Indagationes Mathematicae-New Series最新文献

We give a complete classification of $Q_{\ell }$-cohomology projective planes with isolated ADE-singularities and numerically trivial canonical bundle in odd characteristic. This leads to a beautiful relation with certain Enriques surfaces which parallels the situation in characteristic zero, yet displays intriguing subtleties.

For an elliptic curve $E/Q$ we show that there are infinitely many cyclic sextic extensions $K/Q$ such that the Mordell–Weil group $E\left(K\right)$ has rank greater than the subgroup of $E\left(K\right)$ generated by all the $E\left(F\right)$ for the proper subfields $F\subset K$. For certain curves $E/Q$ we show that the number of such fields $K$ of conductor less than $X$ is $\gg \sqrt{X}$.

We study the Weil representation $\rho$ of a curve over a $p$-adic field with potential reduction of compact type. We show that $\rho$ can be reconstructed from its stable reduction. For superelliptic curves of the form $y^n=f\left(x\right)$ at primes $p$ whose residue characteristic is prime to the exponent $n$ we make this explicit.

Let $K$ be a complete discretely valued field of residue characteristic not 2 and $O_K$ its ring of integers. We explicitly construct a regular model over $O_K$ with strict normal crossings of any hyperelliptic curve $C/K:y^2=f\left(x\right)$. For this purpose, we introduce the new notion of MacLane cluster picture, that aims to be a link between clusters and MacLane valuations.

In this paper we inspect from closer the local and global points of the twists of the Klein quartic. For the local ones we use geometric arguments, while for the global ones we strongly use the modular interpretation of the twists. The main result is providing families with (conjecturally infinitely many) twists of the Klein quartic that are counterexamples to the Hasse Principle.

Let $X$ denote a K3 surface over an arbitrary field $k$. Let $k^{\text{s}}$ denote a separable closure of $k$ and let $X^{\text{s}}$ denote the base change of $X$ to $k^{\text{s}}$. Let $O(PicX)$ and $O(Pic{X}^{\text{s}})$ denote the group of isometries of the lattices $PicX$ and $Pic{X}^{\text{s}}$, respectively. Let $R_X$ denote the Galois invariant part of the Weyl group of $Pic{X}^{\text{s}}$. One can show that each element in $R_X$ can be restricted to an element of $O(PicX)$. The following question arises: Is the image of the restriction map $R_X\to O(PicX)$ a normal subgroup of $O(PicX)$ for every K3 surface $X$? We show that the answer is negative by giving counterexamples over $k=Q$.

Let $a,Q\in Q$ be given and consider the set $G(a,Q)=\{a{Q}^i:i\in N\}$ of terms of geometric progression with 0th term equal to $a$ and the quotient $Q$. Let $f\in Q(x,y)$ and $V_f$ be the set of finite values of $f$. We consider the problem of existence of $a,Q\in Q$ such that $G(a,Q)\subset V_f$. In the first part of the paper we describe certain classes of rational functions for which our problem has a positive solution. In the second, experimental, part of the paper we study the stated problem for the rational function $f(x,y)=(y^2-x^3)/x$. We relate the problem to the existence of rational points on certain elliptic curves and present interesting numerical observations which allow us to state several questions and conjectures.

Many interesting combinatorial sequences, such as Apéry numbers and Franel numbers, enjoy the so-called Lucas property modulo almost all primes $p$. Modulo prime powers $p^r$ such sequences have a more complicated behaviour which can be described by matrix versions of the Lucas property called $p$-linear schemes. They are generalizations of finite $p$-automata. In this paper we construct such $p$-linear schemes and give upper bounds for the number of states which, for fixed $r$, do not depend on $p$.

Although structural maps such as subductions and inductions appear naturally in diffeology, one of the challenges is providing suitable analogues for submersions, immersions, and étale maps (i.e., local diffeomorphisms) consistent with the classical versions of these maps between manifolds. In this paper, we consider diffeological or plotwise versions of submersions, immersions, and étale maps as an adaptation of these maps to diffeology by a nonlinear approach. We study their diffeological properties from different aspects in a systematic fashion with respect to the germs of plots.

In order to characterize the considered maps from their linear behaviors, we introduce a class of diffeological spaces, so-called diffeological étale manifolds, which not only contains the usual manifolds but also includes irrational tori. We state and prove versions of the rank and implicit function theorems, as well as the fundamental theorem on flows in this class. As an application, we use the results of this work to facilitate the computations of the internal tangent spaces and diffeological dimensions in a few interesting cases.