A truncated second main theorem for algebraic tori with moving targets and applications

Guo, Jing; Sun, Chia‐Liang; Wang, Julie Tzu-Yueh

doi:10.48550/arxiv.2103.16087

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2021

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“…Finally we note that in the complex analogue, the corresponding results of Theorem 1, Theorem 2 and Theorem 3 over C were proved by Noguchi, Winkelmann and Yamonoi [19] and [20] in more generality; and the case of Theorem 1 and Theorem 2 with hypersurfaces over "small functions" were established in [12].…”

Section: Introductionmentioning

confidence: 80%

Vojta's abc Conjecture for algebraic tori and applications over Function Fields

Guo¹,

Sun²,

Wang³

2021

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We first formulate a function field version of Vojta's 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 G n m . In particular, it includes the case P n \ D, where D is a hypersurface over a function field in P n with n + 1 irreducible components and deg D ≥ n + 2. The main tools include generalizations of the techniques developed by Corvaja and Zannier in [7] and [8] 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 Levin's methods for number fields in [17] via a weaker version of Schmidt's subspace theorem over function fields (cf. [33]), which we derive with the use of Vojta's machine in [26] in a setting over the constant fields.

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Section: Introductionmentioning

confidence: 80%