| Résume | 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 has been extended to triple products by the $p$-adic GGP formula of 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 $f\times g \times g$. 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 $GSp(4) \times GL(2) \times GL(2)$, where a new double wall-crossing phenomenon arises and is required to explain a $p$-adic Artin formalism for families of the form $F \times g \times g$. This suggests a higher $p$-adic GGP formula concerning second-order derivatives. |
