Logarithmic abelian varieties. III: Logarithmic elliptic curves and modular curves (Q2840044)

This is the third in a series of papers [J. Math. Sci., Tokyo 15, No. 1, 69--193 (2008; Zbl 1156.14038); Nagoya Math. J. 189, 63--138 (2008; Zbl 1169.14031)] whose aim is to re-work the theory of degenerations of abelian varieties by systematically using logarithmic geometry. In their framework, a family of abelian varieties degenerates to a ``log abelian variety'', which is in an appropriate sense both smooth, proper, and a group (which is of course impossible outside the log world!). As an example of the advantages of their set-up, one can consider the problem of obtaining modular interpretations of toroidal compactifications. After the work of Alexeev and Olsson (see [\textit{M. C. Olsson}, Compactifying moduli spaces for abelian varieties. Berlin: Springer (2008; Zbl 1165.14004)]), there is such an interpretation for the second Voronoi compactification of \(A_g\); however, the approach of Kajiwara-Kato-Nakayama provides in a sense a uniform modular interpretation of all toroidal compactifications at the same time. For all this we refer to the first two papers in the series (which treat the analytic and the algebraic theory, respectively).NEWLINENEWLINEIn this paper the authors illustrate their theory by recasting the work of [\textit{P. Deligne} and \textit{M. Rapoport}, Lect. Notes Math. 349, 143--316 (1973; Zbl 0281.14010)] in their language -- that is, the case of degenerating elliptic curves. Deligne and Rapaport (working over a base where the level is invertible) provided a moduli-theoretic interpretation of compactified modular curves in terms of ``generalized elliptic curves''; a degenerate elliptic curve is for them a polygon of N rational curves, where \(N\) needs to depend on what kind of level structure the degenerate curve should support. In contrast, the log approach is far less ad hoc: Kajiwara-Kato-Nakayama can simply define \(X(N)\) as the moduli space parametrizing log elliptic curves equipped with an isomorphism of their N-torsion with \((\mathbb Z/N)^2\), and similarly for other kinds of level structure. (They repeatedly emphasize that the log approach allows one to treat smooth and degenerate elliptic curves on equal footing.) The main result is that the scheme \(X(N)\) defined by Deligne-Rapoport represents also the moduli functor defined using level structures on log elliptic curves.NEWLINENEWLINEIn an appendix they explain precisely the relationship between the universal log elliptic curve over \(X(N)\) and the universal generalized elliptic curve.