Limits of traces of singular moduli

Dohoon Choi, Subong Lim

Research output: Contribution to journalArticle

Abstract

Let f and g be weakly holomorphic modular functions on Γ0(N) with the trivial character. For an integer d, let Trd(f) denote the modular trace of f of index d. Let r be a rational number equivalent to i∞ under the action of Γ0(4N). In this paper, we prove that when z goes radially to r, the limit QH (f)(r) of the sum H(f)(z) =Σd>0 Trd(f)e2πidz is a special value of a regularized twisted L-function defined by Trd(f) for d ≤ 0. It is proved that the regularized L-function is meromorphic on C and satisfies a certain functional equation. Finally, under the assumption that N is square free, we prove that if QH(f)(r) = QH (g)(r) for all r equivalent to i∞ under the action of Γ0(4N), then Trd(f) = Trd(g) for all integers d.

Original languageEnglish
Pages (from-to)185-227
Number of pages43
JournalTransactions of the American Mathematical Society
Volume373
Issue number1
DOIs
Publication statusPublished - 2020 Jan 1

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L-function
Modulus
Trace
Modular Functions
Integer
Square free
Meromorphic
Functional equation
Analytic function
Trivial
Denote
Character

Keywords

  • Eichler-Shimura cohomology theory
  • Modular traces
  • Regularized L-functions

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Limits of traces of singular moduli. / Choi, Dohoon; Lim, Subong.

In: Transactions of the American Mathematical Society, Vol. 373, No. 1, 01.01.2020, p. 185-227.

Research output: Contribution to journalArticle

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N2 - Let f and g be weakly holomorphic modular functions on Γ0(N) with the trivial character. For an integer d, let Trd(f) denote the modular trace of f of index d. Let r be a rational number equivalent to i∞ under the action of Γ0(4N). In this paper, we prove that when z goes radially to r, the limit QH (f)(r) of the sum H(f)(z) =Σd>0 Trd(f)e2πidz is a special value of a regularized twisted L-function defined by Trd(f) for d ≤ 0. It is proved that the regularized L-function is meromorphic on C and satisfies a certain functional equation. Finally, under the assumption that N is square free, we prove that if QH(f)(r) = QH (g)(r) for all r equivalent to i∞ under the action of Γ0(4N), then Trd(f) = Trd(g) for all integers d.

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