Canonical Local Heights on Low Genus Curves
Q. Donker (TU Delft - Electrical Engineering, Mathematics and Computer Science)
B. Janssens – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Robin de Jong – Mentor (Universiteit Leiden)
D.C. Gijswijt – Graduation committee member (TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
In this thesis, we construct explicit formulas for canonical local height function on low genus curves. We do this based on an abstract definition of canonical local heights. In the case of elliptic curves over number fields there is a full ‘formulaire’ by Néron and Tate. We present a new proof of the formula for the canonical local height at a non-archimedean place, if the curve has multiplicative reduction at that place. We do this by evaluating the normalized tropical Riemann theta function on the Jacobian of the reduction graph.
In the case of a Jacobian of a genus 2 curve with semistable reduction, we can again evaluate the normalized tropical Riemann theta function, to obtain formulas for the Néron correction term, at non-archimedean places. This yields seven formulas, one for every semistable reduction type of the curve. These formulas are dependent on the tropical theta characteristic, which is generally hard to find. Under certain assumptions on the tropical theta characteristic, these formulas resemble formulas for canonical local heights on the Kummer surface. This leads us to explicitly calculate the intersection multiplicity of a point on the Jacobian, with the Theta divisor, by evaluating functions on the Kummer surface. This suggests that our assumptions may be correct, and should bring us closer to understanding the tropical theta characteristic