This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.

Let B(\Omega )$B(\Omega )$ be a Banach space of holomorphic functions on a bounded connected domain \Omega $\Omega$ in {{\mathbb C}^n}${{\mathbb C}^n}$. In this paper, we establish a criterion for B(\Omega )$B(\Omega )$ to be reflexive via evaluation functions on B(\Omega )$B(\Omega )$, that is, B(\Omega )$B(\Omega )$ is reflexive if and only if the evaluation functions span the dual space (B(\Omega ))^{*} $(B(\Omega ))^{*}$.

We study the asymptotic behaviour of the least energy solutions to the following class of nonlocal Neumann problems:

\begin{align*} \begin{cases} { d(-\Delta)^{s}u+ u= \vert u\vert^{p-1}u } & \text{in } \Omega, \\ {u>0} & \text{in } \Omega, \\ { \mathcal{N}_{s}u=0 } & \text{in } \mathbb{R}^{n}\setminus \overline{\Omega}, \end{cases} \end{align*} $$\begin{align*} \begin{cases} { d(-\Delta)^{s}u+ u= \vert u\vert^{p-1}u } & \text{in } \Omega, \\ {u>0} & \text{in } \Omega, \\ { \mathcal{N}_{s}u=0 } & \text{in } \mathbb{R}^{n}\setminus \overline{\Omega}, \end{cases} \end{align*}$$

where \Omega \subset \mathbb {R}^{n}$\Omega \subset \mathbb {R}^{n}$ is a bounded domain of class C^{1,1}$C^{1,1}$, 1<p<({n+s})/({n-s}),\,n>\max \{1, 2s \}, 0<s<1, d>0$1<p<({n+s})/({n-s}),\,n>\max \{1, 2s \}, 0<s<1, d>0$ and \mathcal {N}_{s}u$\mathcal {N}_{s}u$ is the nonlocal Neumann derivative. We show that for small d,$d,$ the least energy solutions u_d$u_d$ of the above problem achieve an L^{\infty }$L^{\infty }$-bound independent of d.$d.$ Using this together with suitable L^{r}$L^{r}$-estimates on u_d,$u_d,$ we show that the least energy solution u_d$u_d$ achieves a maximum on the boundary of \Omega $\Omega$ for d sufficiently small.

Let L=-\Delta +V$L=-\Delta +V$ be a Schrödinger operator in {\mathbb R}^n${\mathbb R}^n$ with n\geq 3$n\geq 3$, where \Delta $\Delta$ is the Laplace operator denoted by \Delta =\sum ^{n}_{i=1}({\partial ^{2}}/{\partial x_{i}^{2}})$\Delta =\sum ^{n}_{i=1}({\partial ^{2}}/{\partial x_{i}^{2}})$ and the nonnegative potential V belongs to the reverse Hölder class (RH)_{q}$(RH)_{q}$ with q>n/2$q>n/2$. For \alpha \in (0,1)$\alpha \in (0,1)$, we define the operator

\begin{align*} T_N^{L^{\alpha}} f(x) =\sum_{j=N_1}^{N_2} v_j(e^{-a_{j+1}L^\alpha} f(x)-e^{-a_{j}L^\alpha} f(x)) \quad \mbox{for all }x\in \mathbb R^n, \end{align*} $$\begin{align*} T_N^{L^{\alpha}} f(x) =\sum_{j=N_1}^{N_2} v_j(e^{-a_{j+1}L^\alpha} f(x)-e^{-a_{j}L^\alpha} f(x)) \quad \mbox{for all }x\in \mathbb R^n, \end{align*}$$

where \{e^{-tL^\alpha } \}_{t>0}$\{e^{-tL^\alpha } \}_{t>0}$ is the fractional heat semigroup of the operator L, \{v_j\}_{j\in \mathbb Z}$\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequence and \{a_j\}_{j\in \mathbb Z}$\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence.

We investigate the boundedness of the operator T_N^{L^{\alpha }}$T_N^{L^{\alpha }}$ and the related maximal operator T^*_{L^{\alpha }}f(x):=\sup _N \vert T_N^{L^{\alpha }} f(x)\vert $T^*_{L^{\alpha }}f(x):=\sup _N \vert T_N^{L^{\alpha }} f(x)\vert$ on the spaces L^{p}(\mathbb {R}^{n})$L^{p}(\mathbb {R}^{n})$ and BMO_{L}(\mathbb {R}^{n})$BMO_{L}(\mathbb {R}^{n})$, respectively. As extensions of L^{p}(\mathbb {R}^{n})$L^{p}(\mathbb {R}^{n})$, the boundedness of the operators T_N^{L^{\alpha }}$T_N^{L^{\alpha }}$ and T^*_{L^{\alpha }}$T^*_{L^{\alpha }}$ on the Morrey space L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ and the weak Morrey space WL^{\rho ,\theta }_{1,\kappa }(\mathbb {R}^{n})$WL^{\rho ,\theta }_{1,\kappa }(\mathbb {R}^{n})$ has also been proved.

For a prime p and a rational elliptic curve E_{/\mathbb {Q}}$E_{/\mathbb {Q}}$, set K=\mathbb {Q}(E[p])$K=\mathbb {Q}(E[p])$ to denote the torsion field generated by E[p]:=\operatorname {ker}\{E\xrightarrow {p} E\}$E[p]:=\operatorname {ker}\{E\xrightarrow {p} E\}$. The class group \operatorname {Cl}_K$\operatorname {Cl}_K$ is a module over \operatorname {Gal}(K/\mathbb {Q})$\operatorname {Gal}(K/\mathbb {Q})$. Given a fixed odd prime number p, we study the average nonvanishing of certain Galois stable quotients of the mod-p class group \operatorname {Cl}_K/p\operatorname {Cl}_K$\operatorname {Cl}_K/p\operatorname {Cl}_K$. Here, E varies over all rational elliptic curves, ordered according to height. Our results are conditional, since we assume that the p-primary part of the Tate–Shafarevich group is finite. Furthermore, we assume predictions made by Delaunay for the statistical variation of the p-primary parts of Tate–Shafarevich groups. We also prove results in the case when the elliptic curve E_{/\mathbb {Q}}$E_{/\mathbb {Q}}$ is fixed and the prime p is allowed to vary.