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Abstract Let $$K$$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $$K$$. We study images of semialgebraic subsets of $K^n$ under this map from a general point of view. For a semialgebraic set $$S \subseteq K^n$$ we define a space $$S_r^{{\operatorname{an}}}$$ called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of $$S$$. We prove a real analogue of the tropical fundamental theorem and show that the tropicalization of any semialgebraic set is described by tropicalization of finitely many inequalities, which are valid on the semialgebraic set. We also study the topological properties of real analytification and tropicalization. If $$X$$ is an algebraic variety, we show that $$X_r^{{\operatorname{an}}}$$ can be canonically embedded into the real spectrum $$X_r$$ of $$X$$, and we study its relation with the Berkovich analytification of $$X$$.
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Reductions of abelian surfaces over global function fields
Let $$A$$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $$A$$ does not have real multiplication by any real quadratic field with discriminant a multiple of $$p$$ . We prove that there are infinitely many places modulo which $$A$$ is isogenous to the product of two elliptic curves.
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- PAR ID:
- 10366113
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 158
- Issue:
- 4
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 893 to 950
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X22007473
