Completing an elliptic curve E at its identity section yields the formal group ˆE, a refine-ment of the Lie algebra ofE. In this section, we will define precisely what a formal group is and show how to get a morphism from the moduli stack of elliptic curves to the mod-uli stack of formal groups. This will be essential for the definition of the spectrum of topological modular forms TMF.
Definition 2.8.1. Let S be a scheme. A formal scheme over S is a functor (Sch /S)op → Set, which is a (small) filtered colimit of functors representable by schemes over S. The cartesian product in the functor category restricts to a product on the category of formal schemes overS, denoted by×S. A commutative group object in formal schemes over Sis called anabstract formal group over S.
Example 2.8.2. Let Abe a ring with a chosen ideal I. Then SpfA:=colimnSpec(A(In))is a formal scheme. If we have a morphism f: A→ Bsuch that f(I)⊂ J for a chosen ideal J ⊂ B, then we get an induced map SpfB→SpfA.
Aformal group lawover a ring Rconsists of a power series F ∈ R[[X,Y]]satisfying the axioms of a commutative group in a formal way (see [Rav86], Appendix B, for a precise definition). The formal spectrum SpfR[[X]] := colim SpecR[x]/xi is a formal scheme and F induces a morphism SpfR[[X]]×SpecRSpfR[[Y]] ∼= SpfR[[X,Y]] → SpfR[[X]] (by sendingX to F), which defines an abstract formal group over R; here, the chosen ideal of R[[X,Y]] is the augmentation ideal (X,Y). A (1-dimensional, commutative) formal group over a scheme S is an abstract formal group F which comes Zariski locally on S from a formal group law (i.e., we can cover S as SUi with Ui ∼= SpecRi such that F|Ui is isomorphic to an abstract formal group coming from a formal group law overRi).
37 Definition 2.8.3. Themoduli stack of formal groupsMFG is given by associating to each ring Rthe groupoid of formal groups over R.
The moduli stack of formal group laws FGL (without morphisms between them) is much simpler: It is isomorphic to SpecL forL(uncanonically) isomorphic toZ[x1,x2, . . .] and L carries an universal formal group law Funiv.14 In concrete terms, this means the map HomRings(L,R) → FGL(R)given by φ 7→ φ∗(Funiv) is a bijection. The fiber product SpecL×MFG SpecL is equivalent to SpecW for W = L[u±1,b1,b2, . . .]. As explained in [Nau07], this shows that MFG is algebraic and (by a theorem of Quillen) QCoh(MFG)' (MU∗,MU∗MU)-comod (where the comodules are graded).
For future purposes, we want to be a bit more explicit: We setH =SpecZ[u±1,b1,b2, . . .] and identifyH(SpecR)with power series of the formux+b1x2+b2x3+· · · withb1,b2,· · · ∈ Rand u ∈ R× a unit. Composition of power series defines a natural group structure on H(SpecR) and thus the structure of a group scheme on H. The scheme SpecL ∼= FGL gets the structure of an H-torsor overMFG with H acting as follows: Forh ∈ H(SpecR) andF ∈ FGL(SpecR), define a formal group law h·F over RbyhF(h−1(x),h−1(y)). This defines an action of H on FGL. This can be extended to an action of H on FGL ∼= SpecL over MFG in the sense of Section 2.6: Forh∈ H(SpecR), F ∈ FGL(SpecR)the elementh defines an isomorphism between the underlying formal groups of F andh·F, which we take as our αF,h. That SpecL×H → SpecL×MFGSpecL is an equivalence boils down to the fact that an isomorphism between formal groups associated to formal group laws is given by a power series.
LetFbe a formal group law overRandgbe an automorphism of the associated formal group. Then we can writeg as power seriesϕ(g)∈ R[[x]]with
ϕ(g)−1F((ϕ(g))(x),(ϕ(g))(y)) =F.
This defines a morphism from the automorphism group of the underlying formal group of FintoH(SpecR). The check that this morphism fulfills the conditions on ϕin Proposition 2.6.6 is analogous to the example of the moduli stack of elliptic curves with level structures.
To every elliptic curve E/S, we can associate a formal group as follows: Denote by e: S → E the unit section and by I the ideal sheaf on E corresponding to the reduced subscheme structure on im(e), i.e., im(e) equals the vanishing locus V(I) of I. A map f: X →Efactors overV(I)iff the ideal sheaf f∗I is zero. It factors over ˆE:=colimV(In) iff f∗Iis locally nilpotent, hence, iff the morphismXredfactors overV(I). Suppose now, we have points a,b ∈ Eˆ(X). Via the canonical map ˆE → E, these induce maps a0,b0: X → E.
We get a diagram
Xred //
im(e)×Sim(e) //
im(e)
X a0×b0 //E×SE //E
The map Xred → E corresponds to a point a·b ∈ Eˆ(X), inducing a group structure on E. This defines indeed a formal group (as can be seen, e.g., in the Weierstrass form). Forˆ further information, see also [Rez02, 11.4].
14The ringLis called theLazard ring.
Locally, the corresponding formal group law can be concretely calculated up to arbi-trary precision using a Weierstraß form; either by hand, as in [Sil09], Chapter IV.1, or by Magma or similar programs.
Theorem 2.8.4. The assignment E7→ E induces a morphismˆ M → MFG, which is flat.
The author is not aware of a published proof of the flatness statement, but, at least, this theorem is stated in Lecture 15 of [Lur10]. Furthermore, it can probably be deduced from [BL10, 8.1.6] and the Serre–Tate theorem (stating that elliptic curves have the same deformation theory as p-divisible groups, see [BL10, 7.2.1]).
Chapter 3
Vector Bundles
Our aim in this chapter is the study of vector bundles over the moduli stack of elliptic curves. Recall the following definition:
Definition 3.0.1. Avector bundleon a Deligne–MumfordX stack is an OX-module that is locally free of finite rank in the étale topology.
As noted before, every vector bundle is a quasi-coherent (even coherent) sheaf since it has locally a presentation.
Recall the notationMR for the moduli stack of elliptic curves overR. As a shorthand, denote by M(p) the moduli stack of elliptic curves over Z(p). Furthermore, we denote structure sheaves in general byO(with subscript if it is not clear from the context).
ThePicard group Picof a stack is the group of isomorphism classes of line bundles (with group structure given by the tensor product and the inverses by duals). The classification of line bundles onMR is already known:
Theorem 3.0.2 ([FO10]). Every line bundle overMR, for R a reduced ring, is a tensor power of ω and we haveω12∼=O. Therefore, the Picard groupPic(MR)is isomorphic toZ/12.
We will prove that every vector bundle splits into line bundle on MQ using an argu-ment by Angelo Vistoli. In general, the situation is more complicated and we will mainly restrict to the caseM(3). A particularly accessible class of vector bundles is the following:
Definition 3.0.3. We define the notion of astandard vector bundlefor a prime pinductively:
Every line bundle on M(p) is calledstandard. Furthermore, a vector bundle E on M(p) is called standard if there is an injection L ,→ E from a line bundle on M(p) such that the cokernel is a standard vector bundle.
Thus, standard vector bundles are those vector bundles which can be built as iterated extension of line bundles.
Lemma 3.0.4. 1. LetE be a vector bundle with a surjective morphismE → L to a line bundle such that the kernelF is a standard vector bundle. ThenE is a standard vector bundle.
2. LetE be a standard vector bundle. Then alsoEˇ is a standard vector bundle.
Proof. 1. LetE be of rankn. By induction, we assume that we have shown the first part of the lemma for all smaller ranks. By definition, we have an injectionL0 ,→ F from
39
a line bundle such that the cokernelF0 is a standard vector bundle again. Consider the (snake lemma) diagramm
0
0
0 //L0 = //
L0
//0
0 //F //
E //
L //
=
0
0 //F0 //
E0 //
L //
0
0 0 0
Here,E0 is defined as the cokernel ofL0 → F → E. It is a vector bundle since it is an extension of two vector bundles (see Remark 2.3.10). Furthermore, it is of rankn−1 and has a surjective morphism toLwhose kernelF0is a standard vector bundle. By induction,E0 is thus a standard vector bundle. This implies that alsoE is standard.
2. By induction, we assume that we know the statement for all standard vector bundles of smaller rank thanE. Consider a sequence
0→ L → E → F →0
whereLis a line bundle andF is standard. Dualizing gives 0→F →ˇ E →ˇ L →ˇ 0.
Note that the sequence is short exact because the Ext-sheafExti(F,O)vanishes for i>0 sinceF is a vector bundle. The morphism ˇE →Lˇ is surjective and its kernel is standard by induction. Thus, we can use the first part of the lemma.
The main aim of this chapter is to show the following theorem:
Theorem 3.0.5. Every standard vector bundle overM(3) is isomorphic to a sum of copies of the vector bundlesO, Eα or Eα,eα (and tensor products of line bundles with them). Here, the latter two are vector bundles of rank2and3, respectively, to be introduced in Section 3.4.
Conjecture 3.0.6. Every vector bundle onM(3)is standard.
In addition, we prove that there are infinitely many indecomposable vector bundles on M(2).
As a warm up, we will recall the classification of integral representations of the cyclic groupC2or, what is equivalent, vector bundles over SpecZ//C2– this is easier but in some ways analogous to classification results on vector bundles on the moduli stack of elliptic curves. We must stress that the classification of integral C2-representations is already known for a long time – if not since the beginning of time or the era of Archimedes, then at least since [Die40].
41 After that, we will classify vector bundles on MQand show a few basic properties of the category of vector bundles onM(3). Then we go on and study the vector bundles O, Eα andEα,eα in detail. In the last section, we will prove the main theorem of this chapter.
3.1 Vector bundles over Spec Z //C
2In this section, we will classify integral representations of the cyclic group with two el-ements, C2. We remark that this category is both equivalent to the category of vector bundles over SpecZ//C2 (by Galois descent) and to the category of modules overZ[C2] that are free of finite rank as abelian groups.
Lemma 3.1.1. EveryQ[C2]-module is a direct sum of one-dimensional representations.
Proof. Denote by t the generator of C2. Then e1 = 1+2t and e2 = 1−2t are orthogonal idempotents inQ[C2]. Therefore,Q[C2]∼=Qe1×Qe2 andQ[C2]-mod'Q-mod×Q-mod.
Lemma 3.1.2. Every one-dimension C2-representation overZorQis either the trivial or the sign representation. In particular, every C2-representation over Qis of the form M⊗Qfor an integral C2-representation M.
Proof. The multiplicative groupsQ×andZ×have only one non-trivial element of order 2, the element−1.
Lemma 3.1.3. Every integral C2-representation M of dimension m sits in an extension 0→L→ M→ N→0,
where L is a one-dimensional representation and N is an(m−1)-dimensional one.
Proof. By Lemma 3.1.1, we have an injection L0 → L0⊗Q→ M⊗QofZ[C2]-modules for some 1-dimensional integralC2-representation L0. Multiply this map by a natural number to get an injection L0 → M with cokernel C. Divide out the torsion of C to get a Z[C2] -moduleN, which is free as an abelian group. Denote the kernel of M→ NbyL, which is obviously also free as an abelian group. Since L⊗Q ∼= L0⊗Q, we have that L is of rank 1.
Example 3.1.4(Examples ofC2-representations). We have the two 1-dimensional represen-tations Z and Z0 (the sign representation) and the representation Z[C2] of rank 2. We know that Ext1Z[C
2](Z,Z) ∼= Ext1Z[C
2](Z[C2],Z) = 0 and Ext1Z[C
2](Z0,Z) ∼= F2, where the non-trivial element corresponds to the extension
0→Z→Z[C2]→Z0 →0.
Proposition 3.1.5. Every integral representation of C2 is a direct sum of (several copies of) the trivial representation, the sign representation and the free representation.
Proof. For rankn= 1 this is true by Lemma 3.1.2. Assume by induction that the assertion of the proposition is true for representations of rank smaller thann, for somen∈N. Now let Mbe aC2-representation of rank n+1 and choose an extension
0→L→ M→ N→0
as above. We can assume that L is the trivial representation – else we could tensor the exact sequence with the sign representation. The extension above corresponds to a classx in Ext1Z[C
2](N,Z). By assumptionN∼=Za⊕(Z0)b⊕(Z[C2])c. We see that Ext1Z[C
2](N,Z)∼= F2b. By a change of basis, we can assume that x = (1, 0, . . . , 0) or x = 0. So either M∼=Za⊕Z0b−1⊕Z[C2]c+1or M∼=Za+1⊕Z0b⊕Z[C2]c.