Derived Hecke algebra and automorphic ℒ-invariants

Gehrmann, Lennart

doi:10.1090/tran/7815

Derived Hecke algebra and automorphic ${\mathcal {L}}$-invariants
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Trans. Amer. Math. Soc. 372 (2019), 7767-7784 Request permission

Abstract:

Let $\pi$ be a cohomological cuspidal automorphic representation of PGL$_2$ over a number field of arbitrary signature. Under the assumption that the local component of $\pi$ at a prime ${\mathfrak {p}}$ is the Steinberg representation, the automorphic ${\mathcal {L}}$-invariant of $\pi$ at ${\mathfrak {p}}$ has been defined using the lowest degree cohomology in which the system of Hecke eigenvalues associated with $\pi$ occurs.

In this article we define automorphic ${\mathcal {L}}$-invariants for each cohomological degree and show that they behave well with respect to the action of Venkatesh’s derived Hecke algebra. As a corollary, we show that these ${\mathcal {L}}$-invariants are (essentially) the same if the following conjecture of Venkatesh holds: the $\pi$-isotypic component of the cohomology is generated by the minimal degree cohomology as a module over the $p$-adic derived Hecke algebra.

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