scholarly journals Lang–Vojta conjecture over function fields for surfaces dominating $${{\mathbb {G}}}_m^2$$

2021 ◽  
Author(s):  
Laura Capuano ◽  
Amos Turchet
Keyword(s):  
General Type ◽  
Function Fields ◽  
Ramified Covers

AbstractWe prove the nonsplit case of the Lang–Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $${{\mathbb {G}}}_m^2$$ G m 2 . This extends the results of Corvaja and Zannier (J Differ Geom 93(3):355–377, 2013), where the conjecture was proved in the split case, and the results of Corvaja and Zannier (J Algebr Geom 17(2):295–333, 2008), Turchet (Trans Amer Math Soc 369(12):8537–8558, 2017) that were obtained in the case of the complement of a degree four and three component divisor in $${{\mathbb {P}}}^2$$ P 2 . We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.

scholarly journals Double Kodaira fibrations with small signature

2020 ◽  
Vol 31 (07) ◽  
pp. 2050052 ◽  
Author(s):  
Ju A Lee ◽  
Michael Lönne ◽  
Sönke Rollenske
Keyword(s):  
General Type ◽  
Fiber Bundles ◽  
Surfaces Of General Type ◽  
Positive Signature ◽  
Fix Point ◽  
Ramified Covers

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fiber bundles. They are known to have positive signature divisible by [Formula: see text]. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent fibrations. Special attention is paid to ramified covers of product of curves which we analyze by studying the monodromy action for bundles of punctured curves. As a by-product, we obtain a classification of all fix-point-free automorphisms on curves of genus at most [Formula: see text].

scholarly journals Quadratic sequences of powers and Mohanty’s conjecture

2018 ◽  
Vol 14 (02) ◽  
pp. 479-507 ◽  
Author(s):  
Natalia Garcia-Fritz
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progression ◽  
General Type ◽  
Function Fields ◽  
Hyperelliptic Curves ◽  
Arithmetic Progressions ◽  
Surfaces Of General Type ◽  
Genus Zero ◽  
Curves Of Low Genus

We prove under the Bombieri–Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in arithmetic progression. This answers a question proposed in 2010 by Browkin and Brzezinski, and independently by Gonzalez-Jimenez and Xarles. We also show that under the Bombieri–Lang conjecture for surfaces, for every [Formula: see text] there is an absolute bound on the length of sequences formed by [Formula: see text]th powers with constant second differences. This gives a conditional result on one of Mohanty’s conjectures on arithmetic progressions in Mordell’s elliptic curves [Formula: see text]. Moreover, we obtain an unconditional result regarding infinite families of such arithmetic progressions. We also study the case of hyperelliptic curves of the form [Formula: see text]. These results are proved by unconditionally finding all curves of genus zero or one on certain surfaces of general type. Moreover, we prove the unconditional analogues of these arithmetic results for function fields by finding all the curves of low genus on these surfaces.

scholarly journals Ramified covers of abelian varieties over function fields

1994 ◽  
Vol 1 (4) ◽  
pp. 465-467 ◽  
Author(s):  
José Felipe Voloch
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Ramified Covers

scholarly journals Nonspecial varieties and generalised Lang–Vojta conjectures

2021 ◽  
Vol 9 ◽  
Author(s):  
Erwan Rousseau ◽  
Amos Turchet ◽  
Julie Tzu-Yueh Wang
Keyword(s):  
Number Field ◽  
Function Field ◽  
General Type ◽  
Function Fields ◽  
Rational Points ◽  
Special Variety ◽  
Recent Method ◽  
Dense Set

Abstract We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

2021 ◽  
pp. 1-16
Author(s):  
Alexander Dabrowski
Keyword(s):  
General Type ◽  
Differential Operators ◽  
Laplacian Operator ◽  
Variational Characterization ◽  
Repeated Eigenvalues ◽  
Direct Extension ◽  
Asymptotic Formulae ◽  
Elliptic Differential Operators ◽  
Domain Perturbations ◽  
Boundary Deformation

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

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