William Duke

      UCLA Mathematics Dept.

      • email: wdduke at g.ucla.edu or wdduke at ucla.edu (DO NOT USE wdduke at math.ucla.edu. I wont see it.)



        • Math Reviews (You must have a subscription to view)

          • Papers by W. Duke 


      • On the arithmetic of polynomials over a number field (2024)
      • A restricted divisor problem (2024)
      • Integral and rational representation of forms (2023) to appear in Journal of Experimental Math.
      • On a method of Hurwitz and its applications with O. Imamoglu and A. Toth, Advances in Math. 458, (2024).
      • Ramanujan and modular forms in Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence (2024).
      • Sums of fractional parts Acta Arithmetica (Iwaniec birthday volume) 214 (2024), 173--177
      • Higher Rademacher symbols Experimental Math 33, no. 4, (2024) (published online 2023)
      • On the analytic theory of isotropic ternary quadratic forms Appendix by R. Schulze-Pillot, J. d'Analyse Math., 151 (2023), 115-137. Part II
      • Non-convex geometry of numbers and continued fractions with N. Andersen, Z. Hacking and A. Woodall, Funct. Approx. Comment. Math. 69, no.2. (2023) 137--159
      • Some arithmetic applications of invariant theory   Proceedings of the Indian Academy of Sciences 132, (2022) Paper No. 51, 12 pp.
      • A Diophantine problem about Kummer surfaces Essential Number Theory 1, (2022), 51--56.
      • Asymptotic distribution of traces of singular moduli with N. Andersen   Discrete Anal., Paper No. 4, 14 pp. (2022)
      • On elliptic curves and binary quartic forms   IMRN (2022), no. 23, 19078–19104.
      • The index of a modular differential operator Pacific J. Math. 315 (2021), no. 1, 45–73. dedicated to Raja Varadarajan 
      • On a theorem of Davenport and Schmidt with N. Andersen, Acta Arithmetica 198 (2021), no. 1, 37–75. 
      • On a class number formula of Hurwitz with O. Imamoglu and A. Toth,   JEMS, 23, Issue 12, (2021), pp. 3995–4008.
      • The Minkowski chain and Diophantine approximation, with N. Andersen, J. Théor. Nombres Bordeaux 32 (2020), no. 2, 503–523. 
      • Modular invariants for real quadratic fields and Kloosterman sums (+ corrections) with N. Andersen, J. Algebra and Number Theory, Vol. 14 (2020), No. 6, 1537–1575  
      • Riesz means of certain arithmetic functions with H. N. Nguyen, Journal of Number Theory, 210, (2020), 132-141.  
      • Markov spectra for modular billiards with N. Andersen, Math. Ann. 373 (2019), no. 3-4, 1151–1175.  
      • Kronecker's first limit formula, revisited with O. Imamoglu and A. Toth..In the Zagier Issue, Res. Math. Sci. 5 (2018), no. 2, Paper No. 20, 21 pp.  
      • Fourier series of modular graph functions with Eric D'Hoker, J. Number Theory 192 (2018), 1–36.  
      • Linking numbers and modular cocycles with O. Imamoglu and A. Toth. Duke Math J. 166 (2017), no. 6, 1179-1210. 
      • Geometric invariants for real quadratic fields with O. Imamoglu and A. Toth. Annals of Math. 184 (2016), no. 3, 949-990. 
      • Regularized inner products of modular functions with O. Imamoglu and A. Toth, Ramanujan J. 41 (2016), no. 1-3, 13-29. 
      • Harmonic Maass forms of weight one with Y. Li, Duke Math. J. 164 (2015), no. 1, 39-113. 
      • The graphic nature of Gaussian periods with S.R. Garcia and B. Lutz, Proc. Amer. Math. Soc. 143 (2015), no. 5, 1849-1863. 
      • Infinite products of cyclotomic polynomials with Ha Nguyen, Bull. Aust. Math. Soc. 91 (2015), no. 3, 400-411. 
      • Almost a century of answering the question: What is a mock theta function? Notices Amer. Math. Soc. 61 (2014), no. 11, 1314-1320. 
      • Weyl sums for quadratic roots with J. Friedlander and H. Iwaniec , Int Math Res Notices (2012) Vol. 2012 2493-2549 (see Corrections Not erratum to corrections!)
      • Cycle integrals of the j-function and mock modular forms with O. Imamoglu and A. Toth, Ann. of Math. (2) 173 (2011), no. 2, 947-981. 
      • Real quadratic analogues of traces of singular invariants with O. Imamoglu and A. Toth , Int. Math. Res. Not. IMRN 2011, no. 13, 3082-3094 
      • Rational period functions and cycle integrals with O. Imamoglu and A. Toth , Abh. Math. Semin. Univ. Hambg. 80 (2010), no. 2, 255-264 
      • Integral traces of singular values of weak Maass forms with P. Jenkins, Algebra and Number Theory, 2 (2008), 573-593. 
      • The zeros of the Weierstrass P--function and hypergeometric series, with O. Imamoglu,Math. Ann. 340 (2008), no. 4, 897--905. 
      • Some entries in Ramanujan's notebooks, Proc. Cambridge Phil. Soc. 144, 255 (2008) 
      • On the zeros and coefficients of certain weakly holomorphic modular forms with P. Jenkins, Pure Appl. Math. Q. 4 (2008), no.~4, 1327--1340. 
      • A combinatorial problem related to Mahler's measure Bull. Lond. Math. Soc. 39 (2007), no. 5, 741--748. 
      • On a formula of Bloch, with O. Imamoglu, the Deshouillers Birthday Volume Funct. Approx. Comment. Math. 37 (2007) , part 1, 109--117. 
      • An introduction to the Linnik problems, in Equidistribution in Number Theory, an Introduction, Springer, (2007), 197-216. 
      • Special values of multiple gamma functions, with O. Imamoglu, J. Théorie des Nombres Bordeaux 18 (2006), 113--123. 
      • Modular functions and the uniform distribution of CM points, Math. Ann. 334 (2006), 241--252 
      • Continued fractions and modular functions , Bull. Amer. Math. Soc. (N.S.) 42 (2005), 137--162 
      • Quadratic reciprocity in a finite group, with K. Hopkins, Amer. Math. Monthly 112 (2005), no. 3, 251--256. 
      • On ternary quadratic forms , J. Number Theory 110 (2005), no. 1, 37--43. 
      • Lattice points in cones and Dirichlet series, with O. Imamoglu, IMRN 53 (2004) 
      • Number fields with large class groups, in Number Theory, CNTA VII, CRM Proceedings, 2004 
      • Reductions of an elliptic curve and their Tate-Shafarevich groups , with A. Cojocaru, Math. Annalen 329, 513-534 (2004) 
      • Extreme values of Artin L-functions and class numbers , Compositio Math. 136, 103-115 (2003) 
      • Almost all reductions of an elliptic curve have a large exponent, Comptes Rendus. 337/11, 689-692 (2003) 
      • Rational points on the sphere , The Ramanujan Journal 7 , 235-239 (2003) 
      • The splitting of primes in division fields of elliptic curves , with A. Toth, Experimental Math. 11,555-565 (2002) .
      • The subconvexity problem for Artin L–functions, with J. Friedlander and H. Iwaniec, Inventiones Math. 149,489-577 (2002) .
      • Bounds for automorphic L–functions. III, with J. Friedlander and H. Iwaniec, Inventiones Math. 148, 221-248 (2001) .
      • A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, with E. Kowalski, Inventiones Math. 139, 1-39 (2000).
      • When is the product of two Hecke eigenforms an eigenform?,in Number theory in progress, Vol. 2, 737–741 (1999).
      • Automorphic L-functions in level aspect ,in Number theory (Ottawa, ON, 1996), 73–79, CRM Proc. Lecture Notes, (1999).
      • Probabilistic Galois theory of reciprocal polynomials , with S. Davis and X. Sun, Expositiones Math. 16, 263-270 (1998).
      • Siegel modular forms of small weight , with O. Imamoglu, Math. Annalen 310, 73-82 (1998).
      • A nonnegativity restriction: a Monthly problem, American Math. Monthly 105, 469-470 (1998).
      • Properties of a quadratic Fibonacci recurrence. , with S. Greenfield and E. Speer, J. Integer Seq. 1 (1998), Article 98.1.8.
      • Bounds for arithmetic multiplicities. ICM Berlin 1998
      • Some old problems and new results about quadratic forms, Notices Amer. Math. Soc. 44 (1997), no. 2, 190–196.
      • The distribution of the eigenvalues of Hecke operators, with J.B. Conrey and D. Farmer, Acta Arith. LXXVIII.4 405-409 (1997).
      • Elliptic curves with no exceptional primes, C.R. Acad. Sci. 325,813-818 (1997) 405-409.
      • Bilinear forms with Kloosterman fractions, with J. Friedlander and H. Iwaniec, Inventiones Math. 128, 23-43 (1997) .
      • A converse theorem and the Saito-Kurokawa lift., with With O. Imamoglu, IMRN 1996, no. 7, 347–355.
      • The critical order of vanishing of automorphic L-functions with large level , Invent. Math. 119 (1995), no.1, 165–174.
      • Equidistribution of roots of a quadratic congruence to prime moduli., with J. Friedlander and H. Iwaniec, Ann. of Math. (2) 141 (1995), no. 2, 423–441.
      • Representations by the determinant and mean values of L-functions., with J. Friedlander and H. Iwaniec, Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), 109–115.
      • Class group L-functions, with J. Friedlander and H. Iwaniec, Duke Math. J. 79 (1995), no. 1, 1–56..
      • The dimension of the space of cusp forms of weight one. , IMRN 1995, no. 2, 99–109.
      • Bounds for automorphic L–functions. II , with J. Friedlander and H. Iwaniec, Invent. Math. 115 (1994), no. 2, 219–239 (see Erratum for Bounds for automorphic L–functions. II
      • Convolution L-series, with H. Iwaniec, Compositio Math. 91, 145-158 (1994) .
      • Density of integer points on affine homogeneous varieties, with Z. Rudnick and P. Sarnak, Duke Math. J. 71, 143-179 (1993) .
      • On codes and Siegel modular forms,IMRN 1993, no. 5, 125–136.
      • A relation between cubic exponential and Kloosterman sums., with H. Iwaniec, A tribute to Emil Grosswald: number theory and related analysis, 255–258, 1993.  
      • Bounds for automorphic L-functions, with J. Friedlander and H. Iwaniec, Invent. Math. 115 (1994), no. 2, 219–239 (see the end of the paper II of this series for some corrections.)  
      • A quadratic divisor problem. with J. Friedlander and H. Iwaniec, Invent. Math. 115 (1994), no. 2, 209–217.  
      • Estimates for coefficients of L-functions. IV., with H. Iwaniec, Amer. J. Math. 116 (1994), no. 1, 207–217.  
      • An estimate for the Hecke eigenvalues of Maass forms. , with D. Bump, J. Hoffstein and H. Iwaniec, IMRN 1992, no. 4, 75–81.  
      • On multiple Salié sums. Proc. Amer. Math. Soc. 114 (1992), no. 3, 623–625. 
      • Estimating Hecke eigenvalues of Siegel modular forms. , with R. Howe and J.S.Li, Duke Math. J. 67 (1992), no. 1, 219–240. 
      • Sums over primes of the Fourier coefficients of half-integral weight cusp forms. with H. Iwaniec, Journées Arithmétiques, 1989 (Luminy, 1989). Astérisque No. 198-200 (1991), 121–125 (1992). 
      • Bilinear forms in the Fourier coefficients of half-integral weight cusp forms and sums over primes, with H. Iwaniec, Math. Ann.286(1990), no.4, 783–802.  
      • Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, with R. Schulze-Pillot, Inventiones Math. 99 (1990),49-57.  
      • Estimates for coefficients of L-functions. III., with H. Iwaniec, Séminaire de Théorie des Nombres, Paris, 1989–90, 113–120.  
      • Estimates for coefficients of L-functions. II., with H. Iwaniec, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), 71–82, Salerno, 1992., 43–47. 
      • Estimates for coefficients of L-functions. I., with H. Iwaniec, Automorphic forms and analytic number theory (Montreal, PQ, 1989), 43–47.  
      • Some problems in multidimensional analytic number theory, Acta Arithmetica LII 203-228 (1989). 
      • Lattice points on ellipsoids, Séminaire de Théorie des Nombres, 1987–1988
      • Fourth moments of L-functions attached to newforms,Comm. Pure Appl. Math. 41 (1988), no. 6, 815–831 .
      • Hyperbolic distribution problems and Maass forms, Inventiones Math. 92, 73-90 (1988) .


    Miscellaneous

      • Birthday conference at ETH 2019