Fourier coefficients of Jacobi Eisenstein series over number fields

Intercity Number Theory Seminar, Utrecht, Hollanda, 24 Mart 2017, ss.2

Abstract. In recent work we computed, for any totally real number field K with ring of

integers o, the Fourier coefficients of the Jacobi Eisenstein series of integral weight

and lattice index of rank one and with modified level one on SL(2,o) attached to the

cusp at infinity. This result has a number of important consequences: it provides the

first concrete example for the expected lift from Jacobi forms over K to Hilbert

modular forms, it shows that a Waldspurger type formula holds true in this concrete

case (as also expected for the general lifting), and finally it gives us a clue for the

Hecke theory still to be developed by giving a concrete example for the action of

Hecke operators on Fourier coefficients.

In this talk we recall the basic notions of the theory of Jacobi forms over number

fields as developed in [BoBo], discuss the general theory of Jacobi Eisenstein series

over number fields, and explain in more detail those points in the deduction of our

formulas which are not straight forward and require some new ideas. Finally we

discuss the indicated implications concerning the arithmetic theory of Jacobi forms

over number fields.

References: [BoBo] Boylan, H., "Jacobi forms, finite quadratic modules and Weil

representations over number fields", Lecture Notes in Mathematics, volume 2130,

Springer International Publishing 2015.