The fundamental germ is a generalization of $\pi_{1}$, first defined for laminations which arise through group actions [4]. In this paper, the fundamental germ is extended to any lamination having a dense leaf admitting a smooth structure. In addition, an amplification of the fundamental germ called the mother germ is constructed, which is, unlike the fundamental germ, a topological invariant. The fundamental germs of the antenna lamination and the $PSL(2, \mathbb{Z})$ lamination are calculated, laminations for which the definition in [4] was not available. The mother germ is used to give a new proof of a Nielsen theorem for the algebraic universal cover of a closed surface of hyperbolic type.

Let $K$ be a field, $S$ a polynomial ring and $E$ an exterior algebra over $K$, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in $S$ and $E$ when passing to their generic initial ideals. First, we prove that if the graded Betti numbers $\beta_{ii+k}^{S}(S/I) = \beta_{ii+k}^{S}(S/\operatorname{Gin}(I))$ for some $i > 1$ and $k \geq 0$, then $\beta_{qq+k}^{S}(S/I) = \beta_{qq+k}^{S}(S/\operatorname{Gin}(I))$ for all $q \geq i$, where $I \subset S$ is a graded ideal. Second, we show that if $\beta_{ii+k}^{E}(E/I) = \beta_{ii+k}^{E}(E/\operatorname{Gin}(I))$ for some $i > 1$ and $k \geq 0$, then $\beta_{qq+k}^{E}(E/I) = \beta_{qq+k}^{E}(E/\operatorname{Gin}(I))$ for all $q \geq 1$, where $I \subset E$ is a graded ideal. In addition, it will be shown that the graded Betti numbers $\beta_{ii+k}^{R}(R/I) = \beta_{ii+k}^{R}(R/\operatorname{Gin}(I))$ for all $i \geq 1$ if and only if $I_{\langle k \rangle}$ and $I_{\langle k+1 \rangle}$ have a linear resolution. Here $I_{\langle d \rangle}$ is the ideal generated by all homogeneous elements in $I$ of degree $d$, and $R$ can be either the polynomial ring or the exterior algebra.

We study the Fourier coefficients of cusp forms of half integral weight and generalize the Kohnen-Zagier formula to the case of `$4 \times$ general odd$' level by using results of Ueda. As an application, we obtain a generalization of the result of Luo-Ramakrishnan [11] to the case of arbitrary odd level.

In this paper we compute and continue meromorphically to the entire complex plane the zeta functions of twisted quaternionic Shimura varieties of dimension 3. The twist of the quaternionic Shimura varieties is done by a mod $\wp$ representation of the absolute Galois group.

Let $G$ be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic $p$. Assume that $p$ is good for $G$. In this note we consider particular classes of connected reductive subgroups $H$ of $G$ and show that the cocharacters of $H$ that are associated to a given nilpotent element $e$ in the Lie algebra of $H$ are precisely the cocharacters of $G$ associated to $e$ that take values in $H$. In particular, we show that this is the case provided $H$ is a connected reductive subgroup of $G$ of maximal rank; this answers a question posed by J. C. Jantzen.

Let $F$ be an algebraically closed field and let $G$ be a semisimple $F$-algebraic group for which the characteristic of $F$ is very good. If $X \in \operatorname{Lie}(G) = \operatorname{Lie}(G)(F)$ is a nilpotent element in the Lie algebra of $G$, and if $C$ is the centralizer in $G$ of $X$, we show that (i) the root datum of a Levi factor of $C$, and (ii) the component group $C/C^{o}$ both depend only on the Bala-Carter label of $X$; i.e. both are independent of very good characteristic. The result in case (ii) depends on the known case when $G$ is (simple and) of adjoint type.

The proofs are achieved by studying the centralizer $\mathcal{C}$ of a nilpotent section $X$ in the Lie algebra of a suitable semisimple group scheme over a Noetherian, normal, local ring $\mathcal{A}$. When the centralizer of $X$ is equidimensional on $\operatorname{Spec}(\mathcal{A})$, a crucial result is that locally in the étale topology there is a smooth $\mathcal{A}$-subgroup scheme $L$ of $\mathcal{C}$ such that $L_{t}$ is a Levi factor of $\mathcal{C}_{t}$ for each $t \in \operatorname{Spec}(\mathcal{A})$.

Let $C$ be a smooth irreducible complete curve of genus $g \geq 2$ over an algebraically closed field of characteristic $0$. An ample $K3$ extension of $C$ is a $K3$ surface with at worst rational double points which contains $C$ in the smooth locus as an ample divisor.

In this paper, we prove that all smooth curve of genera $2 \leq g \leq 8$ have ample $K3$ extensions. We use Bertini type lemmas and double coverings to construct ample $K3$ extensions.