2013/10/29 by Rafael von Känel, von Känel, Rafael
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.1310.7980
Comments are very welcome
arxiv created 2013/10/29 · openalex publication_date 2013/10/29 · arxiv updated 2013/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider generalizations of Szpiro's classical discriminant conjecture to hyperelliptic curves over a number field K, and to smooth, projective and geometrically connected curves X over K of genus at least one. The main results give effective exponential versions of the generalized conjectures for some curves, including all curves X of genus one or two. We obtain in particular exponential versions of Szpiro's classical discriminant conjecture for elliptic curves over K. In course of our proofs we establish explicit results for certain Arakelov invariants of hyperelliptic curves (e.g. Faltings' delta invariant) which are of independent interest. The proofs use the theory of logarithmic forms and Arakelov theory for arithmetic surfaces.