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The P-Adic Gross-Zagier Formula on Shimura Curves

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The P-Adic Gross-Zagier Formula on Shimura Curves THE p-ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES by Daniel Disegni Abstract. — We prove a general formula for the p-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above p. The formula is in terms of the cyclotomic derivative of a Rankin–Selberg p-adic L-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author, to the context of the work of Yuan– Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is +1 rather than −1, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a p-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately. Contents 1. Introduction ..................................... ............................... 2 1.1. Heegner points and multiplicity one . ...................... 3 1.2. The p-adic L-function .......................................... .......... 5 1.3. p-adic Gross–Zagier formula ........................... ................... 8 1.4. Anticyclotomic theory ........................... ......................... 9 1.5. Applications ................................... ........................... 12 1.6. History and related work .......................... ...................... 14 1.7. Outline of proofs and organisation of the paper ....... ................... 14 1.8. Acknowledgements ............................... ....................... 16 1.9. Notation ....................................... ......................... 16 arXiv:1510.02114v3 [math.NT] 30 Jul 2019 2. p-adic modular forms .................................. .......................... 18 2.1. Modular forms and their q-expansions .................................... 18 2.2. Hecke algebra and operators Uv .......................................... 22 2.3. Universal Kirillov and Whittaker models . .................... 23 2.4. p-crtical forms and the p-adic Petersson product ........................ 24 3. The p-adic L-function .......................................... ................ 26 3.1. Weil representation ............................. ......................... 26 3.2. Eisenstein series ............................... ........................... 28 3.3. Eisenstein family ............................... ......................... 30 3.4. Analytic kernel ................................. ......................... 31 3.5. Waldspurger’s Rankin–Selberg integral .. .......... ...................... 33 3.6. Interpolation of local zeta integrals .............. ........................ 34 3.7. Definition and interpolation property .............. ...................... 36 4. p-adic heights ....................................... ............................. 37 2 DANIEL DISEGNI 4.1. Local and global height pairings ................... ....................... 37 4.2. Heights and intersections on curves ................ ...................... 40 4.3. Integrality crieria ............................. ........................... 42 5. Generating series and strategy of proof ............... ........................... 43 5.1. Shimizu’s theta lifting . .......................... 43 5.2. Hecke correspondences and generating series .. ...... .................... 45 5.3. Geometric kernel function ........................ ........................ 47 5.4. Arithmetic theta lifting and kernel identity . ...................... 49 6. Local assumptions ................................. ............................. 50 6.1. Assumptions away from p ................................................ 50 6.2. Assumptions at p ................................................... ..... 51 7. Derivative of the analytic kernel .................... ............................ 52 7.1. Whittaker functions for the Eisenstein series . ..................... 52 7.2. Decomposition of the derivative ................... ....................... 53 7.3. Main result on the derivative ...................... ...................... 54 8. Decomposition of the geometric kernel and comparison ... ....................... 55 8.1. Vanishing of the contribution of the Hodge classes . .................. 55 8.2. Decompositon ................................... ......................... 56 8.3. Comparison of kernels ............................ ........................ 56 9. Local heights at p ................................................... ........... 58 9.1. Local Hecke and Galois actions on CM points ........... ................. 59 9.2. Annihilation of local heights ..................... ......................... 61 10. Formulas in anticyclotomic families . ............................ 64 10.1. A local construction ............................ ........................ 64 10.2. Gross–Zagier and Waldspurger formulas ............ .................... 67 10.3. Birch and Swinnerton-Dyer formula ................ .................... 69 Appendix A. Local integrals ........................... ............................. 70 A.1. Basic integral .................................. .......................... 70 A.2. Interpolation factor ............................ .......................... 71 A.3. Toric period .................................... ........................ 72 References ......................................... ................................. 73 1. Introduction The main results of this paper(1) are the general formula for the p-adic heights of Heegner points of Theorem B below, and its version in anticyclotomic families (contained in Theorem C). They are preceded by a flexible construction of the relevant p-adic L-function (Theorem A), and complemented by a version of the Waldspurger formula in anticyclotomic families (presented in Theorem C as well). In Theorem D, we give an application to a version of the p-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. In Theorem E, we state a result on the generic non-vanishing of p-adic heights on CM abelian varieties, as a special case of a theorem to appear in joint work with A. Burungale. Our theorems are key ingredients of a new Gross–Zagier formula for exceptional zeros [Dis], and of a universal p-adic Gross–Zagier formula specialising to analogues of Theorem B in all weights. These will be given in separate works. Here we would just like to mention that all of them, as well as Theorem E, make essential use of the new generality of the present work. (1)2010 Mathematics Subject Classification: 11G40 (primary), 11F33, 11F41 (secondary). Keywords: Gross–Zagier formula, p-adic L-function, Heegner points, p-adic heights, Birch and Swinnerton-Dyer conjec- ture. Note added after publication: the main theorem is off by a factor of 2. This and other errata are noted in the author’s The p-adic Gross–Zagier formula on Shimura curves, II: nonsplit primes, Appendix B. THE p-ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES 3 The rest of this introductory section contains the statements of our results, followed by an outline of their proofs. To avoid interrupting the flow of exposition, the discussion of previous and related works (notably by Perrin-Riou and Howard) has mostly been concentrated in §1.6. 1.1. Heegner points and multiplicity one. — Let A be a simple abelian variety of GL2-type over a totally real field F ; recall that this means that M := End 0(A) is a field of dimension equal to the dimension of A. One knows how to systematically construct points on A when A admits parametrisations by Shimura curves in the following sense. Let B be a quaternion algebra over the adèle ring A = AF of F , and assume that B is incoherent, i.e. that its ramification set ΣB has odd cardinality. We further assume that ΣB contains all the archimedean places of F . Under these conditions there is a tower of Shimura curves {XU } over F indexed by the open compact subgroups ∞× U ⊂ B ; let X = X(B) := lim X . For each U, there is a canonical Hodge class ξ ∈ Pic (X )Q ←−U U U U having degree 1 in each connected component, inducing a compatible family ιξ = (ιξ,U )U of quasi- embeddings(2) ι : X ֒→ J := Alb X . We write J := lim J . The M-vector space ξ,U U U U ←− U π = π = π (B) := lim Hom 0(J , A) A A −→U U is either zero or a smooth irreducible admissible representation of B∞×. It comes with a natural stable lattice πZ ⊂ π, and its central character × × × ωA : F \A → M corresponds, up to twist by the cyclotomic character, to the determinant of the Tate module under the class field theory isomorphism. When πA is nonzero, A is said to be parametrised by X(B). Under the conditions we are going to impose on A, the existence of such a parametrisation, for a suitable choice of B (see below), is equivalent to the modularity conjecture. Recall that the latter asserts the existence of a unique M-rational (Definition 1.2.1 below) automorphic representation σA of weight 2 such that there is an equality of L-functions L(A, s +1/2) = L(s, σA). The conjecture is known to be true for “almost all” elliptic curves A (see [LH]), and when AF has complex multiplication. Heegner points. — Let A be parametrised by X(B) and let E be a CM extension of F admitting an ∞ ∞ A -embedding EA∞ ֒→ B , which we fix; we denote by η the associated quadratic
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