Algebraic geometry : introduction to schemes - download pdf or read online

By I.G. Macdonald

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Because v is discrete, so is is a complete discretely valued field of unequal By construction of the Raynaud extension, the Barsotti-Tate groups associated to A / $ resp. to the 1-motive [M/$ ~P~G/$] coincide. t. { t Gauss ' Since A / $ has good reduction modulo the valuation ideal of $ (indeed its reduction is the generic fiber of the reduction of A modulo v , which is proper when v ~ v ) , this pairing has to be trivial: {qij{ Gauss = 1. g) An example: let us consider the Legendre elliptic pencil with parameter x = A , given by the affine equation v2=u(u-1)(u-x).

Xn]] for some v , and if y has non-zero radius of convergence at every place of E . (Such series form a regular noetherian ring with residue field E , and the filtered 42 union of these rings over a~ finite extensions of E , is strictly henselian). One can show that the (x 1 ... xn)mqij's are globally bounded series (for suitable m) . The problem is to show that the v-adic radius of converge is not 0 for any v ] v . Using the compactification of Siegel modular stacks over ~ , one can find a semi-abelian extension of ~1 over an ~ - m o d e l of some covering of ~1 ' and afterwards, one has to use the 2-step construction of [CF] III 10 to keep track of the possible variation of the torus rank of the reduction, after replacing the divisor x 1 ...

T, bases of HD1R(AE , _ ) HI(Aq:,Q)) . _ [This deep problem is solved only for Abelian varieties isogeneous to some power of an elliptic curve with complex multiplication (Chudnovsky). 33 The conjecture can also be formulated as follows: every polynomial relation between periods, with coefficients in E , comes from Hodge classes. ] 2. t, v ; ]~ denotes the algebraic closure of E in l~. Let us choose an embedding 7 : 1~ ¢----,~ and denote by t its restriction to K . M. HDR(AE) @E BDR ,%/ ~ , H1R(AK) OK BDR ' H~t(AE) @gp BDR ~ ' Hlet(AET)®gp BDR ~'' HI(AE @7 ¢,9) ®9 BDR ~ , HI(AE ® t E,9) ®9 BDR Blasius---Ogus [B1] and independently Wintenberger have recently proved the following striking result: Theorem 1.

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Algebraic geometry : introduction to schemes by I.G. Macdonald

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