Adelic geometry on arithmetic surfaces, I: Idelic and adelic interpretation of the Deligne pairing

Paolo Dolce

Research output: Contribution to journalArticlepeer-review

Abstract

For an arithmetic surface X → B = SpecOK, the Deligne pairing 〈, 〉 : Pic(X)×Pic(X)→Pic(B) gives the "schematic contribution"to the Arakelov intersection number. We present an idelic and adelic interpretation of the Deligne pairing; this is the first crucial step for a full idelic and adelic interpretation of the Arakelov intersection number. For the idelic approach, we show that the Deligne pairing can be lifted to a pairing 〈, 〉i : ker(d1×)×ker(d1×)→Pic(B), where ker(d1×) is an important subspace of the two-dimensional idelic groupA× X. On the other hand, the argument for the adelic interpretation is entirely cohomological.

Original languageEnglish
Pages (from-to)433-470
Number of pages38
JournalKyoto Journal of Mathematics
Volume62
Issue number2
DOIs
StatePublished - 1 Jun 2022
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics (all)

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