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

Abstract

The principal aim of this article is to attach and study p-adic L-functions to cohomological cuspidal automorphic representations Π of GL2n 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∞. Moreover we work under a weaker Panchishkine type condition on Πp 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 Π whose standard L-function admits at least two critical points, and given a prime p such that Πp is ordinary, the central critical value L( 1 2 , Π ⊗ χ) is non-zero for all except finitely many Dirichlet characters χ of p-power conductor.

On the Bloch-Kato conjecture for GSp(4)

“…Having chosen τ , the construction of [DJR18] shows that for each sign we can find constants Ω p (Θ, τ ) ∈ L × /E × , and Ω ∞ (Θ, τ ) ∈ C × /E × , such that the following proposition holds: By comparing the interpolating properties of the p-adic L-functions, we obtain the following: Corollary 17.5.6. Suppose that L p,ν (Π) is not identically 0 (which is automatic if r 1 − r 2 > 0).…”

“…Instead, we use an alternative argument, relying on the existence of a p-adic L-function for functorial liftings to GL 4 of Siegel-type families, a refinement of the results of [DJR18]. (Details of this will appear in forthcoming work.)…”

On the Bloch--Kato conjecture for the symmetric cube

“…These results are due to Dimitrov, Januszewski and Raghuram [DJR18], as a special case of general theorems applying to any automorphic representation Π of GL(4) admitting a Shalika model. An alternative proof using coherent cohomology of Siegel Shimura varieties is given in [LPSZ19, Theorem A].…”

On the Bloch--Kato conjecture for GSp(4) x GL(2)

“…This is not accessible by the methods of our earlier work [LPSZ19], since the version of higher Hida theory used in that paper (based on the earlier work [Pil20]) is only applicable to 1-parameter families in which r 1 varies for a fixed r 2 . A similar issue arises in our earlier work [LZ20a], but in that setting, we were able to bypass the problem by applying the functorial lift from GSp 4 to GL 4 , and applying the results of [DJR18,BDW21] on p-adic L-functions for GL 2n . However, this does not work for GSp 4 × GL 2 , since there appears to be no known construction of p-adic L-functions for GL 4 × GL 2 .…”