Wall-crossing, GGP, and Artin Formalism

Abstract

The celebrated BDP formula evaluates Rankin–Selberg p-adic L-functions at points outside their interpolation range in terms of Generalised Heegner cycles (a phenomenon referred to as wall-crossing). This principle has been extended to triple products by Bertolini–Seveso–Venerucci and Darmon–Rotger, who relate values of Hsieh’s unbalanced p-adic L-functions on the balanced range to diagonal cycles. I will report on a result where wall-crossing is used to factor a triple product p-adic L-function with an empty interpolation range, to yield a p-adic Artin formalism for families of the form f × g × g. The key input is the arithmetic Gan–Gross–Prasad (Gross–Kudla) conjecture, linking central derivatives of (complex) triple product L-functions to Bloch–Beilinson heights of diagonal cycles and their comparison with their GL(2) counterpart (Gross–Zagier formulae). I will also discuss an extension to families on GSp(4) × GL(2) × GL(2), where a new double wall-crossing phenomenon arises and is required to explain a p-adic Artin formalism for families of the form F x g x g. This suggests a higher BDP/arithmetic GGP formula concerning second-order derivatives.