In this paper, the author introduces the Morrey-Campanato spaces $L^{s}_{p}(X)$ and the spaces $C^{s}_{p}(X)$ on spaces of homogeneous type including metric spaces and some fractals, and establishes some embedding theorems between these spaces under some restrictions and the Besov spaces and the Triebel-Lizorkin spaces. In particular, the author proves that $L^{s}_{p}(X) = B^{s}_{\infty, \infty}(X)$ if $0<s<\infty$ and $\mu(X)<\infty$. The author also introduces some new function spaces $A^{s}_{p}(X)$ and $B^{s}_{p}(X)$ and proves that these new spaces when $0<s<1$ and $1<p<\infty$ are just the Triebel-Lizorkin space $F^{s}_{p, \infty}(X)$ if $X$ is a metric space, and the spaces $A^{1}_{p}(X)$ and $B^{1}_{p}(X)$ when $1<p\le\infty$ are just the Hajlasz-Sobolev spaces $W^{1}_{p}(X)$. Finally, as an application, the author gives a new characterization of the Hajlasz-Sobolev spaces by making use of the sharp maximal function.

We characterize the tight closure of a homogeneous primary ideal in a normal homogeneous coordinate ring over an elliptic curve by a numerical condition and we show that it is in positive characteristic the same as the plus closure.

In this paper, we investigate the lower bound $s_{HK}(p, d)$ of Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension $d$ containing a field of characteristic $p>0$. Especially, we focus on three-dimensional local rings. In fact, as a main result, we will prove that $s_{HK}(p, 3) = 4/3$ and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity $4/3$ is isomorphic to the non-degenerate quadric hypersurface $k[[X, Y, Z, W]]/(X^{2}+Y^{2}+Z^{2}+W^{2})$ under mild conditions.

Furthermore, we pose a generalization of the main theorem to the case of $\dim A \ge 4$ as a conjecture, and show that it is also true in case $\dim A = 4$ using the similar method as in the proof of the main theorem.

For an abelian totally real number field $F$ and an odd prime number $p$ which splits totally in $F$, we present a functorial approach to special "$p$-units" previously built by D. Solomon using "wild" Euler systems. This allows us to prove a conjecture of Solomon on the annihilation of the $p$-class group of $F$ (in the particular context here), as well as related annihilation results and index formulae.

We continue to study $l$-adic iterated integrals introduced in the first part. We shall show that the $l$-adic iterated integrals satisfy essentially the same functional equations as the classical complex iterated integrals. Next we are studying $l$-adic analogs of classical polylogarithms.

Fiber cones of $0$-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi's bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of $0$-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.