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G-functions, motives, and unlikely intersections -- old and new
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In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more ``motivic" symmetries; and how Bombieri's principle of global relations can be used to control the height of such parameters (which was also one of the origins of the Andr\'e-Oort conjecture). At the end, we sketch the recent revival of the G-function method in the context of unlikely intersections and the Zilber-Pink conjecture.
Forward citations
Cited by 3 Pith papers
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On Siegel's problem and Dwork's conjecture for $G$-functions
G-functions of order two exist that are not polynomial expressions in algebraic pullbacks of hypergeometric functions, answering Siegel's problem negatively and adding counterexamples to Dwork's conjecture.
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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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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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