By Marina Cohen

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Most of the rigid analytic spaces we will deal with in this article will come along with a natural (rigid analytic) morphism to WN (for some appropriate tame level N). We shall refer to this morphism by the phrase projection to weight space. All the rigid analytic morphisms we encounter will preserve projections to weight space. 5. The eigencurve as the Fredholm closure of the classical modular locus. (Statement of the main theorems) For a prime number p and N (a tame level) an integer prime to p , let S denote the finite set of places of Q consisting of the infinite place, p, and the prime divisors of N.

It follows that hn(X) converges on B[0, s]. Let C = Il/D(X)I A[,,,]- It follows that for large n, I hn (X) I is at most Csn on the circle A[s, s] and hence (by the maximum modulus principle) on the disk B[0, s]. It follows, since IAnIt < on, that the sum S converges to an element of A° (B [t]) [XI t as claimed. 1 of [C-CCS]: Suppose b,,. , bd is a basis for A°(Zt) over Zp[X]t. We may write, G = a,(X )b1 + a2(X)b2 + ... + ad(X)bd where ai(X) E A°(B[t])[X]t. Then as DG E At(ZB, /13n)°, it follows, since b1i...

TM, so vt(a) = m° logp+ mi (- log ti). Thus the lemma follows if we take B(t) = max{log p, - log t1i ... ,-logtn}. As a corollary we get that a series over An which is convergent on 13n x B[0,1] is overconvergent over the open unit n-polydisk l3n if and only if it is overconvergent over any single affinoid subpolydisk around the origin. 1 of [C-CCS] anticipated above. Let l3n be the n-dimensional unit polydisk. Then An may be regarded as the ring of rigid analytic functions on Bn defined over Qp and bounded by 1.