[Paper Review] Special values of automorphic $L$-functions for $GL_{n} imes GL_{n'}$ over CM fields, factorization and functoriality of arithmetic automorphic periods

Jie Lin|arXiv (Cornell University)|Nov 11, 2015

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper generalizes Michael Harris's theory of arithmetic automorphic periods to general CM fields and establishes their factorization into local components. It proves that critical values of $L$-functions for $\mathrm{GL}_n \times \mathrm{GL}_{n'}$ over CM fields are expressible in terms of these periods, demonstrating functoriality under automorphic induction and cyclic base change, and confirming compatibility with Deligne's conjecture for motives.

ABSTRACT

Michael HARRIS defined the arithmetic automorphic periods for certain cuspidal representations of $GL_{n}$ over quadratic imaginary fields in his Crelle paper 1997. He also showed that critical values of automorphic L-functions for $GL_{n} imes GL_{1}$ can be interpreted in terms of these arithmetic automorphic periods. In the thesis, we generalize his results in two ways. Firstly, the arithmetic automorphic periods have been defined over general CM fields. We also prove that these periods factorize as products of local periods over infinity places. Secondly, we show that critical values of automorphic $L$ functions for $GL_{n} imes GL_{n'}$ can be interpreted in terms of these automorphic periods in many situations. Consequently we show that the automorphic periods are functorial for automorphic induction and cyclic base change. We also define certain motivic periods if the motive is restricted from a CM field to the field of rational numbers. We can calculate Deligne's period for tensor product of two such motives. We see directly that our automorphic results are compatible with Deligne's conjecture for motives.

Motivation & Objective

• To extend Harris's theory of arithmetic automorphic periods from quadratic imaginary fields to general CM fields.
• To prove that critical $L$-values for $\mathrm{GL}_n \times \mathrm{GL}_{n'}$ over CM fields are expressible in terms of these periods.
• To establish the factorization of arithmetic automorphic periods into local components at infinite places.
• To demonstrate functoriality of these periods under automorphic induction and cyclic base change.
• To define motivic periods for motives over $\mathbb{Q}$ restricted from CM fields and verify compatibility with Deligne's conjecture.

Proposed method

• Define arithmetic automorphic periods for cohomological, conjugate self-dual, cuspidal representations over general CM fields.
• Prove that these periods factorize as products of local periods over infinite places using Hodge structures and rational pairings.
• Use Shahidi’s Whittaker period calculations and rational structures on cohomology to relate global periods to $L$-values.
• Apply base change and automorphic induction techniques to establish functoriality of periods under these operations.
• Derive explicit formulas for special values of $L$-functions in terms of products of arithmetic periods and powers of $2\pi i$, with precise exponents determined by spectral parameters.
• Verify compatibility with Deligne’s conjecture by computing Deligne’s period for tensor products of motives arising from Hecke characters over CM fields.

Experimental results

Research questions

• RQ1 How can arithmetic automorphic periods be generalized from quadratic imaginary fields to arbitrary CM fields?
• RQ2 In what way do arithmetic automorphic periods factorize into local components at infinite places?
• RQ3 Can critical $L$-values for $\mathrm{GL}_n \times \mathrm{GL}_{n'}$ over CM fields be expressed in terms of these periods?
• RQ4 Are the arithmetic automorphic periods functorial under automorphic induction and cyclic base change?
• RQ5 Is the automorphic formula for special $L$-values compatible with Deligne’s conjecture for motives?

Key findings

• Critical $L$-values for $\mathrm{GL}_n \times \mathrm{GL}_{n'}$ over CM fields are expressed as products of arithmetic automorphic periods and powers of $2\pi i$, with exponents determined by spectral parameters.
• The arithmetic automorphic periods factorize into products of local periods at infinite places, confirming a key structural property.
• The periods are functorial under automorphic induction and cyclic base change, as shown by period relations across global and local components.
• For $r_1 \equiv r_2 \pmod{2}$, $L(1, \Pi_1 \times \Pi_2) \sim_{E(\Pi_1)E(\Pi_2);K} (2\pi i)^{r_1 r_2} \prod_{j=0}^{r_1} P^{(j)}(\Pi_1)^{sp(j,\Pi_1;\Pi_2)} \prod_{k=0}^{r_2} P^{(k)}(\Pi_2)^{sp(k,\Pi_2;\Pi_1)}$.
• For $r_1 \not\equiv r_2 \pmod{2}$, $L(\frac{1}{2}, (\Pi_1 \otimes \psi) \times \Pi_2) \sim_{E(\Pi_1)E(\Pi_2)E(\psi);K} (2\pi i)^{r_1 r_2 / 2} \prod_{j=0}^{r_1} P^{(j)}(\Pi_1 \otimes \psi)^{sp(j,\Pi_1 \otimes \psi;\Pi_2)} \prod_{k=0}^{r_2} P^{(k)}(\Pi_2)^{sp(k,\Pi_2;\Pi_1 \otimes \psi)}$, with precise exponents derived from spectral parameters.
• The automorphic results are compatible with Deligne’s conjecture, as verified by computing Deligne’s period for tensor products of motives over $\mathbb{Q}$ restricted from CM fields.