Abstract In the present work, we compute quasi-derivations of the Witt algebra and some algebras well related to the Witt algebra. Namely, we prove that each quasi-derivation of the Witt algebra is a sum of a derivation and a $$\frac{1}{2}$$ 1 2 -derivation; a similar result is obtained for the Virasoro algebra. A different situation appears for Lie algebras $$\mathscr {W}(a,b):$$ W ( a , b ) : In the case of $$b=-1,$$ b = - 1 , they do not have interesting examples of quasi-derivations, but the case of $$b\ne -1$$ b ≠ - 1 provides some new non-trivial examples of quasi-derivations. We also completely describe all quasi-derivations of $$\mathscr {W}(a,b).$$ W ( a , b ) . As a corollary, we describe the derivations and quasi-derivations of the Novikov–Witt and admissible Novikov–Witt algebras previously constructed by Bai and his co-authors; and $$\delta $$ δ -derivations and transposed $$\delta $$ δ -Poisson structures on cited Lie algebras. In particular, we proved that each $$\mathscr {W}(a,b)$$ W ( a , b ) admits a non-trivial transposed $$\frac{1}{1-b}$$ 1 1 - b -Poisson structure.

Abstract For every nuclear $${\mathbb {Z}}_\ell $$ Z ℓ -algebra $$\Lambda $$ Λ and every small v-stack X on perfectoid spaces, we construct an $$\infty $$ ∞ -category $$\mathcal {D}_{\textrm{nuc}}(X,\Lambda )$$ D nuc ( X , Λ ) of nuclear (i.e., “ind-Banach”) $$\Lambda $$ Λ -modules on X . We then construct a full 6-functor formalism for these sheaves, generalizing the étale 6-functor formalism for $$\Lambda = \mathbb {F}_\ell $$ Λ = F ℓ . Prominent choices for $$\Lambda $$ Λ are $${\mathbb {Z}}_\ell $$ Z ℓ , $$\mathbb {Q}_\ell $$ Q ℓ and $$\overline{\mathbb {Q}_\ell }$$ Q ℓ ¯ . We also provide and study an abstract notion of ULA sheaves in this setting, whose definition and basic properties can be carried over to any 6-functor formalism. Applied to classifying stacks, we obtain a robust theory of nuclear representations, i.e., continuous representations on filtered colimits of Banach spaces.

Abstract We construct adjoint p -adic L -functions generating the congruence ideal attached to Hida families. These functions interpolate the Petersson norm of any classical ordinary newform, normalized by a product of Shimura’s canonical periods. We show that, after adjusting by suitable Euler factors, they are interpolated by a regular element of Hida’s universal ordinary Hecke algebra. We also establish a link between these p -adic L -functions and the characteristic series of primitive adjoint Selmer groups.