## STNB2019 (33è any)

### Non-cristalline generalized Kato classes and the values of the triple-product p-adic L-function at classical points

Francesca Gatti

#### Resum

Let f, g, h be three Hida families. The triple product p-adic L-function $L_p^g (f , g, h)$ interpolates the central L-values $L(f_k ⊗g_l ⊗h_m , (k+l+m−2)/2)$ for classical weights (k, l, m) such that l ≥ k+m. The point (2, 1, 1) lies outside the region of classical interpolation and $L(f_2 ⊗g_1 ⊗h_1 , s) = L(E⊗ρ, s)$, where E is an elliptic curve over $\mathbb{Q}$ and ρ an Artin representation. Assume that it does not vanish at s = 1. In this setting, we describe the value $L_p^g (f , g, h)(2, 1, 1)$ in terms of a non-cristalline cohomology class which lies in the p-relaxed Selmer group attached to (E, ρ). It is a joint work with X. Guitart, M. Masdeu, V. Rotger.

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