Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-05T16:03:00.037Z Has data issue: false hasContentIssue false
  • English
  • Français

Singular moduli and the distribution of partition ranks modulo 2

Published online by Cambridge University Press:  18 December 2015

RIAD MASRI
RIAD MASRI*
Affiliation:
Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843-3368, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we prove an asymptotic formula with a power saving error term for traces of weight zero weakly holomorphic modular forms of level N along Galois orbits of Heegner points on the modular curve X0(N). We use this result to study the distribution of partition ranks modulo 2. In particular, we give an asymptotic formula with a power saving error term for the number of partitions of a positive integer n with even (respectively, odd) rank. We use these results to deduce a strong quantitative form of equidistribution of partition ranks modulo 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 2 , March 2016 , pp. 209 - 232
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Al]Alfes, C.Formulas for the coefficients of half-integral weight harmonic Maass forms. Math. Z. 277 (2014), 769–795.CrossRefGoogle Scholar
[An]Andrews, G. E.On the theorems of Watson and Dragonette for Ramanujan's mock theta functions. Amer. J. Math. 88 (1966), 454–490.CrossRefGoogle Scholar
[BH]Blomer, V. and Harcos, G.Hybrid bounds for twisted L-functions. J. Reine Angew. Math. 621 (2008), 53–79.Google Scholar
[BO]Bringmann, K. and Ono, K.The f(q) mock theta function conjecture and partition ranks. Invent. Math. 165 (2006), 243–266.CrossRefGoogle Scholar
[BJO]Bruinier, J. H., Jenkins, P. and Ono, K.Hilbert class polynomials and traces of singular moduli. Math. Ann. 334 (2006), 373–393.CrossRefGoogle Scholar
[BrO]Bruinier, J. H. and Ono, K.Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms. Adv. Math. 246 (2013), 198–219.CrossRefGoogle Scholar
[C]Cox, D. A.Primes of the Form x2 + ny2. Fermat, Class Field Theory, and Complex Multiplication. Second edition. Pure and Applied Mathematics (Hoboken). (John Wiley and Sons, Inc., Hoboken, NJ, 2013), xviii+356 pp.CrossRefGoogle Scholar
[D]Duke, W.Modular functions and the uniform distribution of CM points. Math. Ann. 334 (2006), 241–252.CrossRefGoogle Scholar
[Dy]Dyson, F.Some guesses in the theory of partitions. Eureka (Cambridge) 8 (1944), 10–15.Google Scholar
[GKZ]Gross, B., Kohnen, W. and Zagier, D.Heegner points and derivatives of L–series. II. Math. Ann. 278 (1987), 497–562.CrossRefGoogle Scholar
[GZ]Gross, B. and Zagier, D.Heegner points and derivatives of L–series. Invent. Math. 84 (1986), 225–320.CrossRefGoogle Scholar
[HM]Harcos, G. and Michel, P.The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points. II. Invent. Math. 163 (2006), 581–655.CrossRefGoogle Scholar
[HR]Hardy, G. H. and Ramanujan, S.Une formule asymptotique pour le nombre des partitions de n. Comptes Rendus 2 (1917). Found in Collected Papers of Srinivasa Ramanujan (AMS Chelsea Publ., Providence, RI, 2000), 239–241.Google Scholar
[HB]Heath-Brown, D. R.Hybrid bounds for Dirichlet L-functions. II. Quart. J. Math. Oxford Ser. 31 (1980), 157–167.CrossRefGoogle Scholar
[HL]Hoffstein, J. and Lockhart, P.Coefficients of Maass forms and the Siegel zero. With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman. Ann. of Math. 140 (1994), 161–181.CrossRefGoogle Scholar
[I]Iwaniec, H.Introduction to the spectral theory of automorphic forms. Biblioteca de la Revista Matemática Iberoamericana (Revista Matematica Iberoamericana, Madrid, 1995). xiv+247 pp.Google Scholar
[I2]Iwaniec, H.Topics in Classical Automorphic Forms. Graduate Studies in Math. 17 (American Mathematical Society, Providence, RI, 1997) xii+259 pp.CrossRefGoogle Scholar
[IK]Iwaniec, H. and Kowalski, E.Analytic Number Theory. American Mathematical Society Colloquium Publications, 53 (American Mathematical Society, Providence, RI, 2004), xii+615 pp.Google Scholar
[M]Masri, R.Fourier coefficients of harmonic weak Maass forms and the partition function. Amer. J. Math. 137 (2015), 1061–1097.CrossRefGoogle Scholar
[O]Ono, K.Unearthing the visions of a master: harmonic Maass forms and number theory. Current Developments in Math. 2008 (Int. Press, Somerville, MA, 2009), 347–454.Google Scholar
[R]Ramanujan, S., The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988).Google Scholar
[W]Waldspurger, J.-L.Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Composition Math. 54 (1985), 173–242.Google Scholar
[Za]Zagier, D.Traces of singular moduli. Motives, Polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), 211–244. Int. Press Lect. Ser. 3, I (Int. Press, Somerville, MA, 2002).Google Scholar
[Za2]Zagier, D.Ramanujan's Mock Theta Functions and Their Applications (after Zwegers and Bringmann–Ono). Séminaire Bourbaki. Vol. 2007/2008. Astérisque 326 (2009), Exp. No. 986 (2010), vii–viii, 143–164.Google Scholar
[Zh]Zhang, S.Gross–Zagier formula for GL 2. Asian J. Math. 5 (2001), 183–290.CrossRefGoogle Scholar