Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new user$p$-adic Integral Geometry
146
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving a result by Oesterle) and to the study of random $p$-adic polynomial systems of equations.
rate research
Read More
Let $Pin Bbb Q_p[x,y]$, $sin Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{P,s}f)(y):=frac{1}{1-p^{-1}}int_{Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(Bbb Z_p)$. We show that if $P$ is homogeneous then for each character $chi$ of $Bbb Z_p^times$ the characteristic function $det(1-uA_{P,s,chi})$ of the restriction $A_{P,s,chi}$ of $A_{P,s}$ to the eigenspace $L^2(Bbb Z_p)_chi$ is the $q$-Wronskian of a set of solutions of a (possibly confluent) $q$-hypergeometric equation. In particular, the nonzero eigenvalues of $A_{P,s,chi}$ are the reciprocals of the zeros of such $q$-Wronskian.
Given three arbitrary vector bundles on the Fargues-Fontaine curve where one of them is assumed to be semistable, we give an explicit and complete criterion in terms of Harder-Narasimha polygons on whether there exists a short exact sequence among them. Our argument is based on a dimension analysis of certain moduli spaces of bundle maps and bundle extensions using Scholzes theory of diamonds.
We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here random means that the coefficients of the polynomials defining the complete intersections are sampled uniformly form the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spaces on a random complete intersection tends to $1$. In the case of the number of lines on a random cubic in three-space and on the intersection of two random quadrics in four-space, we give an explicit formula for this expectation.
A Borel probability measure $mu$ on a locally compact group is called a spectral measure if there exists a subset of continuous group characters which forms an orthogonal basis of the Hilbert space $L^2(mu)$. In this paper, we characterize all spectral measures in the field $mathbb{Q}_p$ of $p$-adic numbers.
For a proper semistable curve $X$ over a DVR of mixed characteristics we reprove the invariant cycles theorem with trivial coefficients (see Chiarellotto, 1999) i.e. that the group of elements annihilated by the monodromy operator on the first de Rham cohomology group of the generic fiber of $X$ coincides with the first rigid cohomology group of its special fiber, without the hypothesis that the residue field of $cal V$ is finite. This is done using the explicit description of the monodromy operator on the de Rham cohomology of the generic fiber of $X$ with coefficients convergent $F$-isocrystals given in Coleman and Iovita (2010). We apply these ideas to the case where the coefficients are unipotent convergent $F$-isocrystals defined on the special fiber (without log-structure): we show that the invariant cycles theorem does not hold in general in this setting. Moreover we give a sufficient condition for the non exactness.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
