[Paper Review] Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field

Baskar Balasubramanyam, A. Raghuram|arXiv (Cornell University)|Aug 18, 2014

Advanced Algebra and Geometry14 references3 citations

TL;DR

This paper establishes an integrality result for special values of the adjoint L-function $ L(1, \mathrm{Ad}^0, \pi, \varepsilon) $ for cohomological cuspidal automorphic representations $ \pi $ on $ \mathrm{GL}_n $ over a number field $ F $, and proves that primes dividing this algebraic part (outside a finite set of exceptional primes) are congruence primes—i.e., they correspond to congruences between automorphic forms. The results generalize classical theorems of Hida and Ribet to higher rank $ \mathrm{GL}_n $, using cohomological methods and Whittaker periods.

ABSTRACT

We prove an integrality result for the value at s=1 of the adjoint L-function associated to a cohomological cuspidal automorphic representation on GL(n) over any number field. We then show that primes (outside an exceptional set) dividing this special value give rise to congruences between automorphic forms. We also prove a non-vanishing property at infinity for the relevant Rankin-Selberg L-functions on GL(n) x GL(n).

Motivation & Objective

• To extend classical congruence theorems for $ \mathrm{GL}_2 $ to $ \mathrm{GL}_n $ over arbitrary number fields.
• To establish an integrality result for the special value $ L(1, \mathrm{Ad}^0, \pi, \varepsilon) $ of the adjoint L-function associated to a cohomological cuspidal automorphic representation $ \pi $.
• To prove that primes dividing this special value (outside a finite set) are congruence primes, i.e., they induce congruences between automorphic forms of the same level and weight.
• To provide a cohomological interpretation of the algebraic part of the adjoint L-function using Betti–Whittaker periods and rational cohomology classes.

Proposed method

• Use of Rankin–Selberg integrals to express $ L(1, \mathrm{Ad}^0, \pi) $ as a residue of $ L(s, \pi \times \tilde{\pi}) $, linking it to Petersson inner products.
• Application of Poincaré duality in the cohomology of locally symmetric spaces $ S^{G}_{K_f} $ to give an algebraic description of the L-value.
• Construction of Betti–Whittaker periods via comparison of $ \mathbb{Q}(\pi) $-structures on Whittaker models and cohomological realizations.
• Use of Casselman–Wallach representations and derived functors to analyze cohomological pairings and nonvanishing of cohomology classes.
• Adaptation of Sun’s nonvanishing results to ensure the archimedean periods do not vanish, crucial for the integrality theorem.
• Computation of discriminants of cohomology modules to relate algebraic L-values to congruence modules and control congruences via p-adic valuation.

Experimental results

Research questions

• RQ1 Can the classical congruence theorems of Hida and Ribet for $ \mathrm{GL}_2 $ be generalized to $ \mathrm{GL}_n $ over arbitrary number fields?
• RQ2 Is the algebraic part of the special value $ L(1, \mathrm{Ad}^0, \pi, \varepsilon) $ integral, and can it be expressed in terms of cohomological invariants?
• RQ3 Do primes dividing this algebraic L-value (outside a finite set) correspond to congruences between automorphic forms of the same level and weight?
• RQ4 What is the role of Betti–Whittaker periods in relating automorphic periods to cohomological periods in the arithmetic of $ \mathrm{GL}_n $?
• RQ5 How can the nonvanishing of archimedean periods be ensured to make the integrality result effective?

Key findings

• The algebraic part $ L^{\mathrm{alg}}(1, \mathrm{Ad}^0, \pi, \varepsilon) $ is integral, up to known arithmetic factors, as shown in Theorem 3.3.7.
• The nonvanishing of the archimedean period $ \mathfrak{p}_{\infty}(\pi) $ is established via a nonvanishing result (Proposition 3.3.4), generalizing Sun’s work.
• A prime $ p $, outside a finite set of exceptional primes, dividing $ L^{\mathrm{alg}}(1, \mathrm{Ad}^0, \pi, \varepsilon) $ implies the existence of a congruence between $ \pi $ and another automorphic form $ \pi' $, as stated in Theorem 4.3.1.
• The congruence module is controlled by the discriminant of the cohomology module, and the congruence primes are precisely those dividing the algebraic L-value, modulo exceptional primes.
• The Betti–Whittaker periods $ \mathfrak{p}^\varepsilon(\pi) $ and $ \mathfrak{q}^{\tilde{\varepsilon}}(\tilde{\pi}) $ are rational over $ \mathbb{Q}(\pi) $, and their ratio appears in the algebraic L-value.
• The cohomological pairing in degree $ b+t $ is nonzero and independent of $ \varepsilon $, ensuring the nonvanishing of the global period class in the cohomology.