1.
Analytic log Picard varieties - Kajiwara, Takeshi; Kato, Kazuya; Nakayama, Chikara
We introduce a log Picard variety over the complex number field by the method of log geometry in the sense of Fontaine-Illusie, and study its basic properties, especially, its relationship with the group of log version of $\mathbb{G}_{m}$-torsors.

2.
Most of the maps near the exponential are hyperbolic - Wang, Xiumei; Zhang, Gaofei
Let $f_{\lambda}(z) = \lambda e^{z}$. In this short note, we consider those maps $f_{\lambda}$ with $\lambda$ close to $1$. We show that the probability that $f_{\lambda}$ is hyperbolic approaches $1$ as $\lambda \to 1$.

3.
Algebraic surfaces of general type with small ${c}_{1}^{2}$ in positive characteristic - Liedtke, Christian
We establish Noether's inequality for surfaces of general type in positive characteristic. Then we extend Enriques' and Horikawa's classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

4.
The filtered Poincaré lemma in higher level (with applications to algebraic groups) - Le Stum, Bernard; Quirós, Adolfo
We show that the Poincaré lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincaré lemma for transversal crystals of level $m$. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be...

5.
On certain Rankin-Selberg integrals on $GE_{6}$ - Ginzburg, David; Hundley, Joseph
In this paper we begin the study of two Rankin-Selberg integrals defined on the exceptional group of type $GE_{6}$. We show that each factorizes and that the contribution from the unramified places is, in one case, the degree 54 Euler product $L^{S}(\pi \times \tau, E_{6} \times GL_{2}, s)$ and in the other case the degree 30 Euler product $L^{S}(\pi \times \tau, \wedge^{2} \times GL_{2}, s)$.

6.
Some numerical criteria for the Nash problem on arcs for surfaces - Morales, Marcel
Let $(X, O)$ be a germ of a normal surface singularity, $\pi : \tilde X \to X$ be the minimal resolution of singularities and let $A = (a_{i, j})$ be the $n \times n$ symmetrical intersection matrix of the exceptional set of $\tilde X$. In an old preprint Nash proves that the set of arcs on a surface singularity is a scheme $\mathcal{H}$, and defines a map $\mathcal{N}$ from the set of irreducible components of $\mathcal{H}$ to the set of exceptional components of the minimal resolution of singularities of $(X, O)$. He proved that this map is injective and ask...

7.
Every curve of genus not greater than eight lies on a $K3$ surface - Ide, Manabu
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.

8.
The centralizer of a nilpotent section - McNinch, George J.
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...

9.
On cocharacters associated to nilpotent elements of reductive groups - Fowler, Russell; Röhrle, Gerhard
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...

10.
On $L$-functions of twisted 3-dimensional quaternionic Shimura varieties - Virdol, Cristian
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.

11.
On the Kohnen-Zagier formula in the case of `$4 \times$ general odd' level - Sakata, Hiroshi
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.

12.
Rigidity of linear strands and generic initial ideals - Murai, Satoshi; Singla, Pooja
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...

13.
The geometric theory of the fundamental germ - Gendron, T. M.
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...

14.
Analytic jet parametrization for CR automorphisms of some essentially finite CR manifolds - Kim, Sung-Yeon
In this paper we construct analytic jet parametrizations for the germs of real analytic CR automorphisms of some essentially finite CR manifolds on their finite jet at a point. As an application we show that the stability groups of such CR manifolds have Lie group structure under composition with the topology induced by uniform convergence on compacta.

15.
Hecke's integral formula for relative quadratic extensions of algebraic number fields - Yamamoto, Shuji
Let $K/F$ be a quadratic extension of number fields. After developing a theory of the Eisenstein series over $F$, we prove a formula which expresses a partial zeta function of $K$ as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker's type which relates the $0$-th Laurent coefficients at $s=1$ of zeta functions of $K$ and $F$.

16.
Logarithmic abelian varieties - Kajiwara, Takeshi; Kato, Kazuya; Nakayama, Chikara
We develop the algebraic theory of log abelian varieties. This is Part II of our series of papers on log abelian varieties, and is an algebraic counterpart of the previous Part I ([6]), where we developed the analytic theory of log abelian varieties.

17.
Symmetry on linear relations for multiple zeta values - Ihara, Kentaro; Ochiai, Hiroyuki
We find a symmetry for the reflection groups in the double shuffle space of depth three. The space was introduced by Ihara, Kaneko and Zagier and consists of polynomials in three variables satisfying certain identities which are connected with the double shuffle relations for multiple zeta values. Goncharov has defined a space essentially equivalent to the double shuffle space and has calculated the dimension. In this paper we relate the structure among multiple zeta values of depth three with the invariant theory for the reflection groups and discuss the dimension of the double shuffle space in this view point.

18.
Hartogs type theorems for $CR$ $L^{2}$ functions on coverings of strongly pseudoconvex manifolds - Brudnyi, Alexander
We prove an analog of the classical Hartogs extension theorem for $CR$ $L^{2}$ functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a question formulated in the paper of Gromov, Henkin and Shubin [GHS] on holomorphic $L^{2}$ functions on coverings of pseudoconvex manifolds.

19.
Direct summands of syzygy modules of the residue class field - Takahashi, Ryo
Let $R$ be a commutative Noetherian local ring. This paper deals with the problem asking whether $R$ is Gorenstein if the $n$th syzygy module of the residue class field of $R$ has a non-trivial direct summand of finite G-dimension for some $n$. It is proved that if $n$ is at most two then it is true, and moreover, the structure of the ring $R$ is determined essentially uniquely.

20.
Generalized Green functions and unipotent classes for finite reductive groups, II - Shoji, Toshiaki
This paper is concerned with the problem of the determination of unknown scalars involved in the algorithm of computing the generalized Green functions of reductive groups $G$ over a finite field. In the previous paper, we have treated the case where $G = SL_{n}$. In this paper, we determine the scalars in the case where $G$ is a classical group $Sp_{2n}$ or $SO_{N}$ for arbitrary characteristic.