Diagonal restrictions of p -adic Eisenstein families
- PDF / 604,541 Bytes
- 46 Pages / 439.37 x 666.142 pts Page_size
- 21 Downloads / 184 Views
Mathematische Annalen
Diagonal restrictions of p-adic Eisenstein families Henri Darmon1
· Alice Pozzi2 · Jan Vonk3
Received: 30 September 2019 / Revised: 17 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions. Mathematics Subject Classification 11G18 · 14G35
Communicated by Wei Zhang.
B
Henri Darmon [email protected] Alice Pozzi [email protected] Jan Vonk [email protected]
1
McGill University, Montreal, Canada
2
Imperial College, London, UK
3
Leiden University, Leiden, Netherlands
123
H. Darmon et al.
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Diagonal restrictions of Hilbert Eisenstein series . . . . . . 1.1 The weight two specialisation G 1 (ψ). . . . . . . . . . 1.2 Ideals and RM points . . . . . . . . . . . . . . . . . . 1.3 An unfolding lemma for geodesics . . . . . . . . . . . 1.4 The Fourier expansion of G 1 (ψ) . . . . . . . . . . . . 2 The incoherent Eisenstein series and its diagonal restriction 2.1 The overconvergence of G 1 (ψ) . . . . . . . . . . . . . 2.2 The Bruhat–Tits tree and the Drinfeld upper half-plane 2.3 The winding cocycle . . . . . . . . . . . . . . . . . . 2.4 Hecke operators on rigid cocycles . . . . . . . . . . . 2.5 Lifting the winding cocycle . . . . . . . . . . . . . . . 2.6 RM values of the winding cocycle . . . . . . . . . . . 2.7 Diagonal restrictions: the incoherent case . . . . . . . 3 The twisted triple product p-adic L-function . . . . . . . . 3.1 The Schneider–Teitelbaum lift . . . . . . . . . . . . . 3.2 The winding element and the winding cocycle . . . . . 3.3 Spectral expansion of the winding element . . . . . . 3.4 Spectral decomposition: the coherent case . . . . . . . 3.5 Spectral decomposition: the incoherent case . . . . . . 3.6 E
Data Loading...
