I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
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Heights on algebraic curves

Tanush Shaska, Lubjana Beshaj

I-AIMS paper 2014-1 · Published in NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. 41 (2015), 137–175 · 2015

In these lectures we cover basics of the theory of heights starting with the heights in the projective space, heights of polynomials, and heights of the algebraic curves. We define the minimal height of binary forms and moduli height for algebraic curves and prove that the moduli height of superelliptic curves satisfies H(f) ≤ c₀ H̃(f), where c₀ is a constant and H̃ the minimal height of the corresponding binary form. For genus g = 2 and 3 such constant is explicitly determined. Furthermore, complete lists of curves of genus 2 and genus 3 hyperelliptic curves with height 1 are computed.

MSC classes: Primary 11G50; Secondary 14H40

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References

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