GRH and finite flat group schemes over $\mathbf{Z}$
Lassina Dembélé1 ; René Schoof2
1 Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK
2 Dipartimento di Matematica, 2 a Università di Roma, “Tor Vergata”, I-00133 Roma, Italy
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 569-578

Since simple commutative finite flat group schemes $G$ over $\mathbf{Z}$ are killed by a prime number $p$, their order is a power of $p$. Tate asked whether a simple group scheme $G$ is necessarily equal to $\mathbf{Z}/p\mathbf{Z}$ or $\mu _p$. This has been proved for primes $p\le 19$. Under assumption of the Generalized Riemann Hypothesis we extend this result to primes $p\le 37$.

Comme un schéma en groupe commutatif, fini, plat et simple sur $\mathbf{Z}$ est tué par un nombre premier $p$, son ordre est une puissance de $p$. Tate a posé la question de savoir si un schéma en groupes fini, plat et simple $G$ sur $\mathbf{Z}$ est nécessairement isomorphe à $\mathbf{Z}/p\mathbf{Z}$ ou $\mu _p$. La réponse à cette question est affirmative pour $p \le 19$. Sous l’hypothèse de Riemann généralisée, nous étendons ce résultat aux nombres premiers $p \le 37$.

Reçu le :
Accepté le :
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DOI : 10.5802/jtnb.1332
Classification : 14L15
Keywords: Finite group scheme

Lassina Dembélé 1 ; René Schoof 2

1 Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK
2 Dipartimento di Matematica, 2 a Università di Roma, “Tor Vergata”, I-00133 Roma, Italy
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Lassina Dembélé; René Schoof. GRH and finite flat group schemes over $\mathbf{Z}$. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 569-578. doi: 10.5802/jtnb.1332

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