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P-adic L-function for GL(n + 1) Ã GL(n) I

Banff International Research Station for Mathematical Innovation and Discovery

A construction of p-adic L-functions for GL(2) via the modular symbol method is reviewed in this talk. I will summarize some technical points of the construction comparing with the works of F. Januszewski on p-adic L-functions for GL(n + 1) Ã GL(n). In particular, the behavior under the Tate twists is emphasized in the talk, since it is the most important new ingredient in Januszewskiâ s recent preprint.
<br/> Related references (for talks I to IV): <br/> (Main) F.Januszewski. Non-abelian p-adic Rankin-Selberg L-functions and non-vanishing of central L- values, arXiv:1708.02616, 2017.
<br/> K. Namikawa. On p-adic L-functions associated with cusp forms on GL2. manuscr. math. 153, pages 563â 622, 2017. (Sub)
<br/> B.J. Birch. Elliptic curves over Q, a progress report. 1969 Number Theory Institute. AMS Proc. Symp. Pure Math. XX, 396â 400, 1971.
<br/> M. Dimitrov. Automorphic symbols, p-adic L-functions and ordinary cohomology of Hilbert modular varieties. Amer. J. Math. 135, 1117â 1155, 2013.
<br/> F. Januszewski. Modular symbols for reductive groups and p-adic Rankin-Selberg convolutions over number fields, J. Reine Angew. Math. 653, 1â 45, 2011.
<br/> F. Januszewski. On p-adic L-functions for GL(n)Ã GL(nâ 1) over totally real fields, Int. Math. Res. Not., Vol. 2015, No. 17, 7884â 7949.
<br/> F. Januszewski. p-adic L-functions for Rankin-Selberg convolutions over number fields, Ann. Math. Quebec 40, special issue in Honor of Glenn Stevens â 60th birthday, 453â 489, 2016.
<br/> F. Januszewski. On period relations for automorphic L-functions I. To appear in Trans. Amer. Math. Soc., arXiv:1504.06973
<br/> H. Kasten and C.-G. Schmidt. On critical values of Rankin-Selberg convolutions. Int. J. Number Theory 9, pages 205â 256, 2013. D. Kazhdan, B. Mazur, and C.-G. Schmidt. Relative modular symbols and Rankin-Selberg convolutions, J. Reine Angew. Math. 512, 97â 141, 2000.
<br/> K. Kitagawa. On standard p-adic L-functions of families of elliptic cusp forms, p-adic mon- odromy and the Birch and Swinnerton-Dyer conjecture (B. Mazur and G. Stevens, eds.), Con- temp. Math. 165, AMS, 81â 110, 1994.
<br/> J.I. Manin. Non-archimedean integration and p-adic Hecke-Langlands L-series. Russian Math. Surveys 31, 1, 1976.
<br/> B. Mazur, and P. Swinnerton-Dyer. Arithmetic of Weil Curves, Invent. Math. 25, 1â 62, 1974. B. Mazur, J. Tate, and J. Teitelbaum. On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84, 1â 48, 1986.
<br/> C.-G. Schmidt. Relative modular symbols and p-adic Rankin-Selberg convolutions, Invent. Math. 112, 31â 76, 1993.
<br/> C.-G. Schmidt. Period relations and p-adic measures, manuscr. math. 106, 177â 201, 2001. B. Sun. The non-vanishing hypothesis at infinity for Rankin-Selberg convolutions. J. Amer. Math. Soc. 30, pages 1â 25, 2017.
<br/> A. Raghuram. On the Special Values of certain Rankin-Selberg L-functions and Applications to odd symmetric power L-functions of modular forms. Int. Math. Res. Not. 2010, 334â 372, 2010.
<br/> A. Raghuram. Critical values for Rankin-Selberg L-functions for GL(n) Ã GL(n â 1) and the symmetric cube L-functions for GL(2). Forum Math. 28, 457â 489, 2016.

Mathematics; Number theory; Algebraic geometry; Arithmetic number theory

video/mp4

Author affiliation: Kyushu University

2019-04-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69371

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/