Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPositivity of Heights of Semistable Varieties
78
0
0.0
(
0
)
Authors
Roberto G. Ferretti
Ask ChatGPT about the research
No Arabic abstract
We completely solve a problem of S. Zhang about the positivity of a normalized height on the moduli space of semistable varieties of given degree and given dimension.
rate research
Read More
We give an explicit description of the stable reduction of superelliptic curves of the form $y^n=f(x)$ at primes $p$ whose residue characteristic is prime to the exponent $n$. We then use this description to compute the local $L$-factor of the curve and the exponent of conductor at $p$.
We determine the irreducible components of Igusa varieties for Shimura varieties of Hodge type and use that to determine the irreducible components of central leaves. In particular, we show that the discrete Hecke-orbit conjecture is false in general. Our method combines recent work of DAddezio on monodromy of compatible local systems with a generalisation of a method of Hida, using the Honda-Tate theory for Shimura varieties of Hodge type developed by Kisin-Madapusi Pera-Shin. We also determine the irreducible components of Newton strata in Shimura varieties of Hodge type by combining our results with recent work of Zhou-Zhu.
The integral model of a GU(n-1,1) Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the Gillet-Soule arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet L-functions.
Let k be a perfect field of characteristic p>0. We prove the existence of ascending and descending slope filtrations for Shimura p-divisible objects over k. We use them to classify rationally these objects over bar k. Among geometric applications, we mention two. First we formulate Manin problems for Shimura varieties of Hodge type. We solve them if either pGe 3 or p=2 and two mild conditions hold. Second we formulate integral Manin problems. We solve them for certain Shimura varieties of PEL type.
Let $E$ be an elliptic curve without CM that is defined over a number field $K$. For all but finitely many nonarchimedean places $v$ of $K$ there is the reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field $k(v)$ at $v$. The set of $v$s with ordinary $E(v)$ has density 1 (Serre). For such $v$ the endomorphism ring $End(E(v))$ of $E(v)$ is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers $N$ and $M$ there are infinitely many nonarchimedean places $v$ of $K$ such that the discriminant $Delta(v)$ of $End(E(v))$ is divisible by $N$ and the ratio $Delta(v)/N$ is relatively prime to $NM$. We also discuss similar questions for reductions of abelian varieties. The subject of this paper was inspired by an exercise in Serres Abelian $ell$-adic representations and elliptic curves and questions of Mihran Papikian and Alina Cojocaru.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
