]without changing the F -isomorphism class of the Artin-Schreier cover Z i → P y . Thus Z i is isomorphic2o an Artin-Schreier curve with equation z i − z i = h i ( y ) for some h i ( y ) ∈ F [ y ] with degree 2 q + 1 =max { ( q + q ) /q , q + 1 } . For 1 ≤ i ≤ q −
1, the curve Z i is a Z / ∞ , where it is totally ramified. Moreover, the break in the filtration of higher ramification groupsin the lower numbering is at index deg( h i ( y )) = 2 q + 1. By [14, VI.4.1], the genus of Z i is q . Thus g = dim(Jac( S m )) = ( q − Z i )) = q ( q − Remark 3.2.
Consider the Artin-Schreier curve Z i : z i − z i = h i ( y ) from the proof of Lemma 3.1. By[1, Proposition 3.4], since deg( h i ) = 2 q + 1 ≡ a -number of Z i is q /
2. Thus the a -number of ⊕ q − i =1 Jac( Z i ) is q ( q − /
2, exactly half of the genus of S m . The fact that Jac( S m ) is isogenous to ⊕ q − i =1 Jac( Z i )gives little information about the a -number of S m since the a -number is not an isogeny invariant.The Hasse-Weil bound states that a (smooth, projective, connected) curve X of genus g defined over F q must satisfy q + 1 − g √ q ≤ X ( F q ) ≤ q + 1 + 2 g √ q. A curve that meets the upper bound is called an F q -maximal curve.It is easy to check that the number of F q -points on the Suzuki curve is S m (cid:0) F q (cid:1) = q + 1 and so S m is not maximal over F q . Analyzing powers of the eigenvalues of Frobenius shows the following. Lemma 3.3.
The Suzuki curve S m is F q -maximal.Proof. The L -polynomial of S m is L ( S m , t ) = (1+ √ qt + qt ) g [7, Proposition 4.3]. This factors as L ( S m , t ) =(1 − αt ) g (1 − αt ) g where α = q (1+ i ). That implies that S m ( F q ) = q +1 − ( − q ) ( α + α ) g = q +1+2 q g which shows that S m is F q -maximal.A curve which is maximal over a finite field is supersingular, in that the slopes of the Newton polygonof its L -polynomial all equal 1 /
2. Thus S m is supersingular. The supersingularity condition is equivalent tothe condition that Jac( S m ) is isogenous to a product of supersingular elliptic curves. A supersingular curvein characteristic 2 has 2-rank 0. This implies, a priori, that the a -number of S m is at least one. -forms To compute a basis for the vector space H ( S m , Ω ) of regular 1-forms on S m , consider the functions h , h ∈ F ( S m ) given by: h : = z q + y q +1 ,h : = z q y + h q . For any f ∈ F ( S m ), let v ∞ ( f ) denote the valuation of f at P ∞ . Lemma 3.4.
The functions y, z, h , h ∈ F ( S m ) have no poles except at P ∞ where v y : = − v ∞ ( y ) = q, v z : = − v ∞ ( z ) = q + q ,v h : = − v ∞ ( h ) = q + 2 q , v h : = − v ∞ ( h ) = q + 2 q + 1 . The function π = h /h is a uniformizer at P ∞ .Proof. See [8, Proposition 1.3].The function y is a separating variable so dy is a basis of the 1-dimensional vector space of differential1-forms. The next lemma shows that dy is regular. Lemma 3.5.
The differential -form dy satisfies v ∞ ( dy ) = 2 g − and v P ( dy ) = 0 for all points P ∈ S m ( F q ) . roof. Recall that π is a uniformizer at P ∞ . To take the valuation of dy at P ∞ , we first rewrite dy = f ( x, y ) dπ for some f ( y, z ) ∈ F q ( y, z ). Note that dπ = d (cid:18) h h (cid:19) = h dh − h dh h = h y q − h z q h dy. Since v ∞ ( h ) = − q + 2 q + 1) and v ∞ ( h y q − v h z q ) = min {− q v y − v h , − q v z − v h } = − q v z − v h = − q − q , we see that v ∞ ( dy ) = v ∞ h h q − h z q dπ ! = − q + 2 q − − (cid:0) − q − q (cid:1) = 4 q o − q −
2= 2 g − . We next show that dy has no zero or pole at any affine point of S m . Note that, for any a ∈ F q , thepolynomial z q + z + a splits into distinct factors in F q ( z ), so there are exactly q points of S m ( F q ) lying overany y ∈ A y ( F q ). Since (cid:2) F q ( y, z ) : F q ( y ) (cid:3) = q, the F q -Galois cover S m → P y is unramified at all affine points of S m . Consequently, for any point P ∈ S m ( F q )lying over a ∈ A y ( F q ), we see that v P ( y − a ) = 1. Thus, y − a is a uniformizer at P and v P ( dy ) = 0, provingthe proposition.By Lemma 3.5, finding a basis for H ( S m , Ω ) is equivalent to finding a basis for L (( dy )), since ( dy ) =(2 g − P ∞ is the canonical divisor. To do this, we make use of the relations: z = yh + h , h q = z + y q +1 , h q = h + zy q , (2)which can be verified by direct substitution, and the following proposition. Proposition 3.6. [8, Proposition 1.5] Let SG be the semigroup h q, q + q , q + 2 q , q + 2 q + 1 i . Then { n ∈ SG | ≤ n ≤ g − } = g . We now have all the required information to find a basis of H ( S m , Ω ). Proposition 3.7.
The following set is a basis of H ( S m , Ω ) : B := (cid:8) y a z b h c h d dy | ( a, b, c, d ) ∈ E (cid:9) where E is the set of ( a, b, c, d ) ⊂ Z satisfying ≤ b ≤ , ≤ c ≤ q − , ≤ d ≤ q − ,av y + bv z + cv h + dv h ≤ g − . Proof.
To prove linear independence, it suffices to prove that all elements in our basis have distinct valuationsat P ∞ . Suppose that y a z b h c h d dy ∈ B and y a ′ z b ′ h c ′ h d ′ dy ∈ B have the same valuation at P ∞ ; we will showthey are equal. Comparing their valuations at P ∞ , we must have that( a − a ′ ) v y + ( b − b ′ ) v z + ( c − c ′ ) v h + ( d − d ′ ) v h = 0 (3)4ow consider equation (3) modulo q . As q divides v y , v z and v h ,( d − d ′ ) ≡ q . As 0 ≤ d, d ′ < q , it must be the case that d = d ′ . Substituting d − d ′ = 0 into equation (3) and reducingmodulo 2 q yields that ( b − b ′ ) q ≡ q . However, as 0 ≤ b, b ′ ≤
1, it must also be the case that b = b ′ . Simplifying (3) and reducing modulo q = 2 q o yields that ( c − c ′ )( q − q ) = ( c − c ′ )2 q ≡ q . Since 0 ≤ c, c ′ ≤ q −
1, we find that c = c ′ ; so a = a ′ as well.We claim that the above set also spans L (( dy )). Clearly the valuations at P ∞ of (cid:8) y a z b h c h c | ( a − a ′ ) v y + ( b − b ′ ) v z + ( c − c ′ ) v h + ( d − d ′ ) v h ≤ g − (cid:9) are equal to { n ∈ SG | ≤ n ≤ g − } , which is a set of size g by Proposition 3.6. Rewriting elements ofthe above set in terms of our basis will not change their valuation at P ∞ . Thus we can use the relationsin equation (2) to see that B also contains an element for each of the g possible valuations at P ∞ . By theprevious paragraphs, each valuation occurs exactly once. By Riemann-Roch, ℓ (( dy )) = g , so B is a basis. In characteristic 2, the Cartier operator C acts on differential 1-forms according to the following properties:(see e.g., [15, Section 2.2.5]).1. C is 1 / C is additive and C ( f ω ) = f C ( ω ).2. C ( y j dy ) = ( , if j y e − dy if j = 2 e − . C ( ω ) = 0 if and only if ω is exact; i.e., if and only if ω = df for some f ∈ F q ( S m ).4. C ( ω ) = ω if and only if ω = df /f for some f ∈ F q ( S m ).Any 1-form ω ∈ H ( S m , Ω ) can be written in the form ω = ( f + g y ) dy , as char( F q ) = 2. Then C (( f + g y ) dy ) = g dy. (4)By these properties, it is clear that C ( y e + r z e + r h e + r h e + r dy ) = y e z e h e h e C ( y r z r h r h r dy ) . (5)Hence to compute the action of C on H ( S m , Ω ), we need only compute C on the 16 monomials in y, z, h , h of degree less than or equal to one in each variable. The table below shows this action, whereeach C ( f dy ) is written in terms of the original basis using the curve equation (1).5 C ( f dy )1 0 y dyz y q / dyh y q dyh (cid:0) ( yh ) q / + h (cid:1) dyyz h q / dyyh (cid:0) ( yh ) q / + h (cid:1) dyzh ( yh ) q / dyzh ( h h ) q / dyh h ( h + zy q ) dyyzh (cid:0) y q / z + ( h h ) q / (cid:1) dyyzh (cid:16) zh q / + y q / h q / (cid:17) dyzh h (cid:16) zy q / h q / + h q / (cid:17) dyyh h (cid:16) ( yh ) q / z + h q / z (cid:17) dyyzh h (cid:16) y q / h + zh q / h q (cid:17) dy Example 3.8.
We illustrate the computation for zh h dy . Direct computation yields C ( zh h dy ) = C (cid:16) zh (cid:16) yz q + h q (cid:17) dy (cid:17) = z q C ( zyh dy ) + h q C ( zh dy )= (cid:16) y q / z q +1 + h q / h q / z q + h q / h q / h q / (cid:17) dy. To write this expression in terms of the original basis, we identify the monomials with the highest pole orderat infinity. Since v ∞ ( h q / h q / z q ) = v ∞ ( h q / h q / h q / ) = − q − q − q / < − (2 g − , these two terms may be simplified. Using Section 3, h q / h q / z q + h q / h q / h q / = h q / h q / (cid:16) y q / h q / + h q / (cid:17) + y q / h q h q / = h q / h q . The final expression follows by rewriting z q +1 and h q in terms of lower order basis elements using equa-tions (2). Remark 3.9.
To compute C ( ω ) for a general element ω ∈ B , simply apply equation (5) and use the tableabove; in nearly all cases the direct result will again be in terms of the basis B . The only exception is when ω = zh q − h dy . In this case we have: C (cid:16) h q − · zh h dy (cid:17) = h q / − C ( zh h )= (cid:16) zy q / h q / − h q / + h q (cid:17) dy. Using equations (2), one can obtain an expression in terms of the original basis. a -number for Suzuki curves a -number We now have the tools to compute a ( m ). The calculation amounts to counting lattice points in polytopesin R , which is a hard problem in general. In our case, however, the values v y , v h , and v h are so similarthat the polytopes in question are nearly regular; this makes our counting problem much easier.6 heorem 4.1. Let a ( m ) and g ( m ) be the a -number and genus of S m respectively. Then a ( m ) = q ( q + 1)(2 q + 1)6 . In particular, < a ( m ) g ( m ) <
16 + 12 m +1 . Proof.
Recall from Section 2 that a ( m ) is the dimension of the kernel of C on H ( S m , Ω ). By equation (4), a ( m ) is the dimension of the vector space of regular differentials of the form f dy . Since f can have apole only at P ∞ , and since the order of the pole can be at most 2 g −
2, we see that a ( m ) = ℓ (( g − P ∞ ).Moreover, squaring is a homomorphism, so ℓ (( g − P ∞ ) = { ω ∈ B | v ∞ ( ω ) ≤ g − } . By Section 3, this number is exactly { ( a, b, c, d ) ∈ E | v y a + v z b + v h c + v h d ≤ g − } ;here we use the notation E as we did in Proposition 3.7. Recall that b ∈ { , } . When b = 0, we must count { ( a, c, d ) ∈ N | a + c + d ≤ q − } . This follows from the fact that that q − g − v h < g − v h < g − v y < q . For b = 1, we must count { ( a, c, d ) ∈ N | a + c + d ≤ q − } since q − < g − − v z v h < g − − v z v h < g − − v z v y < q − . Using these two facts, we obtain a ( m ) = (cid:8) ( a, c, d ) ∈ N | a + c + d ≤ q − (cid:9) + (cid:8) ( a, c, d ) ∈ N | a + c + d ≤ q − (cid:9) = q +1 X i =2 (cid:18) i (cid:19) + q X i =2 (cid:18) i (cid:19) =1 + q X i =2 (cid:18)(cid:18) i + 12 (cid:19) + (cid:18) i (cid:19)(cid:19) = q X i =1 i , as desired.To prove the second statement, simply note that16 < q ( q + 1)(2 q + 1)6 q ( q −
1) = 16 · q + q + q − < (cid:18) q (cid:19) . Here are two open questions about Jac( S m ). Question 4.2.
What is the decomposition of Jac( S m ) into indecomposable principally polarized abelianvarieties? 7heorem 4.1 gives partial information about Question 4.2, namely an upper bound on the number offactors appearing in the decomposition, because of the following fact. Lemma 4.3.
Suppose A is a principally polarized abelian variety with p -rank and a -number a . If A decomposes as the direct sum of t principally polarized abelian varieties, then t ≤ a .Proof. Write A ≃ ⊕ ti =1 A i where each A i is a principally polarized abelian variety. For 1 ≤ i ≤ t , considerthe p -torsion group scheme A i [ p ]. The a -number of A i [ p ] is at least 1 since its p -rank is 0. Thus the a -numberof A is at least t .To state the second question, we need some more notation.The Ekedahl-Oort type of a principally polarized abelian variety A over k is defined by the interactionbetween the Frobenius F and Verschiebung V operators on the p -torsion group scheme A [ p ]. It determinesthe isomorphism class of A [ p ] and its invariants such as the a -number. To define the Ekedahl-Oort type, recallthat the isomorphism class of a symmetric BT group scheme G over k can be encapsulated into combinatorialdata. This topic can be found in [13]. If G has rank p g , then there is a final filtration N ⊂ N ⊂ · · · ⊂ N g of G as a k -vector space which is stable under the action of V and F − such that i = dim ( N i ). The Ekedahl-Oort type of G , also called the final type , is ν = [ ν , . . . , ν r ] where ν i = dim ( V ( N i )). The Ekedahl-Oort typeof G is canonical, even if the final filtration is not.There is a restriction ν i ≤ ν i +1 ≤ ν i + 1 on the final type. Moreover, all sequences satisfying thisrestriction occur. This implies that there are 2 g isomorphism classes of symmetric BT group schemes ofrank p g . The p -rank is max { i | ν i = i } and the a -number equals g − ν g . Question 4.4.
What is the Ekedahl-Oort type of Jac( S m )[2]? Equivalently, what is the covariant Dieudonn´emodule of Jac( S m )[2]?Theorem 4.1 gives partial information about Question 4.4, by limiting the possible final types. For thegroup scheme Jac ( S m )[2], the Ekedahl-Oort type satisfies that ν = 0 and ν g = q (10 q + 7)( q − /
6. Inparticular, Jac( S m ) is not superspecial since a ( m ) = g ( m ). This implies that Jac( S m ) is not isomorphicto the product of supersingular elliptic curves; it is only isogenous to the product of supersingular ellipticcurves.In the next example, we give some more information about the Ekedahl-Oort type of Jac( S )[2] (the case m = 1). Example 4.5. If m = 1 then q = 2, q = 8, and g = 14. By Section 3.3, the image of C on H ( S m , Ω ) isspanned by the nine 1-forms { dy, ydy, h dy, y dy, yh dy, yh , ( z + y ) dy, h h dy, y zdy } . The image of C on H ( S m , Ω ) is spanned by the four 1-forms { dy, y dy, ( z + y ) dy, ( h + y z ) dy } . Also C trivializes H ( S m , Ω ). Thus ν = ν = ν = ν = 0, and ν = 4, and ν = 9. The combinatorialrestrictions on the final type imply that ν = 5, ν = 6, ν = 7, and ν = 8. This leaves only fivepossibilities for the final type, and thus for the isomorphism class of Jac( S )[2]. References [1] Arsen Elkin and Rachel Pries,
Ekedahl-oort strata of hyperelliptic curves in characteristic 2 ,arXiv:1007.1226.[2] Rainer Fuhrmann and Fernando Torres,
On Weierstrass points and optimal curves , Rend. Circ. Mat.Palermo (2) Suppl. (1998), no. 51, 25–46. MR 1631013 (99e:11081)[3] Arnaldo Garc´ıa and Henning Stichtenoth,
Elementary abelian p -extensions of algebraic function fields ,Manuscripta Math. (1991), no. 1, 67–79. MR 1107453 (92j:11139)84] Massimo Giulietti and G´abor Korchm´aros, On automorphism groups of certain Goppa codes , Des. CodesCryptogr. (2008), no. 1-3, 177–190. MR 2375466 (2009d:94156)[5] Massimo Giulietti, G´abor Korchm´aros, and Fernando Torres, Quotient curves of the Suzuki curve , ActaArith. (2006), no. 3, 245–274. MR 2239917 (2007g:11069)[6] Darren Glass and Rachel Pries,
Hyperelliptic curves with prescribed p -torsion , Manuscripta Math. (2005), no. 3, 299–317. MR 2154252 (2006e:14039)[7] Johan P. Hansen, Deligne-Lusztig varieties and group codes , Coding theory and algebraic geometry(Luminy, 1991), Lecture Notes in Math., vol. 1518, Springer, Berlin, 1992, pp. 63–81. MR 1186416(94e:94024)[8] Johan P. Hansen and Henning Stichtenoth,
Group codes on certain algebraic curves with many rationalpoints , Appl. Algebra Engrg. Comm. Comput. (1990), no. 1, 67–77. MR 1325513 (96e:94023)[9] E. Kani and M. Rosen, Idempotent relations and factors of Jacobians , Math. Ann. (1989), no. 2,307–327. MR 1000113 (90h:14057)[10] H. Kraft,
Kommutative algebraische p -gruppen (mit anwendungen auf p -divisible gruppen und abelschevariet¨aten) , manuscript, University of Bonn, September 1975, 86 pp.[11] K.-Z. Li and F. Oort, Moduli of supersingular abelian varieties , Lecture Notes in Mathematics, vol.1680, Springer-Verlag, Berlin, 1998. MR MR1611305 (99e:14052)[12] B. Moonen,
Group schemes with additional structures and Weyl group cosets , Moduli of abelian varieties(Texel Island, 1999), Progr. Math., vol. 195, Birkh¨auser, Basel, 2001, pp. 255–298. MR MR1827024(2002c:14074)[13] F. Oort,
A stratification of a moduli space of abelian varieties , Moduli of abelian varieties (Texel Island,1999), Progr. Math., vol. 195, Birkh¨auser, Basel, 2001, pp. 345–416. MR 2002b:14055[14] Henning Stichtenoth,
Algebraic function fields and codes , second ed., Graduate Texts in Mathematics,vol. 254, Springer-Verlag, Berlin, 2009. MR 2464941 (2010d:14034)[15] Michael Tsfasman, Serge Vl˘adut¸, and Dmitry Nogin,