We study chirally cosmetic surgeries, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3-manifolds with opposite orientations. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredients are the original and the ${\rm SL}(2,\mathbb{C})$ version of Casson invariants. As applications, we give a complete classification of chirally cosmetic surgeries on alternating knots of genus one.

Let $G/H$ be a reductive symmetric space over a $p$-adic field, defined by an involution $\sigma$ on $G$. We study the induced representations ${\rm Ind}_P^G(\rho)$ for any proper $\sigma$-split parabolic subgroup $P=MU$ and any $M\cap H$-distinguished representation $\rho$ of $M$. Let ${\mathcal X}$ denote the complex torus of unramified characters of $M$ which are trivial on $M\cap H$ and $\rho_{\chi}$ the twist of $\rho$ by $\chi\in {\mathcal X}$. We show that ${\rm Ind}_P^G(\rho_{\chi})$ is not $H$-relatively cuspidal {\it for generic} $\chi$, that is, for $\chi$ in a Zariski-open subset of ${\mathcal X}$.

Let $l$ be the prime $3$, $5$ or $7$, and let $m_{1}$,~$m_{2}$, $n_{1}$ and $n_{2}$ be non-zero rational numbers. We construct an infinite family of pairs of distinct quadratic fields $\mathbb{Q}(\sqrt{m_{1}D+n_{1}})$ and $\mathbb{Q}(\sqrt{m_{2}D+n_{2}})$ with $D\in\mathbb{Q}$ such that both class numbers are divisible by $l$, using rational points on an elliptic curve with positive Mordell-Weil rank to parametrize such quadratic fields.

In a previous paper, the authors proved that Milnor link-homotopy invariants modulo~$n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications of welded string links up to $2n$-move and self-crossing virtualization, and up to $V^{n}$-move and self-crossing virtualization, respectively.

We construct a special $2$-adic Lie extension of a number field as an iterated tower by a conjugate of Joukowski map. If the number field is totally real, the unramified Iwasawa module over such a $2$-adic Lie iterated extension is conjecturally pseudo-null under Greenberg's conjecture for all intermediate cyclotomic $\mathbb{Z}_2$-extensions. We give some examples of such $2$-adic Lie iterated extensions with pseudo-null Iwasawa modules.

Lately, Kiuchi and Minamide studied the mean square of the double zeta-function $\zeta_2(\sigma_{1}+it_1, \sigma_{2}+it_2)$ with respect to $ t_1 $ in the critical region. In this paper, using a new upper bound of $\zeta_{2}(\sigma_{1}+it_1, \sigma_{2}+it_2)$ we improve the results on the mean square of $\zeta_{2}(\sigma_{1}+it_{1}, \sigma_{2}+it_{2})$, obtained by Kiuchi and Minamide.

A weakly reflective submanifold is a minimal submanifold of a Riemannian manifold which has a certain symmetry at each point. In this paper we introduce this notion into a class of proper Fredholm (PF) submanifolds in Hilbert spaces and show that there exist many infinite dimensional weakly reflective PF submanifolds in Hilbert spaces. In particular each fiber of the parallel transport map is shown to be weakly reflective. These imply that in infinite dimensional Hilbert spaces there exist many homogeneous minimal submanifolds which are not totally geodesic, unlike in the finite dimensional Euclidean case.

This is Part II of the author's paper Logarithmic structures of Fontaine-Illusie. We discuss log flat topology and log flat descent. We study the first log flat cohomology $H^1(X_{\log, \text{fl}}, G)$ for various sheaves of groups $G$, for example, $G=GL_n$, finite flat commutative group schemes, the log multiplicative group $M^{\text{gp}}$, etc.

Let $p=2^{e+1}q+1$ be an odd prime number with $2 \nmid q$. Let $K$ be the imaginary cyclic field of conductor $p$ and degree $2^{e+1}$. We denote by $\mathcal{F}$ the imaginary quadratic subextension of the imaginary $(2,\,2)$-extension $K(\sqrt{2})/K^+$ with $\mathcal{F} \neq K$. We determine the Galois module structure of the $2$-part of the class group of $\mathcal{F}$.

In this paper, we define a syntomic complex for modulus pair $(X,D)$, where $X$ is a regular semi-stable family and $D$ is an effective Cartier divisor on $X$ and we compute its cohomology sheaves. We construct a symbol map for this syntomic complex for modulus pair $(X,D)$ and investigate its cokernel by computing its cohomology sheaves. To compute the structure of the cohomology sheaves of syntomic complex with modulus, we define some filtrations on it.

We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not greater than three are explicitly described.

In this paper, we construct a multi-parameter family of self-avoiding walks on the Sierpiński gasket. It includes the branching model, the loop-erased random walk and the loop-erased self-repelling walk. We reproduce in a unified manner the proof of the existence of the continuum limit and the self-avoiding property of the limit processes. Our limit processes include not only all the processes obtained from the previously studied self-avoiding walk models, but also the ones that have not been constructed before. While the paths of limit processes appearing in the previous works were self-avoiding or filled the whole space, our family includes continuous processes whose path is self-intersecting but does not fill the whole space.