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 analytic theory of ternary cubic forms with O. Balkanova and D. Frolenkov (2025)
• A note on Cayley's one sentence paper to appear In Math. Intelligencer (2025)
• A Diophantine divisor problem and Hecke zeta functions with O. Balkanova and D. Frolenkov (2025)
• On the arithmetic of polynomials over a number field (2025)
• Integral and rational representation of forms Journal of Experimental Math. 1 (2025)
• 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.
• 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