Greatest common divisors and Vojta’s conjecture for blowups of algebraic tori

2018 ◽  
Vol 215 (2) ◽  
pp. 493-533 ◽  
Author(s):  
Aaron Levin
Keyword(s):  
Algebraic Tori ◽  
Vojta's Conjecture ◽  
Greatest Common Divisors

scholarly journals Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

2020 ◽  
Vol 2020 (767) ◽  
pp. 77-107 ◽  
Author(s):  
Aaron Levin ◽  
Julie Tzu-Yueh Wang
Keyword(s):  
Diophantine Approximation ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Counting Function ◽  
Upper Bounds ◽  
General Version ◽  
Moving Targets ◽  
Algebraic Tori ◽  
Greatest Common Divisors ◽  
Common Zeros

AbstractWe study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural “asymptotic gcd” inequality of Pasten and the second author, and consider moving targets versions of our results.

scholarly journals On S-class number relations of algebraic tori in Galois extensions of global fields

1991 ◽  
Vol 124 ◽  
pp. 133-144 ◽  
Author(s):  
Masanori Morishita
Keyword(s):  
Class Number ◽  
Quadratic Forms ◽  
Euler Number ◽  
Number Fields ◽  
Binary Quadratic Forms ◽  
Algebraic Number Fields ◽  
Algebraic Tori ◽  
Genus Theory ◽  
Class Number Formula ◽  
Number Formula

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

scholarly journals Corrigenda On a classification of the function fields of algebraic tori: (Nagoya Math. J. 56 (1975), 85–104)

1980 ◽  
Vol 79 ◽  
pp. 187-190 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Prime Divisor ◽  
Dihedral Group ◽  
Quaternion Group ◽  
Algebraic Tori ◽  
Root Of Unity ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.

Integral structures in algebraic tori

1995 ◽  
Vol 59 (5) ◽  
pp. 881-897 ◽  
Author(s):  
V E Voskresenskii ◽  
T V Fomina
Keyword(s):  
Algebraic Tori ◽  
Integral Structures
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