THE FORMAL COMPLETION OF THE NÉRON MODEL OF J o ( p )

THE FORMAL COMPLETION OF THE NÉRON MODEL OF Jo (p)

Theorem .Let U, E M, (Z) and TI E M_,(Z), for all primes l :~p, be the matrices of the Atkin-Lehner operator and the Hecke operators, with respect to any Z-basis of S2(Fo(P), Z) .Since these matrices commute, the formal Dirichlet series : n`= (I9 -Up .p-3)-1 .fl(Iy _TI .p-S + Iy .P1-2s)-1 a is well-defined and An E M9 (7L) for all n.Let L(X,Y) be the g-dimensional formal group law with logarithm: 00 where Xn is the notation for (Xi , . . . ,X9)t .Then, L(X,Y) is defined over and it is isomorphic to the formal completion of ,7 along the zero section.
Honda [4] proved an analogous result for Shimura curves, but a finite (fairly big) set of primes had to be left aside.In fact, our proof follows the same pattern, but we have at our disposal deep results of Deligne-Rapoport [1], Deligne [5, thm .A.1] (which was implicitely used in [2]) and Mazur [5, II, sections 3 and 6], which allow us to deal with the bad primes .
After [2], in order to prove the theorem it is sufficient to show that Lie (,7) and S2 (Fo (p), 7L) are isomorphic as T-modules .To this aim is devoted the rest of the paper .The proof consists on adding some details (checking of some compatibilities, essentially) to certain results of Mazur .
For any integer N >_ 5, let Mo(N) be the curve over Z representing the fine moduli stack classifying generalized elliptic curves over 7L[1/N] with a cyclic subgroup of order N. Let Xo(N) i Mo(N) be its minimal regular resolution.These two curves become isomorphic over Z [1/N] .
The Atkin involution w=wnr extends to an involution of Mo(N) [1, IV, Prop.Here (HI, H.) denotes a cyclic subgroup of E of order pl (canonically) decomposed as a product of its p-primary and l-primary parts .By minimality c raises to a finite morphism between the regular resolutions fitting into a commutative diagram: Let us denote X = Xo(p), X' = Xo(pl), M = MO(p), M' = Mo(pl) .The morphism c : X' > X induces covariant and contravariant homomorphisms : Pic °x,/a c PicX/a .
At the level of invertible sheafs, c* is the usual homomorphism and c* is the norm-homomorphism defined by Grothendieck [3, 6.5].Via the canonical identification of H 1 (X, O) with the tangent space of Pic o /a at the zero-section, c* and c* induce homomorphisms : H' (X', O) ; Hl (X, O) .
In fact, the identification of HI(X, 0) with the tangent space of Pic o can be realized through the exact sequence: By Grothendieck duality we obtain homomorphisms : H°(X',S2x,) H°(X,Qx), where Qx is the dualizing sheaf, that is, the sheaf of regular differentials, which is defined as the only non-vanishing homology group (in degree -1) of the complex R7r'0sp ec2, where n is the structural morphism of X .We need to check the compatibility of these homomorphisms c* , c* with the analogous homomorphisms defined by Mazur at the level of the curves Mo (N) [5, page 88], which we denote by (c*)M, (c*)m .More precisely, we need the following diagrams to commute : where i* is defined from i* by duality.Now, diagram (3) commutes since it is obtained from (2) by taking everywhere the natural homomorphisms induced by i and c.Since the 7L-modules involved are free [5,11,Lemma 3.3 and (3.2)] it is sufiicient to check the commutativity of diagram (4) after tensoring with Q.Then, the commutativity amounts to the fact that the natural homomorphism : H°(X Q, 9 1 )c H°(XQ ) 91) is dual to the trace-homomorphism, Trx,Ix : Hl (X, ()) , H l (XQ, C)), under Serre duality, and this is a consequence of the classical trace-formula [7, page 32] .
We are ready to analize the action of the Hecke algebra .The Hecke algebra T is the subalgebra of EndQ(Jo(p)) generated by all the operators TI and the Atkin involution w.The Hecke operator Ti is, by definition, the endomorphism of Jo(p) induced by correspondence on Xo(p),Q determined by the morphsm :
p , H'(X, O) ep.Hl (X ® 7L [e], O*) -H 1 (X, O*), where 7L[e] is the ring of dual numbers and exp(f) = 1 + fe.The above description of the action of c* and c* can be easily deduced from this sequence, working with Cech cocycles and having in mind that 1 + Trx-1x (f) e is the norm of 1 + fe.
Jo (P), induced by cQ and (cwt)Q on PicX0(N)IQ = JO(N), for N = p, pl .By the universal property of the Néron model, Ti operates on ,7 and on its connected component as) o is the connected component of the Néron model of Jo(pl).By a theorem of Raynaud [6, 8.1.4],the connected component of the Néron model of Jo (N) represents the functor PicXo(N)Ia .Hence the homomorphisms : P¡ex,/a (cc )R Picx/z' induced by the finite morphisms X' ce w `X, coincide with (cwj)*r, (c*)n, sincethey induce the same homomorphism on the generic fiber.Thus, T, acts on Hl (X, C)) and (by duality) on H'(X, 9) .We have a commutat¡ve diagram : )) the same structure of T-module on Hl (M, C)) as the one taken by definition by Mazur.That is, we have isomorphisms as T-modules : Hl (X, C)) -Hl (M, 0), Ho (X, 9) -Ho(M, S2) .Therefore we have T-isomorphisms : Lie(JO ) = To(Jo) A = H1 (X, C)) ^= H0(X, 9) = H0(M, S2), and this last group is isomorphic to S2(Fo(p),7L) as a T-module, as shown by Mazur [5, 11, (4.6) and (6.2)] .