Arithmetic properties of p-adic and π-adic L-functions

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(1) I proved the Finiteness conjecture of Fontaine-Mazur on geometric representations in the potentially abelian case. A similar fact had been proved by Anderson-Blasius-Coleman-Zettler. Their theorem was on compatible systems of l-adic representations, whereas my theorem is on p-adic representations (on one prime p).(2) Jointly with Prof.T.Satoh, we established a fast algorithm for computing the trace of the Frobenius on the Dieudonne modules of ordinary formal groups over a finite field.(3) Jointly with H.Moon, we obtained some partial results on mod 7 semisimple Galois representations. According to a conjecture of Serre, there exists no odd and irreducible mod 7 Galois representations unramified outside 7. On the other hand, there is no known precise conjecture on even representations. Here, we proved the nonexistence of even irreducible mod 7 Galois representations unramified outside 7.