Volume 2016, Issue 719

A henselian valued field K is called a tame field if its algebraic closure K~${\tilde{K}}$ is a tame extension, that is, the ramification field of the normal extension K~|K${\tilde{K}|K}$ is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax–Kochen–Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax–Kochen–Ershov Principles. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.

For an elliptic curve E over a number field K , one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell–Weil rank. Assuming finiteness of Ш⁢(E/K)⁢[p∞]${\Sha(E/K)[p^{\infty}]}$ for a prime p this is equivalent to the p -parity conjecture: the global root number matches the parity of the ℤp${\mathbb{Z}_{p}}$-corank of the p∞${p^{\infty}}$-Selmer group. We complete the proof of the p -parity conjecture for elliptic curves that have a p -isogeny for p>3${p>3}$ (the cases p≤3${p\leq 3}$ were known). Tim and Vladimir Dokchitser have showed this in the case when E has semistable reduction at all places above p by establishing respective cases of a conjectural formula for the local root number. We remove the restrictions on reduction types by proving their formula in the remaining cases. We apply our result to show that the p -parity conjecture holds for every E with complex multiplication defined over K . Consequently, if for such an elliptic curve Ш⁢(E/K)⁢[p∞]${\Sha(E/K)[p^{\infty}]}$ is infinite, it must contain (ℚp/ℤp)2${(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{2}}$.

We give a complete solution of a problem in submanifold theory posed and partially solved by the eminent algebraic geometer Pierre Samuel in 1947. Namely, to determine all pairs of immersions f,g:Mn→ℝN${f,g\colon M^{n}\to\mathbb{R}^{N}}$ into Euclidean space that have the same Gauss map and induce conformal metrics on the manifold Mn${M^{n}}$. The case of isometric induced metrics was solved in 1985 by the first author and D. Gromoll.

Let L/K${L/K}$ be a finite Galois CM-extension of number fields with Galois group G . In an earlier paper, the author has defined a module SKu⁢(L/K)${\mathrm{SKu}(L/K)}$ over the center of the group ring ℤ⁢G${\mathbb{Z}G}$ which coincides with the Sinnott–Kurihara ideal if G is abelian and, in particular, contains many Stickelberger elements. It was shown that a certain conjecture on the integrality of SKu⁢(L/K)${\mathrm{SKu}(L/K)}$ implies the minus part of the equivariant Tamagawa number conjecture at an odd prime p for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above p , provided that Iwasawa’s μ-invariant vanishes. Here, we prove a relevant part of this integrality conjecture which enables us to deduce the minus- p -part of the equivariant Tamagawa number conjecture from the vanishing of μ for the same class of extensions. As an application we prove the non-abelian Brumer and Brumer–Stark conjecture outside the 2-primary part for every monomial Galois extension of ℚ${\mathbb{Q}}$ provided that certain μ-invariants vanish.

Based on Thomas and Yong’s K -theoretic jeu de taquin algorithm, we prove a uniform Littlewood–Richardson rule for the K -theoretic Schubert structure constants of all minuscule homogeneous spaces. Our formula is new in all types. For the main examples of Grassmannians of type A and maximal orthogonal Grassmannians it has the advantage that the tableaux to be counted can be recognized without reference to the jeu de taquin algorithm.

Let X be a standard determinantal scheme X⊂ℙn${X\subset\mathbb{P}^{n}}$ of codimension c , i.e. a scheme defined by the maximal minors of a t×(t+c-1)$t\times(t+c-1)$ homogeneous polynomial matrix 𝒜${\mathcal{A}}$. In this paper, we study the main features of its normal sheaf 𝒩X${{\mathcal{N}}_{X}}$. We prove that under some mild restrictions: (1) there exists a line bundle ℒ${\mathcal{L}}$ on X∖Sing⁢(X)${X\setminus\mathrm{Sing}(X)}$ such that 𝒩X⊗ℒ${{\mathcal{N}}_{X}\otimes\mathcal{L}}$ is arithmetically Cohen–Macaulay and, even more, it is Ulrich whenever the entries of 𝒜${\mathcal{A}}$ are linear forms, (2) 𝒩X${{\mathcal{N}}_{X}}$ is simple (hence, indecomposable) and, finally, (3) 𝒩X${{\mathcal{N}}_{X}}$ is μ-(semi)stable provided the entries of 𝒜${\mathcal{A}}$ are linear forms.

We classify quantum analogues of actions of finite subgroups G of SL2⁢(k)${\mathrm{SL}_{2}(k)}$ on commutative polynomial rings k⁢[u,v]${k[u,v]}$. More precisely, we produce a classification of pairs (H,R)${(H,R)}$ where H is a finite-dimensional Hopf algebra that acts inner faithfully and preserves the grading of an Artin–Schelter regular algebra R of global dimension 2. Remarkably, the corresponding invariant rings RH${R^{H}}$ share similar regularity and Gorenstein properties as the invariant rings k⁢[u,v]G${k[u,v]^{G}}$ in the classical setting. We also present several questions and directions for expanding this work in noncommutative invariant theory.