By the modular degree we mean the degree of a modular parametrization of an elliptic curve, namely the mapping degree of the surjection from a modular curve to an elliptic curve. Its arithmetic significance is discussed by Zagier and Agashe-Ribet-Stein et al. in terms of the congruence of modular forms. Given an elliptic curve $E_f$ attached to a rational newform $f$, we explicitly relate its modular degree to a quotient of special values of some two $L$-functions attached to $f$. We also provide several numerical examples of the formula.

We consider the bilinear estimates of a product of functions in homogeneous Besov spaces showed by Bony [3]. It is seen that if we change the condition of indices denoting differential orders, then we can find examples of functions that never satisfy the bilinear estimates. Such examples can be constructed due to those used in the ill-posedness problem of the Navier-Stokes equations, such as Bourgain-Pavlović [4] and Yoneda [13].

Let $G = GL_N(\mathbf{k})$, where $\mathbf{k}$ is an algebraically closed field of $\operatorname{ch} \mathbf{k} \ne 2$, and $\theta$ an involutive automorphism of $G$ such that $H = (G^{\theta})^0$ is isomorphic to $SO_N(\mathbf{k})$. Then $G^{\iota\theta} = \{ g \in G \mid \theta(g) = g^{-1}\}$ is regarded as a symmetric space $G/G^{\theta}$. Let $G^{\iota\theta}_{\operatorname{uni}}$ be the set of unipotent elements in $G^{\iota\theta}$. $H$ acts on $G^{\iota\theta}_{\operatorname{uni}}$ by the conjugation. As an analogue of the generalized Springer correspondence in the case of reductive groups, we establish in this paper the generalized Springer correspondence between $H$-orbits in $G^{\iota\theta}_{\operatorname{uni}}$ and irreducible representations of various symmetric groups.

This article is a sequel of $\langle$Geometric aspects of Lucas sequences, I$\rangle$, which presents a way of viewing Lucas sequences in the framework of group scheme theory. This enables us to treat the Lucas sequences from a geometric and functorial viewpoint, which was suggested by Laxton $\langle$On groups of linear recurrences, I$\rangle$ and by Aoki-Sakai $\langle$Mod $p$ equivalence classes of linear recurrence sequences of degree 2$\rangle$. It is crucial in this work to employ the notion of quotients though at an elementary level.

Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field $F$ with good supersingular reduction at all primes above $p$. We study both the classical and plus/minus Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of $F$. In particular, we give sufficient conditions for these Selmer groups to not contain a non-trivial sub-module of finite index. Furthermore, when $p$ splits completely in $F$, we calculate the Euler characteristics of the plus/minus Selmer groups over the compositum of all $\mathbb{Z}_p$-extensions of $F$ when they are defined.

With the aid of estimates on widths of trajectory-horns and trajectory-harps from above, we show that the magnetic exponential map at an arbitrary point on a Hadamard Kähler manifold induces a homeomorphism of the unit tangent space onto the ideal boundary if the strength of magnetic field is not greater than the square root of the absolute value of the upper bound of sectional curvatures.

We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds and their generalizations. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called \emph{thin position}. Thin position is defined using orderings of the $n$-dimensional simplices of an $n$-dimensional pseudomanifold. In addition to defining and finding combinatorial minimal surfaces, from thin orderings, we derive invariants of even-dimensional closed simplicial pseudomanifolds called \emph{width} and \emph{trunk}. We study additivity properties of these invariants under connected sum and prove theorems analogous to those in knot theory and 3-manifold theory.

Let $G$ be a cyclic group of order $p^k$ for a prime $p$. There is a natural embedding of $\mathbb{Z}_p[G]$ into a product of rings of integers of cyclotomic fields. We consider the ideals $I$ of $\mathbb{Z}_p[G]$ for which the following statement holds: if $\alpha\in\mathbb{Z}_p[G]$ and $\chi(\alpha)\in\chi(I)$ for all $p$-adic characters $\chi$ of $G$, then $\alpha\in I$. We show that they are precisely the contracted ideals of the embedding. Powers of the augmentation ideal of $\mathbb{Z}_p[G]$ are not contracted in general, and we calculate the obstruction of contraction in this case.

Let $K[X_d,Y_d]=K[x_1,\ldots,x_d,y_1,\ldots,y_d]$ be the polynomial algebra in $2d$ variables over a field $K$ of characteristic 0 and let $\delta$ be the derivation of $K[X_d,Y_d]$ defined by $\delta(y_i)=x_i$, $\delta(x_i)=0$, $i=1,\ldots,d$. In 1994 Nowicki conjectured that the algebra $K[X_d,Y_d]^{\delta}$ of constants of $\delta$ is generated by $X_d$ and $x_iy_j-y_ix_j$ for all $1\leq i<j\leq d$. The affirmative answer was given by several authors using different ideas. In the present paper we give another proof of the conjecture based on representation theory of the general linear group $GL_2(K)$.