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Article Dans Une Revue Transactions of the American Mathematical Society Année : 2022

GLOBAL WEIERSTRASS EQUATIONS OF HYPERELLIPTIC CURVES

Qing Liu

Résumé

Given a hyperelliptic curve C of genus g over a number field K and a Weierstrass model {\mathsrc C} of C over the ring of integers O_K (i.e. the hyperelliptic involution of C extends to {\mathsrc C} and the quotient is a smooth model of P1_K over OK), we give necessary and sometimes sufficient conditions for {\mathsrc C} to be defined by a global Weierstrass equation. In particular, if C has everywhere good reduction, we prove that it is defined by a global Weierstrass equation with invertible discriminant if the class number hK is prime to 2(2g+1), confirming a conjecture of M. Sadek.
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hal-03617410 , version 1 (23-03-2022)

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  • HAL Id : hal-03617410 , version 1

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Qing Liu. GLOBAL WEIERSTRASS EQUATIONS OF HYPERELLIPTIC CURVES. Transactions of the American Mathematical Society, In press. ⟨hal-03617410⟩

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