Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 35 min ago
PreprintarXiv licencearXiv

A Kohnen-Zagier Formula in Level 1

  • Clayton Williams
Open at sourcePDFDOICite

Abstract

In 1985 Kohnen used the Shimura correspondence to prove a general formula relating the coefficients of half-integral weight modular forms of level $ 4N $ transforming with respect to the theta multiplier to central values of $ L $-functions for twists of newforms of level $ N $. This generalized previous work with Zagier. In this paper, we prove a Kohnen-Zagier type formula using Yang's Shimura correspondence for powers of the eta multiplier in level $ 1 $, relating the coefficients of half-integral weight modular forms to central values of $ L $-functions for twists of newforms of level $ 6 $. To do so we develop some of the theory of modular forms transforming with respect to powers of the eta multiplier and introduce Shintani lifts of level $ 6 $, with Fourier coefficients given in terms of cycle integrals.

Related

  1. Zagier's 2-3-2 formula for multiple Eisenstein series
    Henrik Bachmann
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. Jacquet-Zagier treatment of the beyond endoscopy trace formula for GL2
    Pranjal Pandurang Warade
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Tree Lattices and the Neumann-Zagier conjecture
    Joshua Evan Greene, Ahmad Issa, Duncan McCoy
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsGeometry & topologyarXiv
  4. Value-distribution of Euler--Zagier type multiple zeta-functions with identical arguments on the critical line
    Kenta Endo
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Weighted Generalizations of Zagier's Phenomenon
    Dragomir Grozev, Navid Safaei
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{williams2026kohnenzagier,
  title = {{A Kohnen-Zagier Formula in Level 1}},
  author = {Clayton Williams},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38051},
  eprint = {2610.38051},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38051},
}
RIS
TY  - UNPB
TI  - A Kohnen-Zagier Formula in Level 1
AU  - Clayton Williams
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.38051
UR  - https://arxiv.org/abs/2610.38051
AB  - In 1985 Kohnen used the Shimura correspondence to prove a general formula relating the coefficients of half-integral weight modular forms of level $ 4N $ transforming with respect to the theta multiplier to central values of $ L $-functions for twists of newforms of level $ N $. This generalized previous work with Zagier. In this paper, we prove a Kohnen-Zagier type formula using Yang's Shimura correspondence for powers of the eta multiplier in level $ 1 $, relating the coefficients of half-integral weight modular forms to central values of $ L $-functions for twists of newforms of level $ 6 $. To do so we develop some of the theory of modular forms transforming with respect to powers of the eta multiplier and introduce Shintani lifts of level $ 6 $, with Fourier coefficients given in terms of cycle integrals.
ER  -