Inter-universal Teichmüller Theory II: Hodge-Arakelov-theoretic Evaluation

Abstract. In the present paper, which is the second in a series of four papers, we study the Kummer theory surrounding the Hodge-Arakelov-theoretic evaluation — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory established by the author in previous papers — of the [reciprocal of the l-th root of the] theta function at l-torsion points, for l ≥ 5 a prime number. In the first paper of the series, we studied “miniature models of conventional scheme theory”, which we referred to as Θ±ellNF-Hodge theaters, that were associated to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F, together with a prime number l ≥ 5. The underlying Θ-Hodge theaters of these Θ±ellNF-Hodge theaters were glued to one another by means of “Θ-links”, that identify the [reciprocal of the l-th root of the] theta function at primes of bad reduction of EF in one Θ ±ellNF-Hodge theater with [2l-th roots of] the q-parameter at primes of bad reduction of EF in another Θ ±ellNF-Hodge theater. The theor