| 释义 |
- References
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 {{harvs|txt|last=Mochizuki|authorlink=Shinichi Mochizuki|year=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 {{harvs|txt|last=Mochizuki|authorlink=Shinichi Mochizuki|year=1999}} and {{harvs|txt|last=Mochizuki|authorlink=Shinichi Mochizuki|year=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- {{Citation | last1=Mochizuki | first1=Shinichi | title=The Hodge-Arakelov theory of elliptic curves: global discretization of local Hodge theories | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves.pdf | series= Preprint No. 1255/1256, Res. Inst. Math. Sci., Kyoto Univ., Kyoto | year=1999}}
- {{Citation | last1=Mochizuki | first1=Shinichi | editor1-last=Fried | editor1-first=Michael D. | editor2-last=Ihara | editor2-first=Yasutaka | title=Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999) | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Survey%20of%20the%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves%20I.pdf | publisher=American Mathematical Society | location=Providence, R.I. | series=Proc. Sympos. Pure Math. | isbn=978-0-8218-2036-0 |mr=1935421 | year=2002a | volume=70 | chapter=A survey of the Hodge-Arakelov theory of elliptic curves. I | pages=533–569}}
- {{Citation | last1=Mochizuki | first1=Shinichi | title=Algebraic geometry 2000, Azumino (Hotaka) | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Survey%20of%20the%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves%20II.pdf | publisher=Math. Soc. Japan | location=Tokyo | series=Adv. Stud. Pure Math. | isbn=978-4-931469-20-4 |mr=1971513 | year=2002b | volume=36 | chapter=A survey of the Hodge-Arakelov theory of elliptic curves. II | pages=81–114}}
{{DEFAULTSORT:Hodge-Arakelov theory}} 2 : Number theory|Algebraic geometry |