Abstract
We study probability distributions arising from local obstructions to the existence of p-adic points in families of varieties. In certain cases we show that an Erdős–Kac type normal distribution law holds.
Highlights
Of the number of varieties in the family with a rational point. (Here H is the usual naive height on Pn(Q))
For an explicit non-negative (π ) ∈ Q. (Here AQ denotes the adeles of Q.) they conjectured in [14, Conj. 1.6] that the upper bound (1.1) is sharp, under the necessary assumptions that the set being counted is non-empty and that the fibre over every codimension 1 point of Pn contains an irreducible component of multiplicity 1
Where X (1) denotes the set of codimension 1 points of X. These invariants are defined by group theoretic data which can often be calculated in practice
Summary
The authors considered the closely related problem of counting the number of varieties in the family which are everywhere locally soluble They proved an upper bound of the shape. 1.6] that the upper bound (1.1) is sharp, under the necessary assumptions that the set being counted is non-empty and that the fibre over every codimension 1 point of Pn contains an irreducible component of multiplicity 1. As it will occur frequently in our results, we recall the definition of (π ) here. A measure-theoretic interpretation of Theorem 1.2 is as follows: It says that
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