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THE UNIVERSAL P-ADIC GROSS–ZAGIER FORMULA by Daniel

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THE UNIVERSAL P-ADIC GROSS–ZAGIER FORMULA by Daniel THE UNIVERSAL p-ADIC GROSS–ZAGIER FORMULA by Daniel Disegni Abstract.— Let G be the group (GL2 GU(1))=GL1 over a totally real field F , and let be a Hida family for G. Revisiting a construction of Howard× and Fouquet, we construct an explicit section Xof a sheaf of Selmer groups over . We show, answering a question of Howard, that is a universal HeegnerP class, in the sense that it interpolatesX geometrically defined Heegner classes at all the relevantP classical points of . We also propose a ‘Bertolini–Darmon’ conjecture for the leading term of at classical points. X We then prove that the p-adic height of is givenP by the cyclotomic derivative of a p-adic L-function. This formula over (which is an identity of functionalsP on some universal ordinary automorphic representations) specialises at classicalX points to all the Gross–Zagier formulas for G that may be expected from representation- theoretic considerations. Combined with a result of Fouquet, the formula implies the p-adic analogue of the Beilinson–Bloch–Kato conjecture in analytic rank one, for the selfdual motives attached to Hilbert modular forms and their twists by CM Hecke characters. It also implies one half of the first example of a non-abelian Iwasawa main conjecture for derivatives, in 2[F : Q] variables. Other applications include two different generic non-vanishing results for Heegner classes and p-adic heights. Contents 1. Introduction and statements of the main results................................... 2 1.1. The p-adic Be˘ılinson–Bloch–Kato conjecture in analytic rank 1............... 3 1.2. The p-adic Gross–Zagier formula for arbitrary weight........................ 6 1.3. The universal Heegner classes............................................... 9 1.4. The universal formula....................................................... 12 1.5. Applications................................................................ 14 1.6. Outline of the proofs........................................................ 15 1.7. Related contemporary work................................................. 15 1.8. Acknowledgements......................................................... 16 1.9. Notation................................................................... 16 2. Automorphic and Galois representations........................................ 16 2.1. Groups..................................................................... 16 2.2. Algebraic representations................................................... 18 2.3. Shimura varieties and local systems.......................................... 19 2.4. Ordinary automorphic representations...................................... 20 2.5. Galois representations....................................................... 22 3. Sheaves on Hida families......................................................... 23 3.1. Hida theory................................................................ 23 3.2. Galois representations in families............................................ 28 3.3. Universal ordinary representation and local-global compatibility............. 31 4. Pairings......................................................................... 36 4.1. Global dualities............................................................. 36 4.2. Local toric pairings......................................................... 42 4.3. Ordinary toric pairings..................................................... 43 4.4. Interpolation of the toric pairings........................................... 44 2 DANIEL DISEGNI 5. Selmer sheaves and p-adic heights................................................ 46 5.1. Continuous cohomology.................................................... 46 5.2. Specialisations.............................................................. 48 5.3. Selmer complexes and height pairings........................................ 50 6. Universal Heegner class.......................................................... 52 6.1. Tate cycles and Abel–Jacobi maps............................................ 52 6.2. Heegner cycles.............................................................. 52 6.3. Universal Heegner class..................................................... 54 6.4. Local properties of the universal Heegner class............................... 55 7. The main theorems, and a conjecture............................................. 58 7.1. Proofs of the main theorems................................................. 58 7.2. A universal Waldspurger formula............................................ 62 7.3. Bertolini–Darmon conjectures and exceptional zeros......................... 63 Appendix A. p-adic semilocal constructions........................................ 67 A.1. Preliminaries............................................................... 67 A.2. Toric, ordinary, and anti-ordinary parts..................................... 69 A.3. Pairings at p ............................................................... 72 A.4. Pairings at infinity.......................................................... 76 References......................................................................... 79 1. Introduction and statements of the main results A beautiful construction of Heegner and Birch, based on the modularity of elliptic curves and the theory of complex multiplication, attaches to an elliptic curve A=Q and an imaginary quadratic field E a point P A(E). The work of Gross–Zagier [GZ86] related the height of P to the derivative of the L- 2 function L0(AE ,1), with striking applications to the Birch and Swinnerton-Dyer conjecture. An analogous result in p-adic coefficients was proved by Perrin-Riou [PR87a] soon thereafter, if A has good ordinary reduction at the prime p. The decade following those works saw a pair of similar results, by Nekováˇr [Nek95] and Zhang [Zha97], relating Heegner cycles on Kuga–Sato varieties to (p-adic) L-functions of higher-weight modular forms. We may single out two major innovations in the approach to Heegner points and Gross–Zagier formulas since then,(1) both answering the question of what ‘other’ Heegner points there are and how they fit together. The first one starts from the observation by Mazur [Maz84] and Perrin-Riou [PR87b] that Heegner points should vary p-adically in anticyclotomic families, in the same way that the L-function of the elliptic curve AE does; this observation inspired Howard [How05] to prove a generalisation to such families of Perrin-Riou’s formula. Howard later significantly expanded the scope of Mazur and Perrin-Riou’s idea by proving that the Kummer classes of Heegner points also vary in Hida families of modular forms [How07]; the question of the relation of the resulting ‘big’ classes to Heegner cycles was left open. The second innovation was the observation by Gross [Gro04] that Heegner points can be viewed as elements of spaces of H0-invariant linear functionals on an automorphic representation of (G H)0 (these reductive groups will be defined below),(2) so that the tools of representation theory may× be brought in to conceive and prove more general formulas: a programme whose main achievement, in complex coefficients, is the work of Yuan–Zhang–Zhang [YZZ12] on Heegner points on Shimura curves. In this work, we combine those two approaches. We construct Heegner classes for the Galois represen- tation over a Hida family for (G H)0, show that they specialise to (cohomological) Heegner cycles at all × (1)Two other recent ideas that our work does not touch upon are nevertheless too important to be ignored: the conjecture of Darmon and Guitart–Madseu–¸Sengünthat there should exist Heegner points attached to any quadratic extension of number fields (see [Dar01], [GM¸S15]), and the formulas for the p-adic logarithms of Heegner points of [BDP13, LZZ18]. (2)N.B.: the notation G used in the informal abstract differs from the notation of the paper. THE UNIVERSAL p-ADIC GROSS–ZAGIER FORMULA 3 classical points, and prove a formula for their p-adic heights that is universal in the sense that it specialises to all the p-adic formulas suggested by the framework of Gross. (The analogous complex Gross–Zagier formulas are not currently known(3) for motives of higher weight.) We obtain various applications to the arithmetic of motives attached to Hilbert modular forms. In the rest of this first section we state our main theorems, and complete the discussion of their history. We begin in § 1.1 by presenting the results concerning the p-adic Be˘ılinson–Bloch–Kato conjecture (TheoremA); they are applications of the general p-adic Gross–Zagier formula for a fixed representation, stated as TheoremB in § 1.2. In § 1.3 we outline the construction and properties of the universal family of Heegner classes (Theorem C), referring to the “Bertolini–Darmon” conjecture of § 7.3 for a further study of its classical specialisa- tions. In § 1.4 we state the universal formula of the title (TheoremD); a complementary ‘Waldspurger’ analogue will be proved in § 7.2 (TheoremH). Finally, in § 1.5 we discuss some further applications: the first non-abelian example of an Iwasawa main conjecture for derivatives of p-adic L-functions (TheoremE); and two results on the generic non- vanishing of p-adic heights and Heegner cycles: one for CM motives (TheoremF), the other for Hida families containing a rank-0 elliptic curve with split multiplicative reduction (TheoremG). A further application, to a criterion for certain Bloch–Kato Selmer groups to be of rank zero, will appear separately. 1.1. The p-adic Be˘ılinson–Bloch–Kato
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