LLR1 - Invent math 157 455518(2004 DOI...

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DOI: 10.1007/s00222-004-0342-y Invent. math. 157, 455–518 (2004) N´eron models, Lie algebras, and reduction of curves of genus one Qing Liu 1 , Dino Lorenzini 2 ,? , Michel Raynaud 3 1 CNRS, Laboratoire A2X, Universit´e de Bordeaux I, 351, cours de la Lib´eration, 33405 Talence, France (e-mail: [email protected] ) 2 Department of Mathematics, University of Georgia, Athens, GA 30602, USA (e-mail: [email protected] ) 3 Universit´e de Paris-Sud, Bât. 425, 91405 Orsay Cedex, France (e-mail: [email protected] ) Oblatum 28-VIII-2003 & 24-IX-2003 Published online: 9 June 2004 – Springer-Verlag 2004 Dedicated to John Tate Let K be a discrete valuation ±eld. Let O K denote the ring of integers of K ,andle t k be the residue ±eld of O K , of characteristic p 0. Let S := Spec O K .Le t X K be a smooth geometrically connected projective curve of genus 1 over K . Denote by E K the Jacobian of X K t X / S and E / S be the minimal regular models of X K and E K , respectively. In this article, we investigate the possible relationships between the special ±bers X k and E k . In doing so, we are led to study the geometry of the Picard functor Pic X / S when X / S is not necessarily cohomologically ²at. As an application of this study, we are able to prove in full generality a theorem of Gordon on the equivalence between the Artin-Tate and Birch-Swinnerton- Dyer conjectures. Recall that when k is algebraically closed, the special ±bers of elliptic curves are classi±ed according to their Kodaira type, which is denoted by a symbol T ∈{ I n , I n , n Z 0 , II , II , III , III , IV , IV } . Given a type T and a positive integer m , we denote by mT the new type obtained from T by multiplying all the multiplicities of T by m .When k is algebraically closed, the relationships between the type of a curve of genus 1 and the type of its Jacobian can be summarized as follows. Theorem 6.6. Assume that k is algebraically closed. Let X K / K be a smooth, geometrically connected projective curve of genus 1 and let E K / K be its ? D.L. was supported by NSF under Grants No. 0070522 and 0302043.
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456 Q. Liu et al. Jacobian. Let X / S and E / S be the minimal regular models of X K and E K , respectively. Let m denote the order of the element of H 1 ( K , E K ) cor- responding to the torsor X K . If T denotes the type of E k ,thenX k is of type mT. The most difFcult part of this theorem is the case of additive reduction. As a corollary to Theorem 6.6 and of results of B´egueri [5] and Bertapelle [6] on the structure of H 1 ( K , E K ) when K is complete, we prove in 6.7 the existence of torsors X K having reduction of type mT , for any additive type T and integer m = p n and, in case the type T is semi-stable, for any integer m > 0. To prove Theorem 6.6, we Frst show in 3.8 that there exists a canonical map of O K -modules H 1 ( X , O X ) H 1 ( E , O E ) which extends the natural isomorphism H 1 ( X K , O X K ) H 1 ( E K , O E K ) . The existence of this map is the main link between X and E , and is proved in the following general theorem on N´eron models of Jacobians.
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LLR1 - Invent math 157 455518(2004 DOI...

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