GENERALISED HEEGNER CYCLES AND p-ADIC RANKIN L-SERIES
نویسندگان
چکیده
Introduction 2 1. Preliminaries 6 1.1. Algebraic modular forms 6 1.2. Modular forms over C 9 1.3. p-adic modular forms 11 1.4. Elliptic curves with complex multiplication 12 1.5. Values of modular forms at CM points 14 2. Generalised Heegner cycles 15 2.1. Kuga-Sato varieties 15 2.2. The variety Xr and its cohomology 18 2.3. Definition of the cycles 19 2.4. Relation with Heegner cycles and L-series 19 3. p-adic Abel-Jacobi maps 20 3.1. The étale Abel-Jacobi map 20 3.2. The comparison isomorphism 21 3.3. Extensions of filtered Frobenius modules 22 3.4. The p-adic Abel-Jacobi map 23 3.5. de Rham cohomology over p-adic fields 24 3.6. The Coleman primitive 27 3.7. p-adic integration and the p-adic Abel-Jacobi map 28 3.8. Calculation of the Coleman primitive 31 4. Period integrals and central values of Rankin-Selberg L-functions 33 4.1. Rankin L-series and their special values 33 4.2. Differential operators 38 4.3. Period integrals and values at CM points 39 4.4. Explicit theta lifts 42 4.5. Seesaw duality and the Siegel-Weil formula 49 4.6. Local zeta integrals 51 4.7. The explicit form of Waldspurger’s formula 56 5. Anticyclotomic p-adic L-functions 56 5.1. Periods and algebraicity 56 5.2. p-adic interpolation 60 5.3. The main theorem 62 Appendix A. Kuga-Sato schemes by Brian Conrad 64 References 67
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