Submission date: 9 June 2023

Abstract:

We use the differentiability of the arithmetic volume function and an arithmetic Bertini type theorem to classify when one can find a closed point on the generic fiber of an arithmetic variety, whose heights with respect to some finite tuple of arithmetic ℝ-divisors approximate a given tuple of real numbers. We use this result to prove existential closedness of ℚ̅ as a globally valued field (abbreviated GVF). We introduce GVF functionals on the space of arithmetic ℝ-divisors and interpret the essential infimum function as the infimum of values of normalised GVF functionals, at least when the generic part of the arithmetic &Roopf;-divisor is big. We also give a new criterion on equality in one of the Zhang's inequalities.

Mathematics Subject Classification:

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Full text arXiv 2306.06275: pdf, ps.