Hyperelliptic curves and their invariants: Geometric, arithmetic and algorithmic aspects

Abstract

We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We focus on genus 3 hyperelliptic curves. Both geometric and arithmetic aspects are considered. 4. Field of definition 32 4.1. Field of definition and field of moduli: general facts 32 4.2. The hyperelliptic case 33 4.3. The hyperelliptic case with no extra-automorphism 35 4.4. The hyperelliptic case with extra-automorphisms 36 4.5. The hyperelliptic case of genus 3 36 4.6. The case of curves over finite fields 39 5. Conclusion 40 References 40 Appendix A. Stratum equations for the automorphism group D4 42

Generic separating sets for three-dimensional elasticity tensors

“…. , J 7 ) = 0 is the zero polynomial), but J 8 , J 9 , J 10 are subject to algebraic relations involving the first six invariants and known as syzygies, see [35,25].…”

Modular invariants for genus 3 hyperelliptic curves

“…In [19], the authors showed that these invariants are also generators of the algebra of invariants and determine hyperelliptic curves of genus 3 up to isomorphism in characteristic p > 7. Later, in his thesis [3], Basson provided some extra invariants that together with the classical Shioda invariants classify hyperelliptic curves of genus 3 up to isomorphism in characteristics 3 and 7.…”

“…We recall that the discriminant ∆ of a hyperelliptic curve C of genus 3 is an invariant of degree 14 (Section 1.5 of [19]). For our computations, we considered the following absolute 3 invariants, derived using the Shioda invariants: The numerical data in Table 5.1 shows the tight connection between the odd primes appearing in the denominators of these invariants, the odd primes of bad reduction for the hyperelliptic curve, and the odd primes dividing the denominators of j 1 , j 2 and j 3 .…”

On explicit descent of marked curves and maps

“…Now there is a notion of a normalized representative of such a point [16]. We call a polynomial π normalized if its tuple of coefficients is normalized.…”

“…So there exists a finite subset I of this set with this property as well. Let c I = (c i ) i∈I be the corresponding point in the projective space weighted by its indices, and choose a normalized representative c I,0 of c I [16]. There exists a scaling t → αt of t that transforms c I into c I,0 , which is uniquely determined up to a power of the automorphism t → ζ n t since the elements in I generate Z.…”