Smirnov's arithmetic defect conjecture for maps on the compactified spectrum of the integers

Let q=a/bq=a/b be a nonzero rational number, let φq:SpecZSmiSpecZSmi\varphi_q:\overline{\operatorname{Spec} \mathbb Z}^{\mathrm{Smi}}\to\overline{\operatorname{Spec} \mathbb Z}^{\mathrm{Smi}} be the associated map, and let δ[x]\delta_{[x]} denote the arithmetic defect of a point [x][x]. Define

X(q)={[x]SpecZφq([x]) has degree 1}.X(q)=\{[x]\in\overline{\operatorname{Spec} \mathbb Z}\mid \varphi_q([x])\text{ has degree }1\}.

For each ϵ>0\epsilon>0, Smirnov's arithmetic defect conjecture. there exists a constant CC satisfying

[x]X(q)δ[x]2+ϵ+Cdeg(φq).\sum_{[x]\in X(q)}\delta_{[x]}\leq 2+\epsilon+\frac{C}{\operatorname{deg}(\varphi_q)}.

This is an analogue of the function-field abc framework, expressing a uniform bound on the total arithmetic defect of degree-one points. The supplied source does not establish the assertion or provide evidence of its resolution.

Sources & referencesView supporting material

Primary source

Manoel Jarra, “On Smirnov's approach to the ABC conjecture”, arXiv:2306.16637 (2024).

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