Do you want to publish a course? Click here

Singularities and vanishing cycles in number theory over function fields

192   0   0.0 ( 0 )
 Added by Will Sawin
 Publication date 2020
  fields
and research's language is English
 Authors Will Sawin




Ask ChatGPT about the research

This article is an overview of the vanishing cycles method in number theory over function fields. We first explain how this works in detail in a toy example, and then give three examples which are relevant to current research. The focus will be a general explanation of which sorts of problems this method can be applied to.



rate research

Read More

We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.
We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the $j$-invariant in an isogeny class. The second one is an isogeny estimate, providing an explicit bound on the degree of a minimal isogeny between two isogenous elliptic curves. We also give several corollaries of these two results.
151 - Takehiro Hasegawa 2017
Elkies proposed a procedure for constructing explicit towers of curves, and gave two towers of Shimura curves as relevant examples. In this paper, we present a new explicit tower of Shimura curves constructed by using this procedure.
Over any field of characteristic not 2, we establish a 2-term resolution of the $eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the K(1)-local sphere in classical stable homotopy theory. As applications we determine the $eta$-periodized motivic stable stems and the $eta$-periodized algebraic symplectic and SL-cobordism groups. Along the way we construct Adams operations on the motivic spectrum representing Hermitian K-theory and establish new completeness results for certain motivic spectra over fields of finite virtual 2-cohomological dimension. In an appendix, we supply a new proof of the homotopy fixed point theorem for the Hermitian K-theory of fields.
We generalize the classical theory of periodic continued fractions (PCFs) over ${mathbf Z}$ to rings ${mathcal O}$ of $S$-integers in a number field. Let ${mathcal B}={beta, {beta^*}}$ be the multi-set of roots of a quadratic polynomial in ${mathcal O}[x]$. We show that PCFs $P=[b_1,ldots,b_N,bar{a_1ldots ,a_k}]$ of type $(N,k)$ potentially converging to a limit in ${mathcal B}$ are given by ${mathcal O}$-points on an affine variety $V:=V({mathcal B})_{N,k}$ generically of dimension $N+k-2$. We give the equations of $V$ in terms of the continuant polynomials of Wallis and Euler. The integral points $V({mathcal O})$ are related to writing matrices in $textrm{SL}_2({mathcal O})$ as products of elementary matrices. We give an algorithm to determine if a PCF converges and, if so, to compute its limit. Our standard example generalizes the PCF $sqrt{2}=[1,bar{2}]$ to the ${mathbf Z}_2$-extension of ${mathbf Q}$: $F_n={mathbf Q}(alpha_n)$, $alpha_{n}:=2cos(2pi/2^{n+2})$, with integers ${mathcal O}_n={mathbf Z}[alpha_n]$. We want to find the PCFs of $alpha_{n+1}$ over ${mathcal O}_{n}$ of type $(N,k)$ by finding the ${mathcal O}_{n}$-points on $V({mathcal B}_{n+1})_{N,k}$ for ${mathcal B}_{n+1}:={alpha_{n+1}, -alpha_{n+1}}$. There are three types $(N,k)=(0,3), (1,2), (2,1)$ such that the associated PCF variety $V({mathcal B})_{N,k}$ is a curve; we analyze these curves. For generic ${mathcal B}$, Siegels theorem implies that each of these three $V({mathcal B})_{N,k}({mathcal O})$ is finite. We find all the ${mathcal O}_n$-points on these PCF curves $V({mathcal B}_{n+1})_{N,k}$ for $n=0,1$. When $n=1$ we make extensive use of Skolems $p$-adic method for $p=2$, including its application to Ljunggrens equation $x^2 + 1 =2y^4$.
comments
Fetching comments Fetching comments
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا