Bibliographic Information: Bartsch, F., Campana, F., Javanpeykar, A., & Wittenberg, O. (2024). The Weakly Special Conjecture contradicts orbifold Mordell, and thus abc. arXiv preprint arXiv:2410.06643.
Research Objective: This paper aims to disprove the Weakly Special Conjecture, a significant conjecture in arithmetic geometry, by constructing counterexamples using Lafon's Enriques surface and exploring the implications of these findings.
Methodology: The authors utilize Lafon's construction of an Enriques surface over Q(t) with no C((t))-point. They analyze the properties of this surface, particularly the presence of an inf-multiple, non-divisible fiber, to construct weakly special threefolds. They then employ the concepts of orbifold bases, orbifold morphisms, and the Orbifold Mordell Conjecture to demonstrate that these threefolds violate the Weakly Special Conjecture.
Key Findings:
Main Conclusions: The authors definitively disprove the Weakly Special Conjecture by providing explicit counterexamples. They highlight the distinction between "weakly special" and "special" varieties, suggesting a revised conjecture where "special" replaces "weakly special." The findings also contribute to a deeper understanding of the geometry and arithmetic of Enriques surfaces and K3 surfaces.
Significance: This research significantly impacts the field of arithmetic geometry by disproving a long-standing conjecture. It prompts a reassessment of the Weakly Special Conjecture and encourages the exploration of alternative conjectures. The construction of explicit counterexamples provides valuable insights into the properties of weakly special varieties and their relationship to other conjectures in number theory.
Limitations and Future Research: The authors focus on specific types of weakly special threefolds constructed using Lafon's Enriques surface. Further research could explore other potential counterexamples and investigate the implications of these findings for related conjectures in arithmetic geometry. Additionally, exploring the properties of the non-divisible but nowhere reduced degenerations of Enriques surfaces and K3 surfaces could be a fruitful avenue for future research.
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