The effect of convolving families of L-functions on the underlying group symmetries

Proceedings of the London Mathematical Society

Volumn 99, Issue 3, 2009, Pages 787-820

  Dueñez, Eduardo ^a   Miller, Steven J ^b  

^a UNIVERSITY OF TEXAS AT SAN ANTONIO   (United States)

^b WILLIAMS COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70350509186     PISSN: 00246115     EISSN: 1460244X     Source Type: Journal    
DOI: 10.1112/plms/pdp018     Document Type: Article

References (59)

1. S. Arms, A. Lozano-Robledo and S. J. Miller, 'Constructing one-parameter families of elliptic curves over Q(T) with moderate rank', J. Number Theory 123 (2007) 388-402.
2. A. Borel, 'Automorphic L-functions', Automorphic forms, representations, and L-functions, Proceedings of Symposia Pure Mathematics 33(part 2), (American Mathematical Society, Providence, RI 1979) 21-61.
3. C. Breuil, B. Conrad, F. Diamond and R. Taylor, 'On the modularity of elliptic curves over Q: wild 3-adic exercises', J. Amer. Math. Soc. 14 (2001) 843-939. (Pubitemid 33587361)
4. D. Bump, 'The Rankin-Selberg method: a survey', Number theory, trace formulas and discrete groups, Oslo, 1987, (Academic Press, Boston, MA, 1989) 49-109.
5. J. Cogdell and P. Michel, 'On the complex moments of symmetric power L-functions at s = 1', Int. Math. Res. Not. 2004 (2004) 1561-1617.
6. B. Conrey, D. Farmer, P. Keating, M. Rubinstein and N. Snaith, 'Integral moments of L-functions', Proc. London Math. Soc. (3) 91 (2005) 33-104.
7. H. Davenport, Multiplicative number theory, 2nd edn, Graduate Texts in Mathematics 74 (Springer, New York, 1980) revised by H. Montgomery.
8. P. Deligne, 'La conjecture de Weil II', Publ. Math. Inst. Hautes É tudes Sci. 52 (1981) 313-428.
9. E. Dueñez and S. J. Miller, 'The low lying zeros of a GL(4) and a GL(6) family of L-functions', Compos. Math. 142 (2006) 1403-1425.
10. F. J. Dyson, 'Statistical theory of energy levels of complex systems, I, II, and III', J. Math. Phys. 3 (1962) 140-156, 157-165, 166-175.
11. S. Fermigier, 'Étude exṕerimentale du rang de familles de courbes elliptiques sur Q', Exper. Math. 5 (1996) 119-130. (Pubitemid 126138678)
12. E. Fouvry and H. Iwaniec, 'Low-lying zeros of dihedral L-functions, Duke Math. J. 116 (2003) 189-217.
13. P. Gao, 'N-level density of the low-lying zeros of quadratic Dirichlet L-functions', PhD Thesis, University of Michigan, 2005.
14. P. B. Garrett, 'Decomposition of Eisenstein series: Rankin triple products', Ann. of Math. (2) 125 (1987) 209-235.
15. S. Gelbart, Automorphic forms on Adele groups, Annals of Mathematics Studies 83 (Princeton University Press, Princeton, NJ, 1975).
16. S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. É cole Norm. Sup. (4) 11 (1978) 471-552.
17. D. Goldfeld, 'Conjectures on elliptic curves over quadratic .elds', Number theory, Carbondale 1979, Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979, Lecture Notes in Mathematics 751, (Springer, Berlin, 1979) 108-118.
18. A. Güloglu, 'Low lying zeros of symmetric power L-functions', Int. Math. Res. Not. 2005 (2005) 517-550.
19. G. Harcos and P. Michel, 'The subconvexity problem for Rankin-Selberg L-functions', Invent. Math. 163 (2006) 581-655.
20. D. Hejhal, 'On the triple correlation of zeros of the zeta function', Int. Math. Res. Not. 1994 (1994) 294-302.
21. C. Hughes and S. J. Miller, 'Low-lying zeros of L-functions with orthogonal symmetry', Duke Math. J. 136 (2007) 115-172.
22. C. Hughes and Z. Rudnick, 'Linear statistics of low-lying zeros of L-functions', Q. J. Math. 54 (2003) 309-333. (Pubitemid 37446081)
23. H. Iwaniec, W. Luo and P. Sarnak, 'Low lying zeros of families of L-functions', Publ. Math. Inst. Hautes É tudes Sci. 91 (2000) 55-131.
24. H. Jacquet, 'Principal L-functions of the linear group', Automorphic forms, representations, and L-functions, Proceedings of Symposia Pure Mathematics 33(part 2) (American Mathematical Society, Providence, RI, 1979) 63-86.
25. H. Jacquet, I. I. Piatetski-Shapiro and J. A. Shalika, 'Rankin-Selberg convolutions', Amer. J. Math. 105 (1983) 367-464.
26. H. Jacquet and J. A. Shalika, 'On Euler products and the classi.cation of automorphic representations I', Amer. J. Math. 103 (1981) 499-558.
27. N. Katz and P. Sarnak, 'Random matrices, frobenius eigenvalues and monodromy', AMS Colloquium Publications 45 (American Mathematical Society, Providence, RI, 1999).
28. N. Katz and P. Sarnak, 'Zeros of zeta functions and symmetries', Bull. Amer. Math. Soc. (N.S.) 36 (1999) 1-26.
29. J. P. Keating and N. C. Snaith, 'Random matrices and L-functions', J. Phys. A 36 (2003) 2859-2881.
30. H. Kim, 'Functoriality for the exterior square of GL2 and the symmetric fourth of GL2 (with appendix 1 by D. Ramakrishnan and appendix 2 by H. Kim and P. Sarnak)', J. Amer. Math. Soc. 16 (2003) 139-183.
31. H. Kim and F. Shahidi, 'Functorial products for GL2 times; GL3 and the symmetric cube for GL2', Annals of Math. (2) 155 (2002) 837-893.
32. H. Kim and F. Shahidi, 'Cuspidality of symmetric powers with applications', Duke Math. J. 112 (2002) 177-197.
33. A. W. Knapp, Local Langlands correspondence: the Archimedean case', Motives, Proceedings of Symposia Pure Mathematics 55(part 2) (American Mathematical Society, Providence, RI, 1994).
34. W. Luo and P. Sarnak, 'Mass equidistribution for Hecke eigenforms', Comm. Pure Appl. Math. 56 (2003) 874-891. (Pubitemid 36765737)
35. P. Michel, 'Rang moyen de familles de courbes elliptiques et lois de Sato-Tate', Monatsh Math. 120 (1995) 127-136.
36. S. J. Miller, '1- and 2-level densities for families of elliptic curves: evidence for the underlying group symmetries, PhD Thesis, Princeton University, 2002, http://www.williams.edu/go/math/sjmiller/ public html/
37. S. J. Miller, '1- and 2-level densities for families of elliptic curves: evidence for the underlying group symmetries', Compos. Math. 140 (2004) 952-992.
38. S. J. Miller, 'Variation in the number of points on elliptic curves and applications to excess rank', C. R. Math. Rep. Acad. Sci. Can. 27 (2005) 111-120.
39. S. J. Miller, 'Investigations of zeros near the central point of elliptic curve L-functions (with an appendix by E. Dueñez)', Experiment. Math. 15 (2006) 257-279.
40. S. J. Miller, A symplectic test of the L-functions ratios conjecture, Int. Math. Res. Not. 2008 (2008) rnm146, doi:10.1093/imrn/rnm146.
41. S. J. Miller, 'Lower order terms in the 1-level density for families of holomorphic cuspidal newforms', Acta Arith. 137 (2009) 51-98.
42. S. J. Miller, 'An orthogonal test of the L-functions ratios conjecture', Proc. London Math. Soc. 2009, doi:10.1112/plms/pdp009.
43. S. J. Miller, 'Density functions for families of Dirichlet characters', (Williams College, 2009), Preprint.
44. H. Montgomery, 'The pair correlation of zeros of the zeta function', Analytic number theory, Proceedings of Symposia Pure Mathematics 24 (American Mathematical Society, Providence, RI, 1973) 181-193.
45. T. Nagell, Introduction to number theory, (Chelsea, New York, 1981).
46. A. Odlyzko, 'On the distribution of spacings between zeros of the zeta function', Math. Comp. 48 (1987) 273-308.
47. A. Odlyzko, 'The 1022-nd zero of the Riemann zeta function', Proc. Conference on Dynamical, Spectral and Arithmetic Zeta-Functions (ed. M. van Frankenhuysen and M. L. Lapidus; American Mathematical Society, Providence, RI, 2001) Contemporary Mathematics Series, http://www.research.att.com/ ¡«amo/ doc/zeta.html.
48. D. Ramakrishnan, 'Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2)', Ann. of Math. (2) 152 (2000) 45-111.
49. G. Ricotta and E. Royer, 'Statistics for low-lying zeros of symmetric power L-functions in the level aspect', Preprint, 2007, http://arxiv.org/abs/ math/0703760.
50. M. Rosen and J. Silverman, 'On the rank of an elliptic surface', Invent. Math. 133 (1998) 43-67. (Pubitemid 128450201)
51. E. Royer, 'Petits źeros de fonctions L de formes modulaires', Acta Arith. 99 (2001) 47-172.
52. M. Rubinstein, 'Low-lying zeros of L-functions and random matrix theory', Duke Math. J. 109 (2001) 147-181.
53. Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, Duke Math. J. 81 (1996) 269-322.
54. J. Stopple, The quadratic character experiment, Preprint, 2008, http://arxiv.org/abs/0802.4255.
55. J. Tate, Algebraic cycles and the pole of zeta functions, Arithmetical Algebraic Geometry (Harper & Row, New York, 1965) 93-110.
56. R. Taylor and A. Wiles, 'Ring-theoretic properties of certain Hecke algebras', Ann. of Math. (2) 141 (1995) 553-572.
57. A. Wiles, 'Modular elliptic curves and Fermat's last theorem', Ann. of Math. (2) 141 (1995) 443-551.
58. M. Young, 'Lower-order terms of the 1-level density of families of elliptic curves', Int. Math. Res. Not. 2005 (2005) 587-633.
59. M. Young, 'Low-lying zeros of families of elliptic curves', J. Amer. Math. Soc. 19 (2006) 205-250.