The paper proves several results about automorphic \(\mathcal L\)-invariants for reductive groups previously constructed by the first author in [J. Reine Angew. Math. 779, 57–103 (2021; Zbl 1482.11076)], generalizing a construction by M. Spieß in the case of Hilbert modular forms in [Invent. Math. 196, No. 1, 69–138 (2014; Zbl 1392.11027)], which in turn was inspired by H. Darmon’s construction in [Ann. Math. (2) 154, No. 3, 589–639 (2001; Zbl 1035.11027)].
The main result is a generalization of the classical formula \(\mathcal L^\pm(f)=-2\frac{da_p}{dk}\Big|_{k=2}\) for the \(\mathcal L\)-invariant of a classical weight \(2\) newform due to M. Bertolini et al. [Astérisque 331, 29–64 (2010; Zbl 1251.11033)].
In order to state the result, we need to recall the notion of \(\mathcal L\)-invariant under consideration.
Fix a connected, adjoint, semi-simple linear algebraic group \(G\) over a number field \(F\) and a cuspidal automorphic representation \(\pi\) of \(G\) with the following three properties: (i) \(\pi\) is cohomological with respect to trivial coefficients; (ii) \(\pi\) is tempered at all archimedean places; (iii) for a fixed finite place \(\mathfrak{p}\) of \(F\), \(G\) is split over \(F_{\mathfrak{p}}\) and the \(G(F_{\mathfrak{p}})\)-representation \(\pi_\mathfrak{p}\) is the smooth Steinberg representation \(\mathrm{St}_{G(F_{\mathfrak{p}})}^\infty(\mathbb C)\).
Any such \(\pi\) admits a model over a number field \(\mathbb Q(\pi)\) ( see [Zbl 1403.11042]). Fix a finite extension \(E/\mathbb Q_p\) together with an embedding \(\mathbb Q(\pi)\to E\), with respect to which we may consider models of \(\pi\) and \(\pi_{\mathfrak{p}}\) over \(E\).
Then for each simple root \(i\) of \(G\) and a sign character \(\varepsilon\), Gehrmann gave a cohomological definition of a space of \(\mathcal L\)-invariants \(\mathcal L_i(\pi,\mathfrak{p})^\varepsilon\subseteq\operatorname{Hom}^{\mathrm{ct}}(F_{\mathfrak{p}}^\ast,E)\). Strictly speaking, this notion of \(\mathcal L\)-invariant depends on the cohomological degree \(q\). We henceforth assume that strong multiplicity one holds for \(\pi\) (hypothesis SMO). This then implies that the above subspace of \(\mathcal L\)-invariants is of codimension at least one. For details, see [Zbl 1482.11076].
The authors introduce the notion of \(\mathfrak{p}\)-étale point \(\mathfrak{m}_\pi\in\mathrm{Spm}\,\mathbb T_{\mathrm{sph}}\) for a suitable commutative Hecke algebra \(\mathbb T_{\mathrm{sph}}\) which at \(\mathfrak{p}\) is of Iwahori level, assuming that \(G\) satisfies Harish-Chandra’s condition at infinity and that the maximal ideal \(\mathfrak{m}_\pi\) attached to \(\pi\) is non-Eisenstein in the sense that the cohomology of the associated locally symmetric space is supported at \(\mathfrak{m}_\pi\) only in middle degree (hypothesis NE).
Then there is a sufficiently small open affinoid \(\mathcal U\) around \( 1\!\! 1\) in the affinoid space corresponding to the localization \(\mathbb T_{\mathrm{sph},\mathfrak{m}_\pi}\) such that the following holds.
Assuming that \(\mathfrak{m}_\pi\) is \(\mathfrak{p}\)-étale with associated character \(\chi_\alpha:T\to\mathcal O_{\mathcal U}^\times\), for every tangent vector \(v\) of \(\mathcal U\) at \(1\!\! 1\) Gehrmann-Rosso prove that \(\frac{\partial}{\partial v}\chi_{\alpha}\circ i^\vee\in\mathcal L_i(\pi,\mathfrak{p})^\varepsilon\) for every simple root \(i\) of \(G\).
Moreover, the codimension of \(\mathcal L_i(\pi,\mathfrak{p})^\varepsilon\) in \(\operatorname{Hom}^{\mathrm{ct}}(F_{\mathfrak{p}}^\ast,E)\) is precisely one in this case.
The proof is based on an automorphic Colmez-Greenberg-Stevens formula, i.e., an interpretation of \(\mathcal L\)-invariants in terms of infinitesimal deformations, which in turn relates \(\mathcal L\)-invariants to the existence of cohomology classes with values in duals of families of principal series representations.
As an application, the authors show that their theory, applied to inner forms \(G\) of \(\mathrm{PGL}_2\) over the totally real field \(F\), settles two conjectures of Spieß: Assuming \(G\) split at \(\mathfrak{p}\), in this setting the automorphic \(\mathcal L\)-invariants \(\mathcal L(\pi,\mathfrak{p})^\varepsilon\) for the unique positive root are independent of the sign character \(\varepsilon\) and are invariant under Jacquet-Langlands transfer. Moreover, the authors show that in this case the automorphic \(\mathcal L\)-invariant \(\mathcal L(\pi,\mathfrak{p})^\varepsilon\) agrees with the Fontaine-Mazur \(\mathcal L\)-invariant \(\mathcal L^{\mathrm{FM}}(\rho_{\pi,\mathfrak{p}})\) attached to the Galois representation \(\rho_\pi\) associated to \(\pi\).
Similar results hold for adjoints \(G\) of definite unitary groups of rank \(n-1\) over \(F\): Again assuming \(G\) split at \(\mathfrak{p}\), for any cuspidal automorphic representation \(\pi\) of \(G\), trivial at all archimedean places and Steinberg at \(\mathfrak{p}\), again assuming strong multiplicity one and \(\mathfrak{m}_\pi\) being \(\mathfrak{p}\)-étale and \(\rho_\pi\) irreducible the authors prove the identity \(\mathcal L_i^{\mathrm{FM}}(\rho_{\pi,\mathfrak{p}})=\mathcal L_i(\pi,\mathfrak{p})^\varepsilon\) where \(i\) indexes the \(n-1\) canonical two-dimensional subquotients of \(\rho_{\pi,\mathfrak{p}}\) and the \(n-1\) simple roots of \(G\).
Finally, the authors also consider identities and invariants of automorphic \(\mathcal L\)-invariants attached to Bianchi newforms \(f\) of weight \(2\), which simplify in the case when \(f\) is a base change of a classical modular form. When extended to higher weight, Gehrmann-Rosso’s approach makes the construction of Stark-Heegner cycles attached to Bianchi modular forms by G. Venkat and C. Williams [J. Lond. Math. Soc., II. Ser. 104, No. 1, 394–422 (2021; Zbl 1490.11055)] unconditional.