2022 Fiscal Year Annual Research Report
| Project/Area Number |
22F22316
|
|---|
| Allocation Type | Single-year Grants |
|---|
| Research Institution | Kyoto University |
|---|
Principal Investigator |
市野 篤史 京都大学, 理学研究科, 准教授 (40347480)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
CHEN SHIH-YU 京都大学, 理学(系)研究科(研究院), 外国人特別研究員
|
|---|
| Project Period (FY) |
2022-11-16 – 2025-03-31
|
| Keywords | Special L-values / Deligne's conjecture / Betti-Whittaker periods |
|---|
| Outline of Annual Research Achievements |
Our research mainly focuses on the study of algebraicity of critical values of automorphic L-functions. In the literature, most algebraicity results were proved using the integral representation. Our current research project is to provide yet another approach to tackle the algebraicity problem when integral representation is not available. We consider ratios of Rankin-Selberg L-functions. In the previous result of G. Harder and A. Raghuram, the authors consider the ratios for a fixed Rankin-Selberg L-function. On the contrary, we fix a critical point and vary the automorphic representations. In this case, we conjectured that the ratios belong to the rationality field of the automorphic representations.
Assuming the validity of the conjecture, we prove the conjectures proposed by D. Blasius and P. Deligne on the tensor product L-functions and the symmetric power L-functions of modular forms where the algebraicity is expressed in terms of product of motivic periods associated to the modular forms. For the conjecture on the algebraicity of ratios, we can prove it under some parity and regularity conditions on the archimedean component of the automorphic representations.
We also proved a period relation between the Betti-Whittaker periods associated to a regular algebraic cuspidal automorphic representation and its contragredient. This is an automorphic analogue of a relation between the period invariants of motives under duality. As a consequence, we prove the algebraicity of the ratios of successive critical L-values for GSpin(2n) x GL(n').
|
| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
To prove our conjecture on the ratios, currently we need to assume certain parity and regularity conditions. There are three main ingredients in our proof:
(1) Theory of Eisenstein cohomology for arithmetic subgroups.
(2) Period relations between the Betti-Whittaker periods.
(3) Result of Raghuram for GL(n) x GL(n-1).
We consider regular algebraic automorphic representations which implies that its cuspidal summands appear in the cuspidal cohomology of the adelic locally symmetric spaces. A complement of the cuspidal cohomology under the Hecke action is the Eisenstein cohomology whose classes are represented by residues or derivatives of Eisenstein series. The first ingredient describes the restriction to the boundary strata of the locally symmetric spaces. More precisely, the restriction map relates the Eisenstein cohomology classes with the cuspidal cohomology classes associated to the inducing data. Under the parity condition, we can construct an automorphic representation as a fully induced isobaric sum, so that its Betti-Whittaker period can be defined. Based on the first ingredient, we show that the Betti-Whittaker period of this isobaric sum factors into product of the Betti-Whittaker period of its cuspidal summands together with a critical Rankin--Selberg L-value in question. This is the content of the second ingredient. The regularity condition implies the existence of non-central critical point for an auxiliary Rankin-Selberg L-function so that we can further separate the dependence of the algebraicity and prove the conjecture by the third ingredient.
|
| Strategy for Future Research Activity |
The regularity condition was imposed to guarantee the existence of non-central critical point which implies the existence of non-zero critical values. To weaken this assumption, we may assume the existence of non-vanishing for twisted Rankin-Selberg L-functions. To remove the parity condition, it is necessary to consider isobaric automorphic representations which are not fully induced. This happens when the Rankin-Selberg L-function under consideration admits central critical point. We can not define Betti--Whittaker period when the globally induced representation is not irreducible. Nonetheless, still we can consider the algebraicity of the corresponding Eisenstein cohomology classes. We expect the algebraicity also holds in this case, that is, expressed in terms of product of Betti-Whittaker period of the cuspidal summands together with a critical value of the Rankin-Selberg L-function. New difficulties arise since the archimedean component of the isobaric automorphic representations might not be essentially unitary anymore. For instance, to generalize the third ingredient on algebraicity for GL(n) x GL(n-1), we need to prove the non-vanishing of certain archimedean pairing. When the archimedean components are essentially unitary, it was proved by B. Sun. We propose to carry out the remaining difficulties and generalize the result of Sun. We also plan to apply our conjecture to prove automorphic analogue of Deligne's conjecture for the tensor product L-functions for GSp(4) x GL(2) x GL(2) and GSp(4) x GSp(4).
|
