Let $X/R$ be a smooth scheme over a ring $R$. Consider the category of locally free crystals of finite rank on the situs $\mathop{\mathit{Crys}}(X/W_{t}(R))$. We show that it is equivalent to the category of locally free $W_{t}(\mathcal{O}_{X})$-modules of finite rank endowed with a nilpotent, integrable de Rham-Witt connection. In the case where $R$ is a perfect field this was shown by Etesse [E] and Bloch [Bl]We use the result for a construction of the Gauß-Manin connection as a de Rham-Witt connection.

We show that isomorphisms between arithmetic fundamental groups of hyperbolic curves over $p$-adic local fields preserve the decomposition groups of all closed points (respectively, closed points arising from torsion points of the underlying elliptic curve), whenever the hyperbolic curves in question are isogenous to hyperbolic curves of genus zero defined over a number field (respectively, are once-punctured elliptic curves [which are not necessarily defined over a number field]). We also show that, under certain conditions, such isomorphisms preserve certain canonical ``integral structures'' at the cusps [i.e., points at infinity] of the hyperbolic curve.

In this article we derive analytic and Fourier aspects of a Kronecker limit formula for second-order Eisenstein series. Let $\Gamma$ be any Fuchsian group of the first kind which acts on the hyperbolic upper half-space $\mathbf{H}$ such that the quotient $\Gamma \backslash \mathbf{H}$ has finite volume yet is non-compact. Associated to each cusp of $\Gamma \backslash \mathbf{H}$, there is a classically studied {\it first-order\/} non-holomorphic Eisenstein series $E(s, z)$ which is defined by a generalized Dirichlet series that converges for $\Re(s) > 1$. The Eisenstein series $E(s, z)$ admits a meromorphic continuation with a simple pole at $s = 1$. Classically, Kronecker's limit formula is the study of the constant term $\mathcal{K}_{1}(z)$ in the Laurent expansion of $E(s, z)$ at $s = 1$. A number of authors recently have studied what is known as the {\it second-order\/} Eisenstein series $E^{\ast}(s, z)$, which is formed by twisting the Dirichlet series that defines the series $E(s, z)$ by periods of a given cusp form $f$. In the work we present here, we study an analogue of Kronecker's limit formula in the setting of the second-order Eisenstein series $E^{\ast}(s, z)$, meaning we determine the constant term $\mathcal{K}_{2}(z)$ in the Laurent expansion of $E^{\ast}(s, z)$ at its first pole, which is also at $s = 1$. To begin our investigation, we prove a bound for the Fourier coefficients associated to the first-order Kronecker limit function $\mathcal{K}_{1}$. We then define two families of convolution Dirichlet series, denoted by $L^{+}_{m}$ and $L^{-}_{m}$ with $m \in \mathbb{N}$, which are formed by using the Fourier coefficients of $\mathcal{K}_{1}$ and the weight two cusp form $f$. We prove that for all $m$, $L^{+}_{m}$ and $L^{-}_{m}$ admit a meromorphic continuation and are holomorphic at $s = 1$. Turning our attention to the second-order Kronecker limit function $\mathcal{K}_{2}$, we first express $\mathcal{K}_{2}$ as a solution to various differential equations. Then we obtain its complete Fourier expansion in terms of the cusp form $f$, the Fourier coefficients of the first-order Kronecker limit function $\mathcal{K}_{1}$, and special values $L^{+}_{m}(1)$ and $L^{-}_{m}(1)$ of the convolution Dirichlet series. Finally, we prove a bound for the special values $L^{+}_{m}(1)$ and $L^{-}_{m}(1)$ which then implies a bound for the Fourier coefficients of $\mathcal{K}_{2}$. Our analysis leads to certain natural questions concerning the holomorphic projection operator, and we conclude this paper by examining certain numerical examples and posing questions for future study.

We prove the modularity offour rigid and three nonrigid Calabi-Yau threefoldsassociated with the $A_{4}$ root lattice.

Let $M^{n}$ be an immersed umbilic-free hypersurface in the $(n+1)$-dimensional unit sphere $\mathbb{S}^{n+1}$, then $M^{n}$ is associated with a so-called Möbius metric $g$, a Möbius second fundamental form ${\bf B}$ and a Möbius form $\Phi$ which are invariants of $M^{n}$ under the Möbius transformation group of $\mathbb{S}^{n+1}$. A classical theorem of Möbius geometry states that $M^{n}$ $(n \geq 3)$ is in fact characterized by $g$ and ${\bf B}$ up to Möbius equivalence. A Möbius isoparametric hypersurface is defined by satisfying two conditions: (1) $\Phi \equiv 0$; (2) All the eigenvalues of ${\bf B}$ with respect to $g$ are constants. Note that Euclidean isoparametric hypersurfaces are automatically Möbius isoparametric, whereas the latter are Dupin hypersurfaces.

In this paper, we prove that a Möbius isoparametric hypersurface in $\mathbb{S}^{4}$ is either of parallel Möbius second fundamental form or Möbius equivalent to a tube of constant radius over a standard Veronese embedding of $\mathbb{R}P^{2}$ into $\mathbb{S}^{4}$. The classification of hypersurfaces in $\mathbb{S}^{n+1}$ $(n\ge2)$ with parallel Möbius second fundamental form has been accomplished in our previous paper \cite{6}. The present result is a counterpart of Pinkall's classification for Dupin hypersurfaces in $\mathbb{E}^{4}$ up to Lie equivalence.

H. Sato introduced a Schwarzian derivative of a contactomorphism of ${\mathbb{R}}^{3}$ and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a contact projective structure is defined to be a cocycle of the contactomorphism group taking values in the space of sections of a certain vector bundle associated to the contact structure, and measuring the extent to which a contactomorphism fails to be an automorphism of the contact projective structure. For the flat model contact projective structure, this gives a contact Schwarzian derivative associating to a contactomorphism of ${\mathbb{R}}^{2n-1}$ a tensor which vanishes if and only if the given contactomorphism is an element of the linear symplectic group acting by linear fractional transformation.