L-functions of GL(2n): p-adic properties and non-vanishing of twists

Dimitrov, Mladen; Januszewski, Fabian; Raghuram, A.

L-functions of GL(2n): p-adic properties and non-vanishing of twists

M. Dimitrov, F. Januszewski, A. Raghuram, Compositio Mathematica 156 (2021) 2437–2468.

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DOI

10.1112/s0010437x20007551

Journal Article | Published | English

Author

Dimitrov, Mladen; Januszewski, Fabian^LibreCat; Raghuram, A.

Abstract

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\operatorname {GL}_{2n}$ 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 because we draw heavily upon the methods used in the recent and separate works of all 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 $L(\frac {1}{2}, \Pi \otimes \chi )$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$-power conductor.

Keywords

Algebra and Number Theory

Publishing Year

2021

Journal Title

Compositio Mathematica

Volume

156

Issue

Page

2437-2468

ISSN

0010-437X, 1570-5846

LibreCat-ID

53192

Cite this

Dimitrov M, Januszewski F, Raghuram A. L-functions of GL(2n): p-adic properties and non-vanishing of twists. Compositio Mathematica. 2021;156(12):2437-2468. doi:10.1112/s0010437x20007551

Dimitrov, M., Januszewski, F., & Raghuram, A. (2021). L-functions of GL(2n): p-adic properties and non-vanishing of twists. Compositio Mathematica, 156(12), 2437–2468. https://doi.org/10.1112/s0010437x20007551

@article{Dimitrov_Januszewski_Raghuram_2021, title={L-functions of GL(2n): p-adic properties and non-vanishing of twists}, volume={156}, DOI={10.1112/s0010437x20007551}, number={12}, journal={Compositio Mathematica}, publisher={Wiley}, author={Dimitrov, Mladen and Januszewski, Fabian and Raghuram, A.}, year={2021}, pages={2437–2468} }

Dimitrov, Mladen, Fabian Januszewski, and A. Raghuram. “L-Functions of GL(2n): P-Adic Properties and Non-Vanishing of Twists.” Compositio Mathematica 156, no. 12 (2021): 2437–68. https://doi.org/10.1112/s0010437x20007551.

M. Dimitrov, F. Januszewski, and A. Raghuram, “L-functions of GL(2n): p-adic properties and non-vanishing of twists,” Compositio Mathematica, vol. 156, no. 12, pp. 2437–2468, 2021, doi: 10.1112/s0010437x20007551.

Dimitrov, Mladen, et al. “L-Functions of GL(2n): P-Adic Properties and Non-Vanishing of Twists.” Compositio Mathematica, vol. 156, no. 12, Wiley, 2021, pp. 2437–68, doi:10.1112/s0010437x20007551.

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