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Register a new userThe valuation criterion for normal basis generators
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If $L/K$ is a finite Galois extension of local fields, we say that the valuation criterion $VC(L/K)$ holds if there is an integer $d$ such that every element $x in L$ with valuation $d$ generates a normal basis for $L/K$. Answering a question of Byott and Elder, we first prove that $VC(L/K)$ holds if and only if the tamely ramified part of the extension $L/K$ is trivial and every non-zero $K[G]$-submodule of $L$ contains a unit. Moreover, the integer $d$ can take one value modulo $[L:K]$ only, namely $-d_{L/K}-1$, where $d_{L/K}$ is the valuation of the different of $L/K$. When $K$ has positive characteristic, we thus recover a recent result of Elder and Thomas, proving that $VC(L/K)$ is valid for all extensions $L/K$ in this context. When $char{;K}=0$, we identify all abelian extensions $L/K$ for which $VC(L/K)$ is true, using algebraic arguments. These extensions are determined by the behaviour of their cyclic Kummer subextensions.
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For a cuspidal automorphic representation Pi of GL(4,A), H. Kim proved that the exterior square transfer wedge^2Pi is an isobaric automorphic representation of GL(6,A). In this paper we characterize those representations Pi for which wedge^2Pi is cuspidal.
Let f:X->X be a morphism of a variety over a number field K. We consider local conditions and a Bruaer-Manin condition, defined by Hsia and Silverman, for the orbit of a point P in X(K) to be disjoint from a subvariety V of X, i.e., the intersection of the orbit of P with V is empty. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection of the orbit of P with V; this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.
Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := dfrac{(L cdot A) : (L cdot B)}{A cdot B}$, where $A cdot B$ denotes the intersection number of $A$ and $B$. Call $X$ arborescent if all the dual graphs of its resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P{l}oski by proving that whenever $X$ is arborescent, the function $u_L$ is an ultrametric on the set of branches on $X$ different from $L$. In the present paper we prove that, conversely, if $u_L$ is an ultrametric, then $X$ is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on $X$, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which $u_L$ is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing $L$ to be an arbitrary semivaluation on $X$ and by defining $u_L$ on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if $X$ is arborescent, and without any restriction on $X$ we exhibit special subspaces of the space of semivaluations in restriction to which $u_L$ is still an ultrametric.
Using the methods developed for the proof that the 2-universality criterion is unique, we partially characterize criteria for the n-universality of positive-definite integer-matrix quadratic forms. We then obtain the uniqueness of Ohs 8-universality criterion as an application of our characterization results.
Let $K$ be a local function field of characteristic $l$, $mathbb{F}$ be a finite field over $mathbb{F}_p$ where $l e p$, and $overline{rho}: G_K rightarrow text{GL}_n (mathbb{F})$ be a continuous representation. We apply the Taylor-Wiles-Kisin method over certain global function fields to construct a mod $p$ cycle map $overline{text{cyc}}$, from mod $p$ representations of $text{GL}_n (mathcal{O}_K)$ to the mod $p$ fibers of the framed universal deformation ring $R_{overline{rho}}^square$. This allows us to obtain a function field analog of the Breuil--Mezard conjecture. Then we use the technique of close fields to show that our result is compatible with the Breuil-Mezard conjecture for local number fields in the case of $l e p$, obtained by Shotton.
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