Related papers: Perfectoid $C_i$ transfer

Perfectoid versions of Abel Jacobi and Reimann Roch Theorem are proved, and perfectoid Elliptic Curve is constructed. A Perfectoid Tate Curve is defined and its cohomology computed via a \v{C}ech complex. Furthermore, perfectoid Theta…

We prove a general version of the "Stability Theorem": if $K$ is a valued field such that the ramification theoretical defect is trivial for all of its finite extensions, and if $F|K$ is a finitely generated (transcendental) extension of…

Given a field $K$, a rational function $\phi \in K(x)$, and a point $b \in \mathbb{P}^1(K)$, we study the extension $K(\phi^{-\infty}(b))$ generated by the union over $n$ of all solutions to $\phi^n(x) = b$, where $\phi^n$ is the $n$th…

We give a self-contained proof of the fact that, for any prime number $p$, there exists a power series $$\Psi= \Psi_p(T) \in T + T^2\Z[[T]] $$ which trivializes the addition law of the formal group of Witt covectors is $p$-adically entire…

Let $k$ be a perfect field of characteristic $p$, let $f_i:X_i\to\mathbb A_k^1$ $(i=1,2)$ be two $k$-morphism of finite type, and let $f:X_1\times_k X_2\to \mathbb A_k^1$ be the morphism defined by $f(z_1,z_2)=f_1(z_1)+f_2(z_2)$. For each…

We prove that Shimura varieties of abelian type with infinite level at $p$ are perfectoid. As a corollary, the moduli spaces of polarized K3 surfaces with infinite level at $p$ are also perfectoid.

Let $p$ be an odd prime number. We construct explicit uniformizers for the totally ramified extension $\mathbb{Q}_p(\zeta_{p^2},\sqrt[p]{p})$ of the field of $p$-adic numbers $\mathbb{Q}_p$, where $\zeta_{p^2}$ is a primitive $p^2$-th root…

Let $K$ be a totally real number field, $d$ a positive integer, and $Q$ a higher degree form over $K$. We prove that there are at most finitely many totally real extensions $L/K$ of degree $d$ such that $Q$ over $L$ is universal. Further,…

Let $f \colon X \to Y$ be a morphism of concentrated schemes. We characterize $f$-perfect complexes $\mathcal{E}$ as those such that the functor $\mathcal{E} \otimes^{\mathbf{L}}_X \mathbf{L} f^*-$ preserves bounded complexes. We prove, as…

Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$-conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in…

In this article, we extend the van Hamel-Lichtenbaum duality theorem to (not necessarily smooth) proper and geometrically integral varieties defined over a $p$-adic field $k$. More precisely, we prove that for such variety $X$ there exists…

Let $G\subset x{\mathbb F}_q[\![x]\!]$ ($q$ is a power of the prime $p$) be a subset of formal power series over a finite field such that it forms a compact abelian $p$-adic Lie group of dimension $d\ge 1$. We establish a necessary and…

The almost purity theorem is central to the geometry of perfectoid spaces and has numerous applications in algebra and geometry. This result is known to have several different proofs in the case that the base ring is a perfectoid valuation…

We prove model completeness for the theory of addition and the Frobenius map for certain subrings of rational functions in positive characteristic. More precisely: Let $p$ be a prime number, $\mathbb{F}_{p}$ the prime field with $p$…

Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension of $W(k)[\frac{1}{p}]$ of ramification degree $e$. We consider an unramified base ring $R_0$ over $W(k)$ satisfying certain conditions,…

For any two configurations of ordered points $p=(p_{1},...,\p_{N})$ and $q=(q_{1},...,q_{N})$ in Euclidean space $E^d$ such that $q$ is an expansion of $p$, there exists a continuous expansion from $p$ to $q$ in dimension 2d; Bezdek and…

Let K be a local field of characteristic p with perfect residue field k. In this paper we find a set of representatives for the k-isomorphism classes of totally ramified separable extensions L/K of degree p. This extends work of Klopsch,…

In this note, we promote an infinite Kadison transitivity theorem on massive $C^*$-algebras, including the Calkin algebra. This transitivity stems from the analog of countable degree-1 saturation on pure states which is inherited from these…

We show that the category of motivic spaces with transfers along finite flat morphisms, over a perfect field, satisfies all the properties we have come to expect of good categories of motives. In particular we establish the analog of…

In this paper, the author proved that the base change lifting associated to a totally ramified extension of a non-archimedean local field coincides with a map coming from the close fields theory of Kazhdan under some conditions. As a…