Jacobi Eisenstein series over number fields


Boylan H.

JOURNAL OF NUMBER THEORY, vol.248, pp.54-77, 2023 (SCI-Expanded)

  • Publication Type: Article / Article
  • Volume: 248
  • Publication Date: 2023
  • Doi Number: 10.1016/j.jnt.2023.01.007
  • Journal Name: JOURNAL OF NUMBER THEORY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, MathSciNet, zbMATH
  • Page Numbers: pp.54-77
  • Istanbul University Affiliated: Yes

Abstract

For any given totally real number field K, we compute the

Fourier developments of the Jacobi Eisenstein series over K

at the cusp at infinity. As main application we prove, for

any K with class number 1, that the L-series of the Jacobi

Eisenstein series of weight k ≥ 3 for indices with rank and

modified level 1 coincide with the L-series of the Eisenstein

series of weight 2k −2 on the full Hilbert modular group of K.

Moreover, under this correspondence the Fourier coefficients

of the Jacobi Eisenstein series are related to the twisted L-

series of the Hilbert Eisenstein series at the critical point by

a Waldspurger type identity. This is a first step in the proof

that Skoruppa’s and Zagier’s lifting from Jacobi forms over

Q to elliptic modular forms holds true over arbitrary totally

real number fields too.