$L$-functions of ${\mathrm{GL}}(2n):$ $p$-adic properties and non-vanishing of twists

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\mathrm{GL}(2n)$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive since we draw heavily upon the methods used in the recent and separate works of all the three authors. By construction our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover we work under a weaker Panchishkine type condition on $\Pi_p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$-adic $L$-functions at all critical points. This has the striking consequence that, given a unitary $\Pi$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $\Pi_p$ is ordinary, the central critical value $L(\tfrac12, \Pi\otimes\chi)$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$-power conductor.
Comment: Minor changes; accepted for publication in Compositio Mathematica