Derivatives of random matrix characteristic polynomials with applications to elliptic curves

Journal of Physics A: Mathematical and General

Volumn 38, Issue 48, 2005, Pages 10345-10360

  Snaith, N C ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 27844511532     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/38/48/007     Document Type: Article

References (46)

1. Barnes E W 1900 The theory of the G-function Q. J. Math. 31 264-314
2. Bogomolny E B and Keating J P 1995 Random matrix theory and the Riemann zeros: I. Three- and four-point correlations Nonlinearity 8 1115-31
3. Bogomolny E B and Keating J P 1996 Random matrix theory and the Riemann zeros: II. n-point correlations Nonlinearity 9 911-35
4. Breuil C, Conrad B, Diamond F and Taylor R 2001 On the modularity of elliptic curves over : wild 3-adic exercises J. Am. Math. Soc. 14 843-939
5. Conrey J B 2001 L-functions and random matrices Mathematics Unlimited 2001 and Beyond ed B Enquist and W Schmid (Berlin: Springer) pp 331-52 (Preprint math-nt/0005300)
6. Conrey J B 2005 Notes on eigenvalue distributions for the classical compact groups Recent Perspectives on Random Matrix Theory and Number Theory (Cambridge: Cambridge University Press) pp 111-45
7. Conrey J B and Farmer D W 2000 Mean values of L-functions and symmetry Int. Math. Res. Not. 17 883-908 (Preprint math-nt/9912107)
8. Conrey J B, Farmer D W, Keating J P, Rubinstein M O and Snaith N C 2003 Autocorrelation of random matrix polynomials Commun. Math. Phys. 237 365-95 (Preprint math-ph/0208007)
9. Conrey J B, Farmer D W, Keating J P, Rubinstein M O and Snaith N C 2005 Integral moments of L-functions Proc. Lond. Math. Soc. 91 33-104 (Preprint math-nt/0206018)
10. Conrey J B, Keating J P, Rubinstein M O and Snaith N C 2002 On the frequency of vanishing of quadratic twists of modular L-functions Number Theory for the Millennium I: Proc. Millennial Conf. on Number Theory ed M A Bennett et al (Natick: A K Peters, Ltd) pp 301-15 (Preprint math-nt/0012043)
11. Conrey J B, Keating J P, Rubinstein M O and Snaith N C 2004 Random matrix theory and the Fourier coefficients of half-integral weight forms Preprint math-nt/0412083 (2005 Exp. Math. at press)
12. Conrey J B, Pokharel A, Rubinstein M O and Watkins M 2005 Secondary terms in the number of vanishings of quadratic twists of elliptic curve L-functions Preprint math-NT/0509059
13. Conrey J B, Rubinstein M O, Snaith N C and Watkins M 2005 Discretisation for odd quadratic twists Preprint math-NT/0509428
14. David C, Fearnley J and Kisilevsky H 2004 On the vanishing of twisted L-functions of elliptic curves Exp. Math. 13 185-98
15. Delaunay C 2005 Moments of the orders of Tate-Shafarevich groups J. Number Theory 1 243-64
16. Delaunay C 2005 Note on the frequency of vanishing of L-functions of elliptic curves in a family of quadratic twists Preprint
17. Dueñez E 2004 Random matrix ensembles associated to compact symmetric spaces Commun. Math. Phys. 244 29-61
18. Feller W 1971 An Introduction to Probability Theory and its Applications vol 2 2nd edn
19. Forrester P J Log-Gases and Random Matrices at press http://www.ms. unimelb.edu.au/ matpjf/matpjf.html
20. Gonek S M, Hughes C P and Keating J P A hybrid Euler-Hadamard product formula for the Riemann zeta function (in preparation)
21. Hejhal D A 1994 On the triple correlation of zeros of the zeta function Int. Math. Res. Not. 7 293-302
22. Hughes C P 2003 Random matrix theory and discrete moments of the Riemann zeta function J. Phys. A: Math. Gen. 36 2907-17 (Preprint math-nt/0207236)
23. Hughes C P, Keating J P and O'Connell N 2000 Random matrix theory and the derivative of the Riemann zeta function Proc. R. Soc. A 456 2611-27
24. Katz N M and Sarnak P 1999 Random Matrices, Frobenius Eigenvalues and Monodromy (American Mathematical Society Colloquium Publications vol 45) (Providence, RI: American Mathematical Society)
25. Katz N M and Sarnak P 1999 Zeros of zeta functions and symmetry Bull. Am. Math. Soc. 36 1-26
26. Keating J P and Snaith N C 2000 Random matrix theory and L-functions at s ≤ 1/2 Commun. Math. Phys. 214 91-110
27. Keating J P and Snaith N C 2000 Random matrix theory and ζ(1/2 + it)) Commun. Math. Phys. 214 57-89
28. Keating J P and Snaith N C 2003 Random matrices and L-functions J. Phys. A: Math. Gen. 36 2859-81
29. Kohnen W and Zagier D 1981 Values of L-series of modular forms at the center of the critical strip Invent. Math. 64 175-98
30. Mehta M L 2004 Random Matrices 3rd edn (Amsterdam: Elsevier)
31. Miller S J 2005 Investigations of zeros near the central point of elliptic curve L-functions Preprint math-nt/0508150
32. Miller S J 2004 One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries Compos. Math. 140 952-92
33. Montgomery H L 1973 The pair correlation of the zeta function Proc. Symp. Pure Math. 24 181-93
34. Nagao T and Wadati M 1991 Correlation functions of random matrix ensembles related to classical orthogonal polynomials J. Phys. Soc. Japan 60 3298-322
35. 20th zero of the Riemann zeta function and 70 million of its neighbors Preprint http://www.dtc.umn.edu/odlyzko/ unpublished/index.html
36. Rubin K and Silverberg A 2002 Ranks of elliptic curves Bull. Am. Math. Soc. 39 455-74
37. Rubinstein M 1998 Evidence for a spectral interpretation of zeros of L-functions PhD Thesis Princeton University
38. Rudnick Z and Sarnak P 1996 Zeros of principal L-functions and random matrix theory Duke Math. J. 81 269-322
39. Shimura G 1973 On modular forms of half integral weight Ann. Math. 97 440-81
40. Szegö G 1939 Orthogonal Polynomials (American Mathematical Society Colloquium Publications vol 23) (Providence, RI: American Mathematical Society)
41. Taylor R and Wiles A 1995 Ring-theoretic properties of certain Hecke algebras Ann. Math. 141 553-72
42. Voros A 1987 Spectral functions, special functions and the Selberg zeta function Commun. Math. Phys. 110 439-65
43. Waldspurger J-L 1981 Sur les coefficients de Fourier des formes modulaires de poids demi-entier J. Math. Pures Appl. 60 375-484
44. Weyl H 1946 Classical Groups (Princeton: Princeton University Press)
45. Wiles A 1995 Modular elliptic curves and Fermat's last theorem Ann. Math. 141 443-551
46. Young M P 2004 Low-lying zeros of families of elliptic curves Preprint math-nt/0406330