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Algebraic Geometry is a powerful tool of number theory because much of it works over any field. It allows one to translate geometric intuition (algebraic geomet
by kevinventullo 8y ago
Algebraic Geometry is a powerful tool of number theory because much of it works over any field. It allows one to translate geometric intuition (algebraic geometry over the complex numbers) into a more algebraic environment (finite, p-adic, or number fields).
Going back further, algebraic geometry over the complex numbers was shown in the early 20th century to be in many ways equivalent to more classical analytic geometry: https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry https://en.wikipedia.org/wiki/Algebraic_geometry_and_analyti...
- heinrichf 8y agoMore than over any field, over any ring. For example, that allows you to look at families of curves or talk about reduction modulo p (if you do number theory) very nicely/naturally. A mathoverflow thread with cool uses of schemes: https://mathoverflow.net/questions/59071/what-elementary-problems-can-you-solve-with-schemes?answertab=votes#tab-top https://mathoverflow.net/questions/59071/what-elementary-pro...
- selimthegrim 8y agoThe book by Gelfand, Kapranov and Zelevinsky is a magical mystery tour of precisely this.