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Geometric G-functions and Atypicality
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abstract
We describe a general method for giving $p$-adic interpretations of $G$-functions arising from degenerating periods of smooth projective algebraic varieties. Using this, we are able to implement a strategy due to Andr\'e for bounding heights of moduli points where period functions acquire unusual algebraic relations. This leads to new results on Galois lower bounds for special moduli, and new cases of the Zilber-Pink conjecture. In particular, we establish the first Galois-orbit lower bounds on CM moduli in non-Shimura settings. As a more technical contribution, we introduce a refinement of the Pila-Zannier strategy capable of handling Zilber-Pink-type atypical intersection problems in arbitrary dimension and for arbitrary smooth projective families.
Forward citations
Cited by 3 Pith papers
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What makes an algebraic curve special?
A survey of special curves and special subvarieties of moduli space, unifying Hodge-theoretic, Teichmüller, and bi-algebraic perspectives, with a few new results and conjectures.
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G-functions, motives, and unlikely intersections -- old and new
A survey of the G-function method, its link to periods, and its use in proving cases of unlikely intersection conjectures.
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Unlikely intersections in Shimura varieties and beyond: a survey
A survey of unlikely intersections in pure Shimura varieties, covering Andre-Oort, Andre-Pink-Zannier, Zilber-Pink, and the Pila-Zannier strategy.
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