Alice Silverberg

It is well-known that every finite subgroup of GL_d(Q_p) is conjugate to a subgroup of GL_d(Z_p). However, this does not remain true if we replace general linear groups by symplectic groups. In this talk I will report on joint work with Yuri Zarhin. We consider the question of when a finite subgroup of Sp_{2d}(Q_p) is conjugate in GL_{2d}(\Q_p) to a subgroup of Sp_{2d}(Z_p). We give sufficient conditions, and give examples which show that the bounds in our results are sharp. Although a finite subgroup of Sp_{2d}(Q_p) can fail to be conjugate in GL_{2d}(Q_p) to a subgroup of Sp_{2d}(Z_p), we prove that it can nevertheless be embedded in Sp_{2d}(F_p) in such a way that the characteristic polynomials are preserved (mod p), as long as p>3. We also give applications to polarizations of abelian varieties.