Abstract
In this final chapter, we end our story by beginning another. Although we have mostly worked over ℂ, and occasionally over a general algebraically closed field, algebraic geometry can be done over any field. Each field has its own character: transcendental over ℂ, and arithmetic over fields such as ℚ, \(\mathbb{F}_p\),&. It may seem that aside from a few formal similarities, the arithmetic and transcendental sides would have very little to do with each other. But in fact they are related in deep and mysterious ways. We start by briefly summarizing the results ofWeil, Grothendieck, and Deligne for finite fields. Then we return to complex geometry and prove Serre’s analogue of the Weil conjecture. This result inspired Grothendieck to formulate his standard conjectures.We explain some of these along with the closely related Hodge conjecture. These are among the deepest open problems in algebraic geometry.
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© 2012 Springer Science+Business Media, LLC
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Arapura, D. (2012). Analogies and Conjectures. In: Algebraic Geometry over the Complex Numbers. Universitext. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1809-2_19
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DOI: https://doi.org/10.1007/978-1-4614-1809-2_19
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Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4614-1808-5
Online ISBN: 978-1-4614-1809-2
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