A local Riemann hypothesis, I

Mathematische Zeitschrift

Volumn 233, Issue 1, 2000, Pages 1-19

  Bump, Daniel ^a   Choi, Kwok Kwong ^b   Kurlberg, Pär ^c   Vaaler, Jeffrey ^d  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSITY OF HONG KONG   (Hong Kong)

^c UNIVERSITY OF GEORGIA   (United States)

^d UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034347297     PISSN: 00255874     EISSN: None     Source Type: Journal    
DOI: 10.1007/pl00004786     Document Type: Article

References (38)

1. L. Brekke, P. Freund, M. Olson, E. Witten, Non-Archimedean string dynamics, Nuclear Phys. B 302 (1988), 365-402.
2. D. Bump, Automorphic Forms and Representations, Cambridge University Press (to appear).
3. D. Bump, S. Friedberg, On Mellin transforms of unramified Whittaker functions on GL(3, ℂ), J. Math. Anal. Appl. 139 (1989), 205-216.
4. D. Bump, J. Huntley, Unramified Whittaker functions for GL(3, ℝ), J. Anal. Math. 65 (1995), 19-44.
5. D. Bump, E. K.-S. Ng, On Riemann's Zeta function, Math. Zeitschrift 192 (1986).
6. D. Cardon, Zeros of Fourier coefficients of Eisenstein series on the metaplectic groups - the function field case, Dissertation, Stanford University (1996). (In progress.)
7. P. Cartier, Quantum mechanical commutation relations and theta functions, in Algebraic Groups and Discontinuous Subgroups, AMS Proc. Symp. Pure Math. 9, Borel and Mostow, ed. (1966), 361-386.
8. W. Casselman, J. Shalika, The unramified principal series of p-adic groups II: the Whittaker function, Compositio Math. 40 (1980), 207-231.
9. P. Freund, M. Olsen, Non-Archimedean strings, Phys. Lett. B 199 (1987), 186-190.
10. P. Freund, E. Witten, Adelic string amplitudes, Phys. Lett. B 199 (1987), 191-194.
11. S. Gelbart, R. Howe, I. Piatetski-Shapiro, Uniqueness and existence of Whittaker models for the metaplectic group, Israel J. Math. 34 (1979), 21-37.
12. S. Gelbart, I. Piatetski-Shapiro, Distinguished representations and modular forms of half integral weight, Invent. Math. 59 (1980), 145-188.
13. D. Goldfeld, J. Hoffstein, Eisenstein series of half-integral weight and the mean values of real Dirichlet series, Invent. Math. 80 (1985), 185-208.
14. M. Harris, S. Kudla, W. Sweet, Theta dichotomy for unitary groups, J. Amer. Math. Soc., to appear.
15. S. Helgason, Groups and geometric analysis, Academic Press (1984).
16. R. Howe, Theta series and invariant theory, in part 1 of A. Borel and W. Casselman, ed., Automorphic Forms, Representations, and L-functions, AMS Proceedings of Symposia in Pure Mathematics 33 (1979), 275-286.
17. H. Kloosterman, The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups, I and II, Ann. of Math. (2) 47 (1946). 317-375, and 376-447.
18. T. Kubota, Automorphic Forms and Reciprocity in a Number Field, Kyoto University and the Kiyokuniya Book Store (1969).
19. P. Kurlberg, A Local Riemann Hypothesis, II, Math. Z. 233 (2000) 21-37.
20. H. Maass, Konstruktion ganzer Modulformen halbzahliger Dimension, Abh. Math. Semin. Univ. Hamburg 12 (1937), 133-162.
21. I. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press (1979); second edition (1995).
22. C. Moen, The dual pair (U(1), U(1)) over a p-adic field, Pacific J. Math. 158 (1993), 365-386.
23. G. Polyá, Bemerkung über die Integraldarstellung der Riemannsche ξ-Funktion, Acta Mathematica 48 (1926), 305-17.
24. D. Prasad, On an extension of a theorem of Tunnell, Compositio Math. 94 (1994), 19-28.
25. E. Rainville, Special Functions, Macmillan (1960). Reissued by Chelsea (1971).
26. J. Rogawski, The multiplicity formula for A-packets, in The zeta functions of Picard modular surfaces, ed. Langlands and Ramakrishnan, Univ. Montréal, Montreal, PQ, 1992
27. P. Ruelle, E. Thiran, D. Verstegen, J. Weyers, Quantum mechanics on p-adic fields, J. Math. Phys. 30 (1989), 2854-2874.
28. G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), 79-98.
29. T. Shintani, On an explicit formula for class-1 "Whittaker Functions" on GL(n) over P-adic fields, Proc. Japan Acad. 52 (1976), 180-182.
30. J. Tate, Fourier analysis in number fields and Hecke's zeta-functions, Dissertation (1950). Reprinted in Cassels and Fröhlich, Algebraic Number Theory, Academic Press (1967).
31. J. Tate, Fourier analysis in number fields and Hecke's zeta-functions, Dissertation (1950). Reprinted in Cassels and Fröhlich, Algebraic Number Theory, Academic Press (1967).
32. E. Titchmarsh, The Theory of the Riemann Zeta Function, Oxford (1951).
33. V. Vladimirov, I. Volovich, E. Zelenov, p-adic analysis and mathematical physics. World Scientific Publishing Co., (1994).
34. J.-L. Waldspurger, Démonstration d'une conjecture de Dualité de Howe dans le cas p-adic, p ≠ 2, in S. Gelbart, R. Howe, P. Sarnak, ed., Festschrift in honor of I.I. Piatetski-Shapiro on the occasion of his 60-th birthday, two volumes, The Weizmann Science Press of Israel (1990), 267-327.
35. G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press (1944).
36. A. Weil, Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211, Collected Works, Vol. 3.
37. H. Weyl, The Theory of Groups and Quantum Mechanics, Methuen (1931). Reissued by Dover (1950).
38. T. Yang, Eigenfunctions of the Weil representation of unitary group of one variable, Trans. AMS 350 (1998), 2393-2407.