In this paper, we prove that, if a full irreducible infinite dimensional anti-Kaehler isoparametric submanifold of codimension greater than one has $J$-diagonalizable shape operators, then it is an orbit of the action of a Banach Lie group generated by one-parameter transformation groups induced by holomorphic Killing vector fields defined entirely on the ambient Hilbert space.

X.-J.~Li gave in~[4] a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients. In this paper, we investigate exactly how the Li criterion for the Riemann hypothesis fails for a class of Epstein zeta functions. This enables to derive some interesting consequences regarding $c_K=\frac{h_K\log d_K}{\sqrt{d_K}}$ of a quadratic imaginary field $K$ of absolute discriminant $d_K$ and class number $h_K$. Similar results are stated for the period ratios of elliptic curves with complex multiplication.

Let $A$ be an abelian variety defined over a number field $k$ and $p$ a prime number. Under some natural and not-too-stringent conditions on $A$ and $p$ we show that certain invariants associated to Iwasawa-theoretic $p$-adic Selmer groups control the Krull-Schmidt decompositions of the $p$-adic completions of the groups of points of $A$ over finite extensions of $k$.

In~[7], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by compact hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture holds in the context of ideal Coxeter polyhedra in $\mathbb{H}^3$. Our methods allow us to bound from below the growth rates of composite ideal Coxeter polyhedra by the growth rates of its ideal Coxeter polyhedral constituents.

A ravel is a spatial graph which is non-planar but contains no non-trivial knots or links. We characterize when a Montesinos tangle can become a ravel as the result of vertex closure with and without replacing some number of crossings by vertices.

For a locally compact Hausdorff space $X$, let $C_0(X)$ be the Banach space of continuous complex-valued functions on $X$ vanishing at infinity endowed with the supremum norm $\|\cdot\|_X$. We show that for locally compact Hausdorff spaces $X$ and $Y$ and certain (not necessarily closed) subspaces $A$ and $B$ of $C_0(X)$ and $C_0(Y)$, respectively, if $T:A \longrightarrow B$ is a surjective map satisfying one of the norm conditions i) $\|(Tf)^s (Tg)^t\|_Y=\|f^s g^t\|_X$, or ii) $\|\, |Tf|^s+|Tg|^t\,\|_Y=\|\, |f|^s+|g|^t\, \|_X$, \noindent for some $s,t\in \mathbb{N}$ and all $f,g\in A$, then there exists a homeomorphism $\varphi: \mathrm{ch}(B) \longrightarrow \mathrm{ch}(A)$ between the Choquet boundaries of $A$ and $B$ such that $|Tf(y)|=|f(\varphi(y))|$ for all $f\in A$ and $y\in \mathrm{ch}(B)$. We also give a result for the case where $A$ is closed (or, in general, satisfies a special property called Bishop's property) and $T:A \longrightarrow B$ is a surjective map satisfying the inclusion $R_\pi((Tf)^s (Tg)^t) \subseteq R_\pi(f^s g^t)$ of peripheral ranges. As an application, we characterize such maps between subspaces of the form $A_1f_1+A_2f_2+\cdots+A_nf_n$, where for each $1\le i \le n$, $A_i$ is a uniform algebra on a compact Hausdorff space $X$ and $f_i$ is a strictly positive continuous function on $X$. Our results in case (ii) improve similar results in~[30], for subspaces rather than uniform algebras, without the additional assumption that $T$ is $\mathbb{R}^+$-homogeneous.

Let $K$ be a number field. Fix a finite set of analytic functions $\mathbf{f}_{\infty}:=\{f_{1,\infty}(x),\ldots,f_{s,\infty}(x) \}$ defined on $\{x\in \mathbb{C} \mid |x|>1\}$ (resp. $\mathbb{C}_p$-valued functions $\mathbf{f}_{p}:=\{f_{1,p}(x),\ldots,f_{s,p}(x) \}$ defined on $\{x\in \mathbb{C}_p \mid |x|_p>1\}$). For $\beta\in K$, we denote the $K$-vector space spanned by $f_{1,\infty}(\beta),\ldots,f_{s,\infty}(\beta)$ by $V_K(\mathbf{f}_{\infty},\beta)$ (resp. $f_{1,p}(\beta),\ldots,f_{s,p}(\beta)$ by $V_K(\mathbf{f}_{p},\beta)$). In this article, under some assumptions for $\mathbf{f}_{\infty}$ (resp. $\mathbf{f}_{p}$), we give an estimation of a lower bound of the dimension of $V_K(\mathbf{f}_{\infty},\beta)$ (resp. $V_K(\mathbf{f}_{p},\beta)$) (see Theorem~2.4 for Archimedean case and Theorem~8.6 for $p$-adic case). Applying our estimation, we give a lower bound of the dimension of the $K$-vector space spanned by the special values of the Lerch functions over a number field in $\mathbb{C}$ (see Theorem~1.1 and Remark~1.2) and the $p$-adic analog of the above result (see Theorem~1.3 and Remark~1.4). Furthermore, we also give a lower bound of the $K$-vector space spanned by the special values of certain $p$-adic functions related with $p$-adic Hurwitz zeta function (see Theorem~1.5).

Let $X$ be a smooth complex projective variety of dimension $3$ and $L$ an ample line bundle on $X$. In this paper we study the second sectional class $\mathrm{cl}_{2}(X,L)$ of $(X,L)$. First we show the inequality $\mathrm{cl}_{2}(X,L)\geq L^{3}-1$, and we characterize $(X,L)$ with $-1\leq \mathrm{cl}_{2}(X,L)-L^{3}\leq 3$. Furthermore the classification of pairs $(X,L)$ with small second sectional classes is obtained. We also classify $(X,L)$ with $2L^{3}\geq \mathrm{cl}_{2}(X,L)$.

Let $R=k [[X_1, \dots ,X_{n+1}]]$ be a formal power series ring over a perfect field $k$ of characteristic $p>0$, and let $\mathfrak{m} = (X_1 , \dots , X_{n+1})$ be the maximal ideal of $R$. Suppose $0\neq f \in\mathfrak{m}$. In this paper, we introduce a function $\xi_{f}(x)$ associated with a hypersurface $R/(f)$ defined on the closed interval $[0,1]$ in $\mathbb{R}$. The Hilbert-Kunz multiplicity and the F-signature of $R/(f)$ appear as the values of our function $\xi_{f}(x)$ on the interval's endpoints. The F-signature of the pair, denoted by $s(R,f^{t})$, was defined by Blickle, Schwede and Tucker. Our function $\xi_{f}(x)$ is integrable, and the integral $\int_{t}^{1}\xi_{f}(x)dx$ is just $s(R,f^{t})$ for any $t\in[0,1]$.

We are interested in harmonic analysis on $p$-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space $X$ of unitary hermitian matrices of size $m$ over a ${\mathfrak p}$-adic field $k$ mainly for dyadic case, and give the unified description with our previous papers for non-dyadic case. The space becomes complicated for dyadic case, and the set of integral elements in $X$ has plural Cartan orbits. We introduce a typical spherical function $\omega(x;z)$ on $X$, study its functional equations, which depend on $m$ and the ramification index $e$ of $2$ in $k$, and give its explicit formula, where Hall-Littlewood polynomials of type $C_n$ appear as a main term with different specialization according as the parity $m = 2n$ or $2n+1$, but independent of $e$. By spherical transform, we show the Schwartz space ${\mathcal S}(K\backslash X)$ is a free Hecke algebra ${\mathcal H}(G,K)$-module of rank $2^n$, and give parametrization of all the spherical functions on $X$ and the explicit Plancherel formula on ${\mathcal S}(K\backslash X)$. The Plancherel measure does not depend on $e$, but the normalization of $G$-invariant measure on $X$ depends.

We study concentration phenomena of the least energy solutions of the following nonlinear Schrödinger equation: \[ h^2 \Delta u - V(x) u + f( u ) = 0 \quad \text{in} \ \mathbb{R}^N, \ u>0, \ u \in H^1(\mathbb{R}^N)\,, \] for a totally degenerate potential $V$. Here $h>0$ is a small parameter, and $f$ is an appropriate, superlinear and Sobolev subcritical nonlinearity. In~[16], Lu and Wei proved that when the parameter $h$ approaches zero, the least energy solutions concentrate at the most centered point of the totally degenerate set $\Omega = \{ x \in \mathbb{R}^N \mid V(x) = \min_{ y \in \mathbb{R}^N } V(y) \}$ when $f(u) = u^p$. In this paper, we show that this kind of result holds for more general $f$. In particular, our proof does not need a so-called uniqueness-nondegeneracy assumption (see, the next-to-last paragraph in Section~1) on the limiting equation (2.6) in Section~2. Furthermore, in~[16] Lu and Wei made a technical assumption for $V$, that is, \[ V(x) - \min_{y \in \mathbb{R}^N} V(y) \geq C d(x, \partial \Omega)^2 \quad \text{for} \ x \in \Omega^{\rm c}\,, \] where $C$ is a positive constant, but our proof does not need this assumption. In our proof, we employ a modification of the argument which has been developed by del Pino and Felmer in~[9] using Schwarz's symmetrization.