A new tower of Rankin-Selberg integrals
Authors:
David Ginzburg and Joseph Hundley
Journal:
Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 56-62
MSC (2000):
Primary 32N10
DOI:
https://doi.org/10.1090/S1079-6762-06-00160-0
Published electronically:
May 16, 2006
MathSciNet review:
2226525
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We recall the notion of a tower of Rankin-Selberg integrals, and two known towers, making observations of how the integrals within a tower may be related to one another via formal manipulations, and offering a heuristic for how the $L$-functions should be related to one another when the integrals are related in this way. We then describe three new integrals in a tower on the group $E_6,$ and find out which $L$-functions they represent. The heuristics also predict the existence of a fourth integral.
- Stephen Gelbart and Hervรฉ Jacquet, A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$, Ann. Sci. รcole Norm. Sup. (4) 11 (1978), no. 4, 471โ542. MR 533066
- David Ginzburg, A Rankin-Selberg integral for the adjoint representation of ${\rm GL}_3$, Invent. Math. 105 (1991), no. 3, 571โ588. MR 1117151, DOI https://doi.org/10.1007/BF01232279
- David Ginzburg, On standard $L$-functions for $E_6$ and $E_7$, J. Reine Angew. Math. 465 (1995), 101โ131. MR 1344132, DOI https://doi.org/10.1515/crll.1995.465.101
[G-H]G-H D. Ginzburg, J. Hundley, On spin $L$-functions for $GSO_{10}$. To appear in J. Reine Angew. Math. ArXiv:math.NT/0512113.
- David Ginzburg and Dihua Jiang, Periods and liftings: from $G_2$ to $C_3$, Israel J. Math. 123 (2001), 29โ59. MR 1835288, DOI https://doi.org/10.1007/BF02784119
- D. Ginzburg and S. Rallis, A tower of Rankin-Selberg integrals, Internat. Math. Res. Notices 5 (1994), 201 ff., approx. 8 pp.}, issn=1073-7928, review=\MR{1270133}, doi=10.1155/S107379289400022X,.
- David Ginzburg, Stephen Rallis, and David Soudry, On the automorphic theta representation for simply laced groups, Israel J. Math. 100 (1997), 61โ116. MR 1469105, DOI https://doi.org/10.1007/BF02773635
- Hervรฉ Jacquet and Joseph Shalika, Exterior square $L$-functions, Automorphic forms, Shimura varieties, and $L$-functions, Vol. II (Ann Arbor, MI, 1988) Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 143โ226. MR 1044830
- V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190โ213. MR 575790, DOI https://doi.org/10.1016/0021-8693%2880%2990141-6
- Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297โ355. MR 610479, DOI https://doi.org/10.2307/2374219
[Ge-J]Ge-J S. Gelbart, H. Jacquet, A relation between automorphic representations of $\textrm {GL}(2)$ and $\textrm {GL}(3)$. Ann. Sci. Ecole Norm. Sup. (4) 11 (1978), no. 4, 471โ542.
[G1]G1 D. Ginzburg, A Rankin-Selberg integral for the adjoint representation of $GL_3$, Invent. Math. 105 (1991), 571โ588.
[G2]G2 D. Ginzburg, On standard $L$-functions for $E_6$ and $E_7$. J. Reine Angew. Math. 465 (1995), 101โ131.
[G-H]G-H D. Ginzburg, J. Hundley, On spin $L$-functions for $GSO_{10}$. To appear in J. Reine Angew. Math. ArXiv:math.NT/0512113.
[G-J]G-J D. Ginzburg, D. Jiang, Periods and liftings: From $G_2$ to $C_3$. Israel J. of Math. 123 (2001), 29โ59.
[G-R]G-R D. Ginzburg, S. Rallis, A tower of Rankin-Selberg integrals. IMRN 5 (1994), 201โ208.
[G-R-S]G-R-S D. Ginzburg, S. Rallis, D. Soudry, On the automorphic theta representation for simply laced groups. Israel J. of Math. 100 (1997), 61โ116.
[J-S]J-S H. Jacquet, J. Shalika, Exterior square $L$-functions, automorphic forms, Shimura varieties, and $L$-functions, Vol. 2. Academic Press, 1990, pp. 143โ226.
[K]K V. Kac, Some remarks on nilpotent orbits. J. of Algebra 64 (1980), 190โ213.
[S]S F. Shahidi, On certain $L$-functions. Amer. J. Math. 103 (1981), no. 2, 297โ355.
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (2000):
32N10
Retrieve articles in all journals
with MSC (2000):
32N10
Additional Information
David Ginzburg
Affiliation:
School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
Email:
ginzburg@post.tau.ac.il
Joseph Hundley
Affiliation:
Mathematics Department, Penn State University, University Park, PA 16802
MR Author ID:
746477
Email:
hundley@math.psu.edu
Received by editor(s):
October 12, 2005
Published electronically:
May 16, 2006
Communicated by:
Barry Mazur
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
