Stark-Heegner cycles attached to Bianchi modular forms

Abstract

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch--Kato conjecture predicts that this should force the existence of non-trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Barrera Salazar and the second author to construct certain P-adic Abel--Jacobi maps, which we use to propose a construction of such classes via "Stark--Heegner cycles". This builds on ideas of Darmon and in particular generalises an approach of Rotger and Seveso in the setting of classical modular forms.

arXiv comment27 pages; comments welcome! v3: Minor corrections. This version accepted for publication in J. Lond. Math. Soc. [Changes for v2: Minor corrections in the p inert case, and additional comments on evidence for our conjectures]

References in corpus (7)

• l-Adic representations associated to modular forms over imaginary quadratic fields
• Darmon points on elliptic curves over number fields of arbitrary signature
• $P$-adic $L$-functions of Bianchi modular forms
• Overconvergent cohomology and quaternionic Darmon points
• $\mathcal{L}$-invariants and exceptional zeros of Bianchi modular forms
• P-adic L-functions for GL(2)
• Functoriality of automorphic L-invariants and applications