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 language | English |
|---|---|
| Pages (from-to) | 185-227 |
| Number of pages | 43 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 373 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2020 Jan 1 |
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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 journal › Article
}
TY - JOUR
T1 - Limits of traces of singular moduli
AU - Choi, Dohoon
AU - Lim, Subong
PY - 2020/1/1
Y1 - 2020/1/1
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.
AB - 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.
KW - Eichler-Shimura cohomology theory
KW - Modular traces
KW - Regularized L-functions
UR - http://www.scopus.com/inward/record.url?scp=85077387338&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85077387338&partnerID=8YFLogxK
U2 - 10.1090/tran/7890
DO - 10.1090/tran/7890
M3 - Article
AN - SCOPUS:85077387338
VL - 373
SP - 185
EP - 227
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 1
ER -