Project B5: $p$-adic $L$-functions, $\mathcal L$-invariants and the cohomology of arithmetic groups

Project B5 aims to extract the so-called $\mathcal{L}$-invariant from special values of automorphic $L$-functions. This fine invariant appears naturally in the context of the $p$-adic Birch and Swinnerton-Dyer Conjecture: the order of vanishing of the $p$-adic $L$-function may be strictly larger than the order of vanishing of the complex $L$-function at the central critical point. Understanding the so-called exceptional zeroes is a crucial step towards the conjecture of Birch and Swinnerton-Dyer.