Bei Zhang (IAS)

Title: Nonvanishing mod p of Eisenstein series

Ribet's idea about the congruence argument between automorphic forms turns out to be useful in the proof towards one direction of Iwasawa main conjecture. In the proof, a p-integral Eisenstein series E needs to be constructed so that E is nonzero (mod p). I will recall in several cases (the GL₂ case is trivial, GSp(4) by Urban and U(2, 1) by Mainardi), how this problem was solved, then explain the difficulty of obtaining such result for an Eisenstein series on U(3, 1). At last, I will show the result we have so far through the calculation of Fourier-Jacobi coefficient of this Eisenstein series.