Height pairings on Shimura curves and p-adic uniformization

Abstract

In a recent paper [16] of one of us it was shown that there is a close connection between the value of the height pairing of certain arithmetic 0-cycles on Shimura curves and the values at the center of their symmetry of the derivatives of certain metaplectic Eisenstein series of genus 2. On the one hand, the height pairing can be written as a sum of local height pairings. For example, if the 0-cycles have disjoint support on the generic fiber, then their height pairing is a sum of an archimedean contribution and a contribution from each of the (finitely many) finite primes p for which the cycles meet in the fiber at p. On the other hand, it turns out that the non-singular part of the Fourier expansion of the central derivative of the metaplectic Eisenstein series also has a decomposition into a sum of contributions indexed by the places of Q. Then, one would like to compare the height pairing and the Fourier coefficients by proving an identity of local contributions place by place. In loc. cit. the identity for the archimedean place was proved, and it was shown that the identity at a non-archimedean place of good reduction is a consequence of results of Gross and Keating, [10], (for the algebraicgeometric side) and of Kitaoka, [12], (for the analytic side). It then remains to consider the finite primes p where the Shimura curve has bad reduction. These are of three sorts: (i) the primes p at which the quaternion algebra defining the Shimura curve is split, but which divide the level, (ii) the primes p at which the quaternion algebra remains division, but no level structure is imposed, and (iii) the primes p at which the quaternion algebra remains division and where a higher level structure is imposed at p. So far, little is known concerning cases (i) and (iii).