An Algorithm for Computing Local Moduli of Abelian Varieties

Annals of Mathematics

An Algorithm for Computing Local Moduli of Abelian VarietiesAuthor(s): Peter NormanSource: Annals of Mathematics, Second Series, Vol. 101, No. 3 (May, 1975), pp. 499-509Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970937 .

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An algorithm for computing local moduli of abelian varieties

By PETER NORMAN

Over an algebraically closed field of characteristic p our knowledge of explicit local moduli of a polarized abelian variety was tied to separable phenomena: either the polarization had to be separable ([6]) or the abelian variety had to be ordinary ([2]). We present here a method for finding explicitly the local moduli in what was formerly the least tractable case-an abelian variety of p-rank zero with a possibly inseparable polarization.

We compute the local moduli in two steps: (1) We find the formal deformations of the abelian variety and (2) We find the largest subscheme of the space of deformations on which

the polarization extends ([6]). For an abelian variety of p-rank zero these can be translated into a problem about Dieudonne modules. First a result of Serre-Tate ([3]) implies that both steps can be done by working with p-divisible groups. Second Cartier ([1]) has shown that over an arbitrary ring the categories of smooth formal groups and Dieudonne modules are equivalent. Thus the first step can be done in terms of Dieudonne modules. Lastly Mumford ([4]), using biextensions, translated the concept of polarization on p-divisible groups to Dieudonne modules. For a polarized abelian variety of p-rank zero the problem of formal moduli comes out to:

(1) Finding explicitly the space of deformations of a Dieudonne module, (2) Finding the largest subscheme of this space on which a given biex-

tension extends.

In Section 1 we compute explicitly the formal deformations of a Dieudonne module. In Section 2 we construct a universal biextension and apply this to simplifying the concept of a biextension for a special but sufficiently wide class of Dieudonne modules.

In the last section we apply the results of the previous section to compute explicitly the local moduli in two specific cases. The first example is of a polarization on the product of an elliptic curve with itself and gives a local moduli space which is reduced and singular. The second example is also a polarization on the product of an elliptic curve with itself and gives an example of a local moduli space which has two components.



500 PETER NORMAN

I would like to thank D. Mumford for both help and encouragement.

0. Preliminaries and notation

Let R be a ring of characteristic p. Let W(R) be the ring of Witt vectors over R and let

(a0, al, = (a', a, ...

(aop alp ...= (O. aop alp ...

Then AR denotes the ring W(R)[[ VJ][F] modulo the relations: (a) FV= p and VaF== at, (b) Fa= a7F and Va7 = a V.

A Dieudonne module M, or D-module for short, is a left AR module satisfying (i) n ViM== (o), (ii) Vm - 0 implies m = 0 for all m E M, and (iii) M/ VM is a free R module of finite rank.

There is an equivalence of categories between D-modules over R and com- mutative smooth formal groups over R. Often we will refer to a D-module when we actually mean the associated formal group.

For details on biextensions the reader should consult Mumford's paper ([4]).

1. Deformations of Dieudonne modules

Let M be a D-module over an algebraically closed field K corresponding to a smooth p-divisible group. It is well-known that the space of deformations of M is smooth with (dim M) . (codim M) parameters ([7]). We need to know explicitly the deformations of M; further for a forthcoming paper we need to put the deformations of M into a special form.

The D-module M has generators ei, i = 1, ... * g + h and is defined by the relations

(1) Fei =aijej., Y g ej = V(E aijej) g + l . I g + h

for some invertible matrix (aij) with entries in W(K). Let R be a local Artin ring of characteristic p, residue field K, and maximal ideal m.

THEOREM 1. There is a one to one correspondence between isomorphism classes of deformations of M over R and maps

d: VM/pM In OK (M/ VM), e-i E dij 0 8)j

where e-i, e-j are the images of ei, ej in VM/pM and M/ VM respectively. Set



COMPUTING LOCAL MODULI OF ABELIAN VARIETIES 501

dij = (dj6, O. ... ) E W(R) for i > g, j < g and dij = 0 otherwise. The D- module corresponding to the map d is defined by

(2) Fei = E (aii(ej + dJkek)), i = 1, .. g

(2') ej = V(Laij(ej + Ldikek)) = g + 1, , g + h . Furthermore every D -module over R which restricts to M over K is isomorphic to exactly one D-module of the form (2), (2').

This theorem has a use far beyond the classification of deformations. To this end we make the

Definition. A D-module M over an arbitrary ring S of characteristic p is displayed if M is given by generators ei, i = 1, *..., g + h and relations

Fei = Eai6ej i = 1, * *, g

ej = V(Lasiej) ig +1P .. Pg +h

for an invertible matrix (aij) with entries in W(S).

In Section 2 we show that in the case of displayed modules the concept of biextension can be greatly simplified. Thus we have the useful

COROLLARY. Any D-module over an Artin local ring of characteristic p with algebraically closed residue field which specializes to a p-divisible group over the residue field is displayed.

Proof. (This proof, due to Mumford, is much shorter and clearer than the author's original one).

Because the deformation functor for smooth formal groups is smooth [6], it suffices to classify deformations over the ring K[5], 52 = 0. Let s E W(K[5]) be the element s = (5, 0, 0O ...), and let Rt be a D-module over K[S] lifting M.

There is an exact sequence

(3) 0 - 3 VOe(R1/vt) -+OR 7 - M 0

This follows from the facts (a) s eVm = Vepm = 0 and (b) the Witt vector (0, ..., 5, 0 *.*) = VnsFn for the appropriate choice

of n. Split this sequence by lifting Mto M={x E OR I Vkx E FOR + Lk-,' VFeO}.

This splitting comes from noticing that FM can be lifted since FOR n ker w = (0) and extending this lifting. M has the properties

(i) it splits the exact sequence (3), (ii) it is a W(k) and an F submodule of OR,



502 PETER NORMAN

(iii) V2 ;M ' + D. Only (iii) requires any proof. Let m C M, and let k be the smallest integer such that w( Vkm) E FM. Write Vm = m1 + em2 + Vm3 with ml C M and m3 C ker w. We need to show m3 = 0. Now Vkm = Vk-lml + Vk-lem2 + Vkm3. Since 7( Vk-lml) E FM and m1 E Ml}, Vk-lmeFRT + Ek-2 VE C R. But m EC implies VIeM+ y Vi e ' 0 so Vkm3=0and m3,= 0. It is easy to see that M is unique with these properties. We recover Rt from this splitting as follows. Start by writing

(+) O = MED ((D VseM/VM)

as W(k) module. (a) Define F as usual on M and as 0 on the second summands. (b) Define s as 0 on the second summand and s: Me- eM/ VMon the first

summand. (c) Define V as the shift on the second summand.

If x e M, then because of (iii) V(x, 0) (Vx, sd(Vx)) where d: VM/pM MI VM is a i-linear map.

Thus the liftings of M are in one to one correspondence with the maps d. Pick generators {Uf} of M as in (1). Let d: VM/PM-+ Ml VM be given in

terms of {Uf} by the matrix (dij). Let dij = (dij, 0, 0, ... ) E W(k). Let ei = (e-i, 0) be the lifting of e-i to Rt as in (+). We then have from (a) and (c):

Fei= aijej6, g V( aijej) ei + s Ed dijej, > g

Setting e' = ei E dijej gives the form (2) asked for in the statement of the theorem. Q.E.D.

2. Biextensions

The object of this section is to put the concept of a biextension into a more manageable form. (By biextension we mean a biextension by the formal multiplicative group.) Let M, N be D-modules. We show there is an AR- module C which is not necessarily a D-module such that the functors N.- Biext [M x N] and N-o HomAR (N. C) are equivalent.

More specifically let AR = A1 = A2 and let

0- Aln 45Anf- M- ,

be a free resolution of M. Let A = W(R)[[F, V]] with the same relations as AR and let * denote the antiautomorphism of A given by (i) F* = V, (ii) V* = F, and (iii) a* = a for a e W(R). Let HomR (A,, A) denote the maps f from An to A satisfying f(am) = f(m) . ca* for a e AR, m e A". Let C denote




the cokernal of the map s*: HomR (A,, A) HomR (A", A). THEOREM. There is a canonical isomorphism of functors

(N ' Hom (N, C)) ----* (N- Biext [N x M]) .

In particular if C is a D-module there is a canonical biextension B C Biext [C x M] such that f f *B from Hom (N, C) to Biext [N x M] is an isomorphism.

Proof. Let 0 Am Am- N-- 0 be a free resolution of N. We have to show that a map 8,, e Hom (N, C) gives two bihomomorphisms a, ,i into A satisfying p*,a =- o*a as in the diagram:

0 > Am - > An -i N > 0

(4)0-A __ ( ) O >

~~~~A" An M > 0

a A map from N to C induces a map , from Am to C such that p*,fl 0. Next lift 8, to a map from Am to HomR (A", A), i.e., a bihomomorphism f8: Am x

In~~~~~~~~~~~~~~~~~~~~~~

Al HA:

Awn P >4Am > N

HomR (A", A) > HomR (A", A)/s* HomR (A", A) - C The condition p*,S, = 0 gives a bihomomorphism a: Am x A"- A. More

concretely let R1, *, Rm be generators of Am. Since for all i, f,3(pRi) 0 in C, f31(pRj) = s*aj for some map ao: A- A. Define a: Am x A"' A by a(Ri, x) = ai(x) for i = 1, * m , m, and x C A". By construction pfl* = oc and a, fi together give a biextension of N x M.

We need to check that 8, induces a biextension in a well-defined way. In particular let A, 8' be two liftings of fl1. We need to show that fi and 9' correspond to the same biextension; that is, we have to show 9 - /1 = O*a for some -: Am HomR (A,, A). But the image of /9 - l9' in C is zero. This implies /9- /9 = * The map /9 determines a since Hom (M, A) = (0).

Conversely given a biextension of M x N as in (4) we get a map from N to C by following the above process backward. More explicitly, from (4) we see that / induces a map from Am to HomR (An, A) and hence a map fi from Am to C. The condition paoid I*a implies p*- 0 in C, but p*fl 0 0 is exactly the condition needed to insure that /9 induces a map from N to C.

Q.E.D.



504 PETER NORMAN

We can explicitly compute C if M is a displayed module.

PROPOSITION. Let M be a displayed D-module of local-local type with resolution

0 -* A1A An - M-> 0.

Let {Ei} be the generators of A", and {Ri} the generators of A". Let

0(Ri)= FEi - EaijEj ... 1, *,g 0(Ri)-=Es- V(,aijEj) i= g + 1, . * jg +h .

Here (aij) is an invertible matrix with entries in W(R). Then C is a D- module which we denote by Mt, generated by fi, i = 1, ... , g + h, and defined by the relations

fi = V(E bijfj), i= 1 , ,g h Ffi-=I, bijfj p g + 1, .. * * g + h.

Here (b j) = (ajJ-1.

Proof. Let Fi (resp. Pi) be the projections from Al (resp. A") onto the ith component. Set Si = E bijPj.

05*(SJ)(Rk) = (I bijPj)(FEk- E ak.E.) , k ? g, = bikV- a kkg.

Here aik is the Kronecker delta.

) (SiR) = (I bijPj)(Ek - V(E akeEe)) k > g = bik - aikF k > g.

Therefore

(5) O~~~~*Si = ,bij VFj + IbijFj -Fi , i < g

(O*Si = bij VFj + bijFj - FFf, i>g.

On the other hand let li = I bijfj for i < g and li = fi for i > g be a new set of generators of Mt. Since Vli = fi for i < g, these generators satisfy

li = Ej:g bij Vlj + Ej>g bijlj , i < g P Fli = Ej:g bij Vlj + Zj>g bijlj i > g .

Since these are the same relations as in (5), Mt and C are isomorphic. Q.E.D.

3. Examples

In this section we compute explicitly the equations defining the local moduli of polarized abelian varieties for two examples. The examples are of two different polarizations on the product of a supersingular elliptic curve with itself. The method we use is the one outlined in the introduction: we




find the largest subscheme of the space of formal deformations on which the polarization extends. This is done in terms of D-modules and biextensions. Assume p # 2.

Let A be an abelian variety of p-rank zero over the algebraically closed field K and M the D-module of its p-divisible group. Define a quasi-polarization of Mto be an AK linear map M-- induced by a symmetric biextension. It is straightforward to prove:

Let X be a quasi-polarization on M inducing an isomorphism X: Me Mt. Let s be an endomorphism of M. The Rosati involution is defined by s -' = ,-1 o oX where 05t is the transpose of s. Then s' = s if and only if s is a quasi-polarization. If E is a supersingular elliptic curve over K, then its D- module D(E) is generated by elements e1 and e2 with the relations

Fel= e2, e2=Ve1.

If f1, f2 the basis from Section 2, D(E)t is defined by

f1 = Vf2, Ff2= Af1

A simple calculation shows that

A: e1 Xf2

for some \ such that Va - -X are the quasi-polarizations of D(E). In gen- eral an endomorphism of M is given by

f(el) = ael + be2

with a?2 = a and b72 - b. Another direct calculation shows that the Rosati involute of f is f': el awel - be2. Therefore the only polarizations of E are the Z-multiples of A.

The D-module M of E x E is defined by

Fe,= e3 , Fe2= e4

e3 =Ve , e4==Ve2

In the notation of Section 2, M' is given by

fl = Vf3, f2= Vf4,

Ff3 = fl Ff4 = f2

Any endomorphism ep of M is given by a matrix (0)i) where i, j = 1, 2, and where e c End D(E). The Rosati involute of the map 0 is O' = (by). We conclude that q = 5' if and only if l, 0,22 G W(Z/pZ) and 012 = 021

For the first example set 012 to be the Frobenius map from the first copy of E to the second. To get a quasi-polarization we need to let (021 be the negative of this. Set oil =22 = 0. Thus 0(e1) = e4 and 0(e2) =-e3. Composing



506 PETER NORMAN

this with the principal polarization A x A gives the quasi-polarization

P: e, - Xf2

e2 Xf,.

Although P is not a polarization, P + pn(A. x A.) for sufficiently high n is a polarization. Further these two quasi-polarizations have the isomorphic local moduli spaces if n is large enough. To prove this it suffices to show that the equations defining the local moduli space are the same modulo a high power of the maximal ideal. This follows from the fact that if (a1, a2, *.*) is a Witt vector with all its components in the maximal ideal, then p"(a,, a2, * * *)

has all its components in the (pn)th power of the maximal ideal. In order to keep the equations simple we compute the equations defining the local moduli space by using P.

From Section 2 we get the space of formal deformations of M. Let Ti, i = 1, * , 4 be formal parameters, and set t. = (Ti0, , O.*), a Witt vector in W(k[[ T1, ... , T4j]). The universal deformation of Mis a D -module denoted by Ot over k[[ T1, ... , T4j] generated by e1, e2, e3, e4 and satisfying the relations

Fel = e3 + tie, + t2e2,

Fe2= e4 + t3e, + t4e2,

e3= Ve, e4= Ve2

or

(6) Fe1 = Ve1 + t1e,+ t2e2, Fe2 = Ve2 + t3e, + t4e2,

while Mt is generated by f1, ... , f4 and has relations

(7) Ff3= Vf3-tIf3-t3f4, Ff4 = Vf4-t2f3-t4f4 .

Let N be any D-module over a k-algebra S and Ic S an ideal. Let S/I = SO. For n e N we say n = O mod I or n reduces to zero mod I if the image of n in N x AS As, is zero. In particular if m c OR, we say it reduces to zero if it reduces to zero modulo the ideal (T1, T2, T3, T4).

To find the local moduli we must find the largest subscheme spec k[ T1, T4] on which P extends. We know this is a subscheme and that it is

defined by one equation since by the results of Mumford and Serre-Tate outlined in the introduction we are actually computing the local moduli of a polarized abelian variety ([6]).

The extension of P will be defined by




C: e 1 --Xf2+ A, C: e2- Xf1 + B

for some A and B which reduce to zero. In order that 9i be a D-module map the relations (6) must be satisfied:

(F - V)9(e,) t1(el) + t2P(e2)

(F - V)i(e2) t3i(ej) + t4i(e2)

Expanding these gives

(8) -VX(-t2f3 - t4f4) + (F - V)A = tA + t2B + V(Xtlf4 - Xt2f3) , (9) VX(-tlf3 - t3f4) + (F - V)B = t3A + t4B - V(xt3f4 - Xt4f3) .

Let m be the ideal generated by T1, *.., 7T4, and examine these equations mod in2. Notice A = 0 mod m implies FA = 0 mod mW. From the first equation we get

VX(t2f3 + t4f4) - VA 0 mod m2 A X(t23 + t4f4) modm2.

A similar calculation with the second equation (9) shows that

B = -X(tJf3 + t3f4) mod M2 .

Using these approximations of A and B we examine the equation (8). This gives successively

-VX(-t23 - t4f4) - VA tjA + t2B mod (m3, V), t1A + t2B- 0 mod (m3, V).

This is where the assumption p / 2 is used. Upon simplification this gives X(t1t4 - t2t3)f4 0 mod (m3, V). Similarly the equation (9) implies t3A + t4B 0 mod (m3, V) and this in turn implies x(t2t3-t1t4)f3 0 mod (m3, v). The local moduli space is defined by T2T3- TT4 + higher degree terms. In fact it is easy to see that these higher degree terms will all be degree p or greater. Since all power series with this initial term are isomorphic we know the singularity at this point.

The second example is of a diff erent polarization on E x E and gives a local moduli space with more than one component. To get the polarization, compose the polarization A x A with the endomorphism, multiplication by p. The resulting polarization on Ot is

9P(e) = pXf3 + A,

9P(e2)= pXf4 + B

where A, B reduce to zero. For IP to be a map of D-modules we require



508 PETER NORMAN

(10) (F - V)P(ej) = t1g(el) + t2,(e2)

(11) (F - V)I(e2) = t3i(ej) + t4i(e2)

We expand (10) and get

(l0t) -PX(-tlf3 - t3f4) + (F - V)A = tl(pXf3 + A) + t2(pXf4 + B).

If n C M, let ord n = r denote the greatest integer such that n reduces to zero modulo the ideal (T1, * * *, T44)r = Mrn. Now let r = (ord A) + 1 = (ord B) + 1 and examine the above equation (10') mod Mr. You get

pX[tif3 + t3f4] - VA pX[t1f3 + t2f41, mod in,

VFX[t3f4 - t2f4] VA, mod m7 V(-X(ta - ta))Ff4 = VA, mod Mr

Thus

A = X(t2 - t3)Ff4 = X(t2 - t3)( Vf4 - t2f3 - t4f4) mod mr

From equation (11) we get

(11') B= -X(t2 - t3)(Vf3 - t1f3 - t3f4) modMr.

Notice that in the expression for A we have (t - t) Vf4 = V(Vt 2 -ta3)f4 0 mod M +3. A similar thing happens with B. We substitute these values into equation (10') and examine mod (Mr+3, V). We get

t1A + t2B 0 mod (mp+3, V) .

And after a simple calculation

X(-tlt4 + t2t3)(t - t)f4 0 mod (mn+3, V)

Equation (11) leads to a similar result. We can conclude that the local moduli space is defined by (T2? - T3i) x

(?2 ?3 - T, ?4) mod ( T1, * * *, T4)P+3. Since - 2T3 = 0 defines the local moduli space of the polarization A x A (an easy computation), (T2 - T3)P gives a component of the local moduli space of the polarization in this example. The other component has leading term (T2 T3 - T, T4).

THE UNIVERSITY OF MASSACHUSETTS

BIBLIOGRAPHY

[1 ] P. CARTIER, Modules Associes 'a Un Groupe Formel Commutatif, C. R. Acad. Sc. Paris 265 (1967), 129-132.

[2] S. CRICK, Local Moduli for Abelian Varieties, Harvard Thesis, 1973. [ 3 ] W. MESSING, The Crystals Associated to Barsotti-Tate Groups, Lecture notes math. 264,

Springer, Berlin, 1972. [4] D. MUMFORD, Biextensions of Formal Groups, Algebraic Geometry, Oxford, London, 1968. [ 5 1 T. ODA, The First DeRham Cohomology Group and Dieudonne Modules, Ann. Sci. Ecole




Norm. Sup., Paris (1969). [ 6 ] F. OORT, Finite Group Schemes, Local Moduli for Abelian Varieties, and Lifting Problems,

Proc. Fifth Nordic Summer School, Walters-Noordhoff, Amsterdam, 1970. [7] H. UNEMURA, Formal Moduli of p-divisible Groups, Nagoya Jour. Math. 42 (1971).

(Received September 25, 1974) (Revised February 19, 1975)



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An Algorithm for Computing Local Moduli of Abelian Varieties