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

Mladen Dimitrov, Fabian Januszewski+1 more|arXiv (Cornell University)|Feb 27, 2018

Advanced Algebra and Geometry20 references11 citations

TL;DR

This paper constructs p-adic L-functions for cohomological cuspidal automorphic representations of GL_{2n} over a totally real field F that admit a Shalika model, using a modular symbol approach with refined Hecke theory. The key result establishes that, under mild ordinarity conditions, the central twisted L-values L(1/2, Π ⊗ χ) are non-zero for all but finitely many Dirichlet characters χ of p-power conductor, extending non-vanishing results to higher rank L-functions.

ABSTRACT

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $Π$ 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 $Π_p$ 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 $Π$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $Π_p$ is ordinary, the central critical value $L( frac12, Π\otimesχ)$ is non-zero for all except finitely many Dirichlet characters $χ$ of $p$-power conductor.

Motivation & Objective

• To construct p-adic L-functions for cohomological cuspidal automorphic representations of GL_{2n} over a totally real field F with a Shalika model.
• To generalize non-vanishing results for twisted central L-values beyond GL_2 and GL_3, particularly for higher-rank L-functions.
• To establish Manin relations between p-adic L-functions at critical points, enabling non-vanishing conclusions.
• To extend non-vanishing results to the nearly-ordinary case and simultaneous non-vanishing for multiple L-functions.
• To provide arithmetic applications, including new rationality results for symmetric power L-functions of Hilbert cusp forms.

Proposed method

• Uses a modular symbol approach inspired by Ash and Ginzburg, adapted to cohomological automorphic representations of GL_{2n} over a totally real field.
• Constructs p-adic distributions on the Galois group of the maximal abelian extension of F unramified outside p∞, valued in a p-adic field.
• Employs Hecke operators at primes above p to define eigenclasses in compactly supported cohomology, focusing on the top degree t = |Σ∞|(n² + n - 1).
• Implements a Panchishkine-type condition on the local components Π_p, weaker than full ordinarity, to ensure p-adic interpolation.
• Applies the Weierstrass preparation theorem to p-adic power series associated with the p-adic L-functions to control zeros.
• Uses twisted norm maps and pushforwards of measures to extend non-vanishing results to non-ordinary and twisted representations.

Experimental results

Research questions

• RQ1 Can p-adic L-functions be constructed for cohomological automorphic representations of GL_{2n} over a totally real field with a Shalika model?
• RQ2 Under what conditions does the central twisted L-value L(1/2, Π ⊗ χ) remain non-zero for Dirichlet characters χ of p-power conductor?
• RQ3 How do Manin relations between p-adic L-functions constrain the vanishing behavior of twisted L-values?
• RQ4 Can the non-vanishing result be extended to the nearly-ordinary case, where Π_p is not fully ordinary?
• RQ5 Can simultaneous non-vanishing be proven for multiple L-functions twisted by the same character χ?

Key findings

• For a unitary cuspidal automorphic representation Π of GL_{2n}/F with a Shalika model and Q-ordinary at p, L(1/2, Π ⊗ (χ∘N_{F/Q})) ≠ 0 for all but finitely many Dirichlet characters χ of p-power conductor.
• The central twisted L-value remains non-zero even when Π is not fully ordinary, provided it satisfies a weaker Panchishkine-type condition.
• The proof relies on Manin relations between p-adic L-functions, which control the zero locus of the associated p-adic distributions.
• A simultaneous non-vanishing result holds: for finitely many such representations Π_k, the product of their central twisted L-values is non-zero for all but finitely many χ of p-power conductor.
• The results yield new rationality results for symmetric fifth power L-functions of Hilbert cusp forms, including for the Ramanujan Δ-function.
• The construction of p-adic L-functions as measures on Cl_F^+(p^∞) provides a framework for arithmetic applications in Iwasawa theory and special value conjectures.