We consider generalizations of Szpiro's classical discriminant conjecture tohyperelliptic curves over a number field $K$, and to smooth, projective andgeometrically connected curves $X$ over $K$ of genus at least one. The mainresults give effective exponential versions of the generalized conjectures forsome curves, including all curves $X$ of genus one or two. We obtain inparticular exponential versions of Szpiro's classical discriminant conjecturefor elliptic curves over $K$. In course of our proofs we establish explicitresults for certain Arakelov invariants of hyperelliptic curves (e.g. Faltings'delta invariant) which are of independent interest. The proofs use the theoryof logarithmic forms and Arakelov theory for arithmetic surfaces.
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机译：我们考虑了Szpiro的经典判别猜想在数域$ K $上的超椭圆曲线的推广，以及对至少一个属的$ K $的光滑，投影和几何相关的曲线$ X $的推广。主要结果给出了某些曲线的广义猜想的有效指数形式，包括一或两个属的所有曲线$ X $。对于$ K $的椭圆曲线，我们获得Szpiro经典判别猜想的特定指数版本。在我们的证明过程中，我们建立了具有独立利益的某些超椭圆曲线的Arakelov不变量（例如Faltings'delta不变量）的显式结果。证明将对数形式的理论和Arakelov理论用于算术曲面。
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