2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

Abstract

For each group G, (|G| > 2) which acts as a full automorphism group on a genus 3 hyperelliptic curve, we determine the family of curves which have 2-Weierstrass points. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. The 1-dimensional families that we discover have the property that they contain only genus 0 components.

From hyperelliptic to superelliptic curves

“…Weierstrass points of superelliptic curves. Most of this section is summarizing the results in [104] and [102]. Let C g be a smooth superelliptic curve given by an affine equation…”

“…The material of this section can be found in every book on the subject. We mostly refer to [7,30,102,104].…”

On automorphisms of algebraic curves

“…(6). Then i) g ≤ 1 2 (n − 1)(m − 1) ii) If no point of X other than y ∞ is singular, then g = 1 2 (n − 1)(m − 1) For more details about Weierstrass points and their weights see [64], [65] where Weierstrass points of superelliptic curves are studied.…”

“…For more details about Weierstrass points and their weights see [64], [65] where Weierstrass points of superelliptic curves are studied.…”

Higher-order Weierstrass weights of branch points on superelliptic curves

“…In the non-singular models of these curves, there are G = gcd(n, d) = 2 points at infinity P ∞ 1 and P ∞ 2 . If 4b = a 2 , then [13,Lemma 4 and Proposition 3], the authors consider hyperelliptic curves of genus 3 of the form y 2 = f (x) where deg(f ) = 8. In the non-singular models of these curves, there are G = gcd(n, d) = 2 points at infinity P ∞ 1 and P ∞ 2 .…”