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Register a new userSpecialization Method in Krull Dimension two and Euler System Theory over Normal Deformation Rings
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The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where $O$ is the ring of integers of a finite extension of the field of p-adic integers $Q_p$. The specialization method is a technique that recovers the information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$, where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen-Macaulay normal domains by combining the main results in an earlier article of the first named author and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in an article of the first named author.
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We generalize a result of Galatius and Venkatesh which relates the graded module of cohomology of locally symmetric spaces to the graded homotopy ring of the derived Galois deformation rings, by removing certain assumptions, and in particular by allowing congruences inside the localized Hecke algebra.
Let $k,pin mathbb{N}$ with $p$ prime and let $finmathbb{Z}[x_1,x_2]$ be a bivariate polynomial with degree $d$ and all coefficients of absolute value at most $p^k$. Suppose also that $f$ is variable separated, i.e., $f=g_1+g_2$ for $g_iinmathbb{Z}[x_i]$. We give the first algorithm, with complexity sub-linear in $p$, to count the number of roots of $f$ over $mathbb{Z}$ mod $p^k$ for arbitrary $k$: Our Las Vegas randomized algorithm works in time $(dklog p)^{O(1)}sqrt{p}$, and admits a quantum version for smooth curves working in time $(dlog p)^{O(1)}k$. Save for some subtleties concerning non-isolated singularities, our techniques generalize to counting roots of polynomials in $mathbb{Z}[x_1,ldots,x_n]$ over $mathbb{Z}$ mod $p^k$. Our techniques are a first step toward efficient point counting for varieties over Galois rings (which is relevant to error correcting codes over higher-dimensional varieties), and also imply new speed-ups for computing Igusa zeta functions of curves. The latter zeta functions are fundamental in arithmetic geometry.
Let $mathcal{G}$ be a connected reductive almost simple group over the Witt ring $W(mathbb{F})$ for $mathbb{F}$ a finite field of characteristic $p$. Let $R$ and $R$ be complete noetherian local $W(mathbb{F})$ -algebras with residue field $mathbb{F}$. Under a mild condition on $p$ in relation to structural constants of $mathcal{G}$, we show the following results: (1) Every closed subgroup $H$ of $mathcal{G}(R)$ with full residual image $mathcal{G}(mathbb{F})$ is a conjugate of a group $mathcal{G}(A)$ for $Asubset R$ a closed subring that is local and has residue field $mathbb{F}$ . (2) Every surjective homomorphism $mathcal{G}(R)tomathcal{G}(R)$ is, up to conjugation, induced from a ring homomorphism $Rto R$. (3) The identity map on $mathcal{G}(R)$ represents the universal deformation of the representation of the profinite group $mathcal{G}(R)$ given by the reduction map $mathcal{G}(R)tomathcal{G}(mathbb{F})$. This generalizes results of Dorobisz and Eardley-Manoharmayum and of Manoharmayum, and in addition provides an abstract classification result for closed subgroups of $mathcal{G}(R)$ with residually full image. We provide an axiomatic framework to study this type of question, also for slightly more general $mathcal{G}$, and we study in the case at hand in great detail what conditions on $mathbb{F}$ or on $p$ in relation to $mathcal{G}$ are necessary for the above results to hold.
Let $k$ be a perfect field of characteristic $p geq 3$. We classify $p$-divisible groups over regular local rings of the form $W(k)[[t_1,...,t_r,u]]/(u^e+pb_{e-1}u^{e-1}+...+pb_1u+pb_0)$, where $b_0,...,b_{e-1}in W(k)[[t_1,...,t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin.
In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent $mu$ $in$ (1/2, 1) is 2(1 -- $mu$) when $mu$ $ge$ $sqrt$ 2/2, whereas for $mu$ textless{} $sqrt$ 2/2 it is greater than 2(1 -- $mu$) and at most (3 -- 2$mu$)(1 -- $mu$)/(1 + $mu$ + $mu$ 2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when $mu$ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for $mu$ $ge$ 0.565. .. .
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