Appendix to "Congruences for rational points on varieties over finite   fields" by N. Fakhruddin and C. S. Rajan
  We give a positive answer to the questions 4.4 on Hodge type and 4.2 on divisibility of the eigenvalues of Frobenius of the article by N. Fakhruddin and C. S. Rajan (NT/0402230). 
H\'el\`ene Esnault