Congruence properties of Borcherds product exponents

Keenan Monks, Sarah Peluse, Lynnelle Ye

Research output: Contribution to journalArticlepeer-review


In his striking 1995 paper, Borcherds [Automorphic forms on O s+2,2(ℝ) and infinite products, Invent. Math. 120 (1995) 161-213] found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant -d evaluated at the modular j-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, A(n,d), are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of A(n,d) modulo a prime ℓ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial.

Original languageEnglish (US)
Pages (from-to)1563-1578
Number of pages16
JournalInternational Journal of Number Theory
Issue number6
StatePublished - Sep 2013

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


  • Borcherds product
  • Hilbert class polynomial
  • modular forms
  • modular j-function


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