We work with completed adelic structures on an arithmetic surface and justify that the construction under consideration is compatible with Arakelov geometry. The ring of completed adeles is algebraically and topologically self-dual and fundamental adelic subspaces are self orthogonal with respect to a natural differential pairing. We show that the Arakelov intersection pairing can be lifted to an idelic intersection pairing.
We prove a generalisation of Roth's theorem for proper adelic curves, assuming that the logarithmic absolute values of the approximants satisfy a condition similar to the equicontinuity with respect to the places. This work extends Corvaja's results [Cor97] for fields admitting a product formula, and Voita's ones [Voj21] for arithmetic function fields.
Adelic curvesWe will use the following notations throughout the whole paper: log + x := max{0, log x} , log − x := min{0, log x} ; ∀x ∈ R>0
We give an explicit formula for the Deligne pairing for proper and flat morphisms $$f:X\rightarrow S$$
f
:
X
→
S
of schemes, in terms of the determinant of cohomology. The whole construction is justified by an analogy with the intersection theory on non-singular projective algebraic varieties.
We give an explicit formula for the Deligne pairing for a proper and flat morphisms f : X → S of schemes, in terms of the determinant of cohomology. The whole construction is justified by an analogy with the intersection theory on non-singular projective algebraic varieties.
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