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Original Article

Random Matrix Theory and the Fourier Coefficients of Half-Integral-Weight Forms

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Pages 67-82 | Published online: 30 Jan 2011

  • E. Baruch and Z. Mao. "Central Values of Automorphic L-Functions." Preprint, 2003.
  • E.B. Bogomolny and J.P. Keating. "Random Matrix Theory and the Riemann Zeros I: Three- and Four-Point Correlations." Nonlinearity 8 (1995), 1115–1131.
  • E.B. Bogomolny and J.P. Keating. "Random Matrix Theory and the Riemann Zeros II: n-Point Correlations." Nonlinearity, 9 (1996), 911–935.
  • C. Breuil, B. Conrad, F. Diamond, and R. Taylor."On the Modularity of Elliptic Curves over ℚ." J. Amer. Math. Soc. 14:4 (2001), 843–939.
  • J.B. Conrey. "L-Functions and Random Matrices." In Mathematics Unlimited—2001 and Beyond, pp. 331–352. Berlin: Springer, 2001.
  • J.B. Conrey and D.W. Farmer. "Mean Values of L-Functions and Symmetry." Int. Math. Res. Notices 17 (2000), 883–908.
  • J.B. Conrey, J.P. Keating, M.O. Rubinstein, and N.C. Snaith. "On the Frequency of Vanishing of Quadratic Twists of Modular L-Functions." In Number Theory for the Millennium I: Proceedings of the Millennial Conference on Number Theory, edited by, M. A. Bennett et al., pp. 301–315. Natick: A K Peters, Ltd., 2002.
  • J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein, and N.C. Snaith. "Autocorrelation of Random Matrix Polynomials." Comm. Math. Phys. 237:3 (2003), 365–395.
  • J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein, and N.C. Snaith. "Integral Moments of L-Functions." Proceedings of the London Mathematical Society 91 (2005), 33–104.
  • B. Conrey, A. Pokharel, M. Rubinstein, and M. Watkins. "Secondary Terms in the Number of Vanishings of Quadratic Twists of Elliptic Curve LFunctions." In Ranks of Elliptic Curves and Random Matrix Theory, edited by J. B. Conrey, D. W. Farmer, F. Mezzadri, and N. C. Snaith. Cambridge: Cambridge University Press, 2006.
  • J. Cremona. Algorithms for Modular Elliptic Curves. Available at http://www.maths.nott.ac.uk/personal/jec. http://www.maths.nott.ac.uk/personal/jec
  • C. Delaunay. "Heuristics on Tate–Shafarevitch Groups of Elliptic Curves Defined over ℚ." Experimental Math. 10:2 (2001), 191–196.
  • C. Delaunay. "Note on the Frequency of Vanishing of L-Functions of Elliptic Curves in a Family of Quadratic Twists." In Ranks of Elliptic Curves and Random Matrix Theory, edited by J.B. Conrey, D.W. Farmer, F. Mezzadri, and N.C. Snaith. Cambridge: Cambridge University Press, 2006.
  • C.P. Hughes, J.P. Keating, and N. O'Connell. "Random Matrix Theory and the Derivative of the Riemann Zeta Function." Proc. R. Soc. Lond. A 456 (2000), 2611–2627.
  • C.P. Hughes, J.P. Keating, and N. O'Connell. "On the Characteristic Polynomial of a Random Unitary Matrix." Commun. Math. Phys. 220:2 (2001), 429–451.
  • N.M. Katz and P. Sarnak. Random Matrices, Frobenius Eigenvalues and Monodromy, American Mathematical Society Colloquium Publications, 45. Providence: American Mathematical Society, 1999.
  • J.P. Keating and N.C. Snaith. "Random Matrix Theory and ζ(½ + it)." Commun. Math. Phys. 214 (2000), 57–89.
  • J.P. Keating and N.C. Snaith. "Random Matrix Theory and L-Functions at s = ½ ." Commun. Math. Phys. 214 (2000), 91–110.
  • J.P. Keating and N.C. Snaith. "Random matrices and L-functions." J. Phys. A 36:12 (2003), 2859–2881.
  • W. Kohnen and D. Zagier. "Values of L-Series of Modular Forms at the Center of the Critical Strip." Invent. Math. 64 (1981), 175–198.
  • Z. Mao, F. Rodriguez-Villegas, and G. Tornaria. "Computation of Central Value of Quadratic Twists of Modular L-Functions." In Ranks of Elliptic Curves and Random Matrix Theory, edited by J. B. Conrey, D. W. Farmer, F. Mezzadri, and N. C. Snaith, pp. 115–129. Cambridge: Cambridge University Press, 2006.
  • H.L. Montgomery. "The Pair Correlation of the Zeta Function." Proc. Symp. Pure Math. 24 (1973), 181–93.
  • A.M. Odlyzko, "The 1020th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint, 1989.
  • F. Rodriguez-Villegas and G. Tornaria. Private communication, 2004.
  • M. Rubinstein. "Evidence for a Spectral Interpretation of Zeros of L-functions." PhD diss., Princeton University, 1998.
  • M. Rubinstein. "The L-Function Database." Available at http://www.math.uwaterloo.ca/~mrubinst.
  • Z. Rudnick and P. Sarnak, "Zeros of Principal L-Functions and Random Matrix Theory." Duke Mathematical Journal 81:2 (1996), 269–322.
  • A. Selberg. "Contributions to the Theory of Dirichlet's L-Functions." Skr. Norske Vid. Akad. Oslo 1 (1946), 1–62.
  • A. Selberg. "Contributions to the Theory of the Riemann Zeta-Function." Arch. for Math. og Naturv. B 48:5), 1946.
  • G. Shimura. "On Modular Forms of Half Integral Weight." Ann. Math. 97:2 (1973), 440–481.
  • R. Taylor and A. Wiles. "Ring- Theoretic Properties of Certain Hecke Algebras." Ann. of Math. (2) 141:3 (1995), 553–572.
  • J.-L. Waldspurger. "Sur les coefficients de Fourier des formes modulaires de poids demi-entier." J. Math. Pures Appl. 60:9 (1981), 375–484.
  • A. Wiles. "Modular Elliptic Curves and Fermat's Last Theorem." Ann. of Math. (2) 141:3 (1995), 443–551.

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