Fourier coefficients of Jacobi Eisenstein series over number fields

Abstract

Abstract. In recent work we computed, for any totally real number field K with ring ofintegers o, the Fourier coefficients of the Jacobi Eisenstein series of integral weightand lattice index of rank one and with modified level one on SL(2,o) attached to thecusp at infinity. This result has a number of important consequences: it provides thefirst concrete example for the expected lift from Jacobi forms over K to Hilbertmodular forms, it shows that a Waldspurger type formula holds true in this concretecase (as also expected for the general lifting), and finally it gives us a clue for theHecke theory still to be developed by giving a concrete example for the action ofHecke operators on Fourier coefficients.In this talk we recall the basic notions of the theory of Jacobi forms over numberfields as developed in [BoBo], discuss the general theory of Jacobi Eisenstein seriesover number fields, and explain in more detail those points in the deduction of ourformulas which are not straight forward and require some new ideas. Finally wediscuss the indicated implications concerning the arithmetic theory of Jacobi formsover number fields.References: [BoBo] Boylan, H., "Jacobi forms, finite quadratic modules and Weilrepresentations over number fields", Lecture Notes in Mathematics, volume 2130,Springer International Publishing 2015.