In this paper we study local indices of systems of p-adic linearly differential equations which arise from p-adic representations of the absolute Galois group of local field of characteristic p with finite monodromy. We show the induction formula of the local index of p-adic differential equations and prove the equality between the local index of differential equations and the Swan conductor of p-adic Galois representations by inductive methods.

In the paper we find a further generalization of congruences of the K. Hardy and K. S. Williams [5] type which seems to be a full generalization of congruences of G. Gras [4]. Moreover we extend results of [5], [7], [8], [9] and in part of [6]. We apply ideas and methods of [.

We give a new proof of Shiota's theorem on Novikov's consecture, which states that the K.P. equation characters Jacobians among all indecomposable principally polarized abelian varieties.

In this paper we take into consideration a conjecture of Bloch equating in a suitable range and under some standard conjectures, the rank of the motivic cohomology of the special fiber of a semistable degeneration over the ring of integers of a number field with the order of zero of the local Euler factor of the L-function over the semistable prime corresponding to the special fiber. We then develop, following the philosophy that the fiber ‘at infinity’ of an arithmetic variety should be considered as ‘maximally degenerate’, a construction that goes in parallel to the one we use for the non-archimedean fiber. Namely, a definition of a double complex and a weighted operator N on it that plays the role of the logarithm of the (local) monodromy map at infinity. We like to see this archimedean theory as the analogue of the limiting mixed Hodge structure theory of a degeneration of projective varieties over a disc. In particular, this yields to a description of the (motivic) Deligne cohomology as homology of the mapping cone of N. The main result arising from this construction is the proof of a conjecture of Deninger. Namely we show that the archimedean Γ-factor of the Zeta-function of the archimedean fiber can be seen as the characteristic polynomial of an archimedean Frobenius acting on the subgroup of the invariants of N into the hypercohomology of our double complex.

Motivated by a result of Bost, we use the relationship between Faltings' heights of abelian varieties with complex multiplication and logarithmic derivatives of Artin L-functions at $s=0$ to investigate these heights. In particular, we prove that the height of an elliptic curve with complex multiplication by $Q√-d$ is bounded from below by an effective affine function of log d.