Bachelorcolloquium Paul Helmink

Bachelorcolloquium Paul Helmink

Begeleider: Jaap Top
Datum: 6 juli
Tijd: 12.00
Zaal: 5161.0289

Titel: Tropical Elliptic Curves

An elliptic curve over any field K can be given in affine form as the zero locus of a cubic polynomial in two variables. Elliptic curves over the field P of complex Puiseux series will be of particular interest in our discourse. These elliptic curves can be tropicalised to obtain a new piece-wise linear curve known as the tropical elliptic curve. Obtaining relations between elliptic curves over P and their tropical counterpart shall be the main goal.

In particular every elliptic curve over the field of Puiseux series has an associated j-invariant: an element of P which all isomorphic elliptic curves have in common. One might wonder whether such an invariant still exists in the tropical world. The answer lies in the honeycomb structure that every tropical elliptic curve has. This structure may or may not contain an isolated complex: a cycle. The length of such a cycle is an invariant of the tropical elliptic curve. This invariant can be linked to the original j-invariant by means of the natural valuation on P.

To prove that these two invariants are related, a technique called reduction is used. Every elliptic curve over P can be reduced to an elliptic curve over the field of complex numbers. The reduction process can however turn a nonsingular elliptic curve into a singular one. If this occurs, then the curve is said to have bad reduction. It can be shown that the tropical versions of elliptic curves with bad reduction have a cycle whose length is equal to the valuation of the j-invariant.