arXiv subjects
Publications and source records attributed to Akihiko Yukie.
We determine all orbits of the prehomogeneous vector space $G = \mathrm{GL}_8,V =\wedge^3\mathrm{Aff}^8$ rationally over an arbitrary perfect field of characteristic not equal to $2$ in this paper.
We determine all orbits of the prehomogeneous vector space $(\mathrm{GL}_5\times \mathrm{GL}_4,\wedge^2\mathrm{Aff}^5 \otimes \mathrm{Aff}^4)$ rationally over an arbitrary perfect field in this paper.
We determine the set which parametrizes the GIT stratification for four prehomogeneous vector spaces in this paper.
We determine all orbits of two prehomogeneous vector spaces rationally over an arbitrary perfect field in this paper.
Let $k$ be a field of characteristic not equal to $2,3$, $\mathbb{O}$ an octonion over $k$ and $\mathcal{J}$ the exceptional Jordan algebra defined by $\mathbb{O}$. We consider the prehomogeneous vector space $(G,V)$ where $G=GE_6\times \mathrm{GL}(2)$ and $V=\mathcal{J}\oplus \mathcal{J}$. We prove that generic rational orbits of this prehomogeneous vector space are in bijective correspondence with $k$-isomorphism classes of pairs $(\mathcal{M},\mathfrak{n})$ where $\mathcal{M}$'s are isotopes of $\mathcal{J}$ and $\mathfrak{n}$'s are cubic étale subalgebras of $\mathcal{M}$. Also we prove that if $\mathbb{O}$ is split, then generic rational orbits are in bijective correspondence with isomorphism classes of separable extensions of $k$ of degrees up to $3$.
Let $\mathcal{J}$ be the exceptional Jordan algebra and $V=\mathcal{J}\oplus \mathcal{J}$. We construct an equivariant map from $V$ to $\mathrm{Hom}_k(\mathcal{J}\otimes \mathcal{J},\mathcal{J})$ defined by homogeneous polynomials of degree $8$ such that if $x\in V$ is a generic point, then the image of $x$ is the structure constant of the isotope of $\mathcal{J}$ corresponding to $x$. We also give an alternative way to define the isotope corresponding to a generic point of $\mathcal{J}$ by an equivariant map from $\mathcal{J}$ to the space of trilinear forms.
Let $k$ be a number field and $\tilde k$ a fixed quadratic extension of $k$. In this paper and its companion, we find the mean value of the product of class numbers and regulators of two quadratic extensions $F,F^*\not=\tilde k$ contained in the biquadratic extensions of $k$ containing $\tilde k$.
This is the second part of a two part paper. In this part, we evaluate the previously unevaluated local densities at dyadic places which appear in the density theorem stated in the first part. For this part, we introduce an invariant, the level, attaced to a pair of ramified quadratic extensions of a dyadic local field. This invariant measures how close two fields are in their arithmetic properties and may be of interest independent of its application here.
In this note, we consider applications of Ratner's theorem to constructions of families of polynomials with dense values on the set of primitive integer points from the viewpoint of invariant theory.
In this note, we prove that if $(G,V)$ is a prehomogeneous vector space over any field $k$ such that the stabilizer of a generic point is reductive, the set of semi-stable points is a single orbit over the separable closure of $k$.
Let H_1=SL(5), H_2=SL(3), H=H_1 \times H_2. It is known that (G,V) is a prehomogeneous vector space (see [22], [26], [25], for the definition of prehomogeneous vector spaces). A non-constant polynomial δ(x) on V is called a relative invariant polynomial if there exists a character χsuch that δ(gx)=χ(g)δ(x). Such δ(x) exists for our case and is essentially unique. So we define V^{ss}={x in V such that δ(x) is not equal to 0}. For x in V_R^{ss}, let H_{x R+}^0 be the connected component of 1 in classical topology of the stabilizer H_{x R}. We will prove that if x in V_R^ss is "sufficiently irrational", H_{x R+}^0 H_Z is dense in H_R.
In this paper, we determine the rational orbit decomposition for two prehomogeneous vector spaces associated with the simple group of type G_2.
We apply M. Ratner's theorem on closures of unipotent orbits to the study of three families of prehomogeneous vector spaces. As a result, we prove analogues of the Oppenheim Conjecture for simultaneous approximation by values of certain alternating bilinear forms in an even number of variables and certain alternating trilinear forms in six and seven variables.
This is part one of a series of papers. In this series of papers, we consider problems analogous to the Oppenheim conjecture from the viewpoint of prehomogeneous vector spaces.
The purpose of the paper is to prove an analogous correspondence for the above prehomogeneous vector space (1)-(3). For case (1) the correspondence is bijective. However, it turns out that the correspondence is not bijective for cases (2), (3). We describe the fiber structures of this correspondence in subsection 2-4. In section 1 we briefly review basic properties of the non-abelian Galois cohomology set and its relation to the orbit space G_k \ V_k^ss. In subsection 2-4 we consider the prehomogeneous vector spaces (1)-(3) respectively.
In this paper, we give an introduction to the rationality of the equivariant Morse stratification, and state the author's results on zeta functions of prehomogeneous vector spaces.
Throughout this paper, k is a number field. We fix a quadratic extension k_1=k(a_0) of k, where a_0=sqrt(B_0) for a certain B_0 in k^\times/(k^\times)^2. In this paper, we consider the zeta function defined for the space of pairs of binary Hermitian forms. This is the prehomogeneous vector space we discussed in section 2 [1] and is a non-split form of the D_4 case in [4]. The purpose of this paper is to determine the principal part of the adjusted zeta function. Our main result is Theorem (8.15). Our case resembles the space of pairs of binary quadratic forms which we discussed in Chapter 5 [5]. However, the meaning of the problem, the pole structure of the adjusted zeta function, and the adjusting terms are different from those of the above case. The zeta function of our case is a counting function of the class number times the regulator of fields of the form k(sqrt(B_0),sqrt(B)) with B_0 fixed and B in k_, whereas the zeta function for the above case is essentially the same as the space of binary quadratic forms. Our case is a first example where we need two adjusting terms whereas the adjusting term of the above case was essentially the same as that of the space of binary quadratic forms. We will not prove the meromorphic continuation of the adjusting terms in this paper. For the sake of the density theorem, we still get the residue of the rightmost pole of the zeta function (without adjusting). If this space appears in a bigger prehomogeneous vector space. So the principal part formula of the adjusted zeta function should be enough for most applications. We have yet to obtain the corresponding density theorem because we have not carried out the local theory.
In this paper, we consider the most non-split parabolic D_4 type prehomogeneous vector space. The vector space is an analogue of the space of Hermitian forms. We determine the principal part of the zeta function.