Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSome cases of Vojtas conjectures related to algebraic tori over function fields
61
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We first formulate a function field version of Vojtas generalized abc conjecture for algebraic tori. We then show a function field analogue of the Lang-Vojta Conjecture for varieties of log general type that are ramified covers of $mathbb G_m^n$. In particular, it includes the case $ mathbb P^nsetminus D$, where $D$ is a hypersurface over a function field in $mathbb P^n$ with $n+1$ irreducible components and $deg Dge n+2$. The main tools include generalizations of the techniques developed by Corvaja and Zannier in 2008 and 2013 and a gcd estimate of two multivariable polynomials over function fields evaluated at $S$-unit arguments. The gcd theorem obtained here is an adaptation of Levins methods for number fields in 2019 via a weaker version of Schmidts subspace theorem over function fields, which we derive with the use of Vojtas machine in a setting over the constant fields.
rate research
Read More
Let F/F_q be an algebraic function field of genus g defined over a finite field F_q. We obtain new results on the existence, the number and the density of dimension zero divisors of degree g-k in F where k is a positive integer. In particular, for q=2,3 we prove that there always exists a dimension zero divisor of degree gamma-1 where gamma is the q-rank of F. We also give a necessary and sufficient condition for the existence of a dimension zero divisor of degree g-k for a hyperelliptic field F in terms of its Zeta function.
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.
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $mathbb{R}$. All are supported by numerical verification to the extent possible.
We describe examples motivated by the work of Serre and Abhyankar.
We study the existence of non-special divisors of degree $g$ and $g-1$ for algebraic function fields of genus $ggeq 1$ defined over a finite field $F_q$. In particular, we prove that there always exists an effective non-special divisor of degree $ggeq 2$ if $qgeq 3$ and that there always exists a non-special divisor of degree $g-1geq 1$ if $qgeq 4$. We use our results to improve upper and upper asymptotic bounds on the bilinear complexity of the multiplication in any extension $F_{q^n}$ of $F_q$, when $q=2^rgeq 16$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
