Overconvergent cohomology and quaternionic Darmon points (Q2922849)

There are open problems in the theory of Heegner-Stark (Darmon) and Darmon-like points in elliptic curves \(E\) over \(\mathbb Q\) of conductor \(N\) with a prime \(p\) such that \(N = pDM,\) where \(D\) is the product of even (possible zero) distinct primes and \((D,M) = 1, \) namely:NEWLINENEWLINESuppose that \(K\) is a real quadratic field in which all prime dividing \(M\) are split, and all primes dividing \(pD\) are inert and let \({\mathcal H}_p = K_p \setminus {\mathbb Q}_p \) be the \( K_p\)-points on the \(p\)-adic upper half plane.NEWLINENEWLINEThen in the case \(D=1\) there is the conjecture by \textit{H. Darmon} [Ann. Math. (2) 154, No. 3, 589--639 (2001; Zbl 1035.11027)] stating that local points \(P_{\tau} \in E(K_p)\) associated to elements \(\tau \in K \cap {\mathcal H}_p \) defined as certain Coleman integrals of a modular form attached to \(E\) should be rational over specific ring class field of \(K,\) and behave in many aspects as the classical Heegner points arising from imaginary quadratic fields.NEWLINENEWLINEIn the case \(D > 1\) there is a construction of Darmon-like points on \(E(K_p),\) by means of certain \(p\)-adic integrals related to modular forms on quaternion division algebras of discriminant \(D.\) Greenberg conjectured that these points behave in many aspects as Heegner points and, in particular, that they are rational over ring class of \(K.\)NEWLINENEWLINEThe authors of the paper under review develop (co)homological techniques for an effective construction of the quaternionic Darmon points on \(E(K_p).\)NEWLINENEWLINEThey use the reinterpretation of Darmon's theory of modular symbols, the mixed period integrals by \textit{M. Greenberg} [Duke Math. J. 147, No. 3, 541--575 (2009; Zbl 1183.14030)] and relate \(p\)-adic integration to certain overconvergent cohomology classes in order to derive an efficient algorithm for the computation of quaternionic Darmon points.NEWLINENEWLINEIn the Introduction the authors describe the contents and fix the notation.NEWLINENEWLINEThe second section is devoted to preliminaries on Hecke operators, the Bruhat-Tits tree, and measures.NEWLINENEWLINEIn Section 3, Greenberg's construction of quaternionic \(p\)-adic Darmon points is dealt with.NEWLINENEWLINEThe main results of the paper are presented in Sections 4--6.NEWLINENEWLINEIn Section 4, the explicit algorithms that allow an effective calculation of the quaternionic \(p\)-adic Darmon points are presented (Theorem 4.1, Theorem 4.2). The theorems and their proofs describe the algorithms and contain a correctness proof of the algorithms. Section 5 treats the integration pairing via overconvergent cohomology. The authors reduce the problem of computing the considered integrals to that of computing moments and provide an algorithm for computing the moments by means of the overconvergent cohomology lifting techniques developed by \textit{D. Pollack} and \textit{R. Pollack} [Can. J. Math. 61, No. 3, 674--690 (2009; Zbl 1228.11074)]. Finally, the last section treats results of extensive calculations and numerical evidence obtained by the authors in support of the conjectured rationality of quaternionic Darmon points.