A $p$-adic approach to rational points on curves
| Autor: | Bjorn Poonen |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Mathematics - Number Theory Applied Mathematics General Mathematics Mathematics::Number Theory Galois module Mathematics - Algebraic Geometry Rational point Genus (mathematics) FOS: Mathematics Number Theory (math.NT) Algebraic Geometry (math.AG) 11G30 (Primary) 11G20 14D07 14D10 14G05 14H25 (Secondary) Mathematics |
| Popis: | In 1922, Mordell conjectured the striking statement that for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983, and again by a different method by Vojta in 1991, but neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of $p$-adic Galois representations; this is the subject of the present exposition. 14 pages. This expository article is associated with a lecture given January 17, 2020 in the Current Events Bulletin at the 2020 Joint Mathematics Meetings in Denver |
| Databáze: | OpenAIRE |
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