Hodge–Arakelov theory

In mathematics, Hodge–Arakelov theory of elliptic curves is an analogue of classical and p-adic Hodge theory for elliptic curves carried out in the framework of Arakelov theory. It was introduced by Mochizuki (1999).

Mochizuki's main comparison theorem in Hodge–Arakelov theory states (roughly) that the space of polynomial functions of degree less than d on the universal extension of a smooth elliptic curve in characteristic 0 is naturally isomorphic (via restriction) to the d²-dimensional space of functions on the d-torsion points. It is called a comparison theorem as it is an analogue for Arakelov theory of comparison theorems in cohomology relating de Rham cohomology to singular cohomology of complex varieties or étale cohomology of p-adic varieties.

In Mochizuki (1999) and Mochizuki (2002a) he pointed out that arithmetic Kodaira-Spencer map and Gauss-Manin connection may give some important hints for Vojta's conjecture, ABC conjecture and so on.

References

• Mochizuki, Shinichi (1999), The Hodge-Arakelov theory of elliptic curves: global discretization of local Hodge theories (PDF), Preprint No. 1255/1256, Res. Inst. Math. Sci., Kyoto Univ., Kyoto
• Mochizuki, Shinichi (2002a), "A survey of the Hodge-Arakelov theory of elliptic curves. I", in Fried, Michael D.; Ihara, Yasutaka, Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999) (PDF), Proc. Sympos. Pure Math. 70, Providence, R.I.: American Mathematical Society, pp. 533–569, ISBN 978-0-8218-2036-0, MR 1935421
• Mochizuki, Shinichi (2002b), "A survey of the Hodge-Arakelov theory of elliptic curves. II", Algebraic geometry 2000, Azumino (Hotaka) (PDF), Adv. Stud. Pure Math. 36, Tokyo: Math. Soc. Japan, pp. 81–114, ISBN 978-4-931469-20-4, MR 1971513