On the Special Values of Certain Rankin-Selberg L-Functions and Applications to Odd Symmetric Power L-Functions of Modular Forms

By A. Raghuram. 2008

63 citations recorded by the source corpus.

We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL n × GL n−1 . This is a generalization and refinement of the results of Harder [14], Kazhdan, Mazur, and Schmidt [23], and Mahnkopf [29]. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands' functoriality, one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blasius and Panchishkin on twisted L-values in some cases. These results, as in the above works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/da98d3b76473f6edc72ffa73330f2e5ee6b1e5f1

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In Algebra and Number Theory

1. H. Grobner
2. Michael Harris
3. Binyong Sun
4. A. Raghuram

In Geometry and Topology

1. H. Grobner
2. Michael Harris
3. Binyong Sun
4. A. Raghuram
5. W. Gan
6. T. Gee
7. Peter Scholze
8. David Geraghty

In Mathematical Physics

1. H. Grobner
2. Michael Harris
3. Binyong Sun
4. A. Raghuram
5. W. Gan
6. T. Gee
7. Peter Scholze
8. David Geraghty

Citations in the corpus

• WHITTAKER PERIODS, MOTIVIC PERIODS, AND SPECIAL VALUES OF TENSOR PRODUCT $L$ -FUNCTIONS (2013, background): “In fact, the central character of Π is the twisted base change of the central character of π, and therefore its restriction to the idèles of Q is trivial! The term G(ωσΠ′ f,0 ) is nevertheless retained in the statement of the above theorem because it applies, with minor modifications, to cohomologic”
• Critical values of Rankin-Selberg L-functions for GL(n) x GL(n-1) and the symmetric cube L-functions for GL(2) (2013, methodology): “math.NT] 4 Mar 2015 CRITICAL VALUES OF RANKIN–SELBERG L-FUNCTIONS FOR GLn ×GLn−1 AND THE SYMMETRIC CUBE L-FUNCTIONS FOR GL2 A. RAGHURAM 1. Introduction and statements of results In a previous article [35] an algebraicity result for the central critical value for L-functions for GLn × GLn−1 over Q wa”
• The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions (2013, result): “many theorems on special values of L-functions have been proved under the assumption that Theorem A (or its variant) is valid. See, for examples, Kazhdan-Mazur-Schmidt [KMS], Mahnkopf [Mah], Raghuram [Rag], Kasten-Schmidt [KS], Januszewski [Jan], Greni´e [Gre], Raghuram-Shahidi [RS], Ash-Ginzburg [A”