The genus of division algebras over discrete valued fields
Srinivasan Srimathy1
1 School of Mathematics, Tata Institute of Fundamental Research, Mumbai- 400005, India
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 535-567

Given a field with a set of discrete valuations $V$, we show how the genus of a division algebra over the field is related to the genus of the residue algebras at various valuations in $V$ and the ramification data. When the division algebra is a quaternion, we show the triviality of genus over many fields which include higher local fields, function fields of curves over higher local fields and function fields of curves over real closed fields. We also consider function fields of curves over global fields with a rational point and show how the genus problem is related to the $2$-torsion of the Tate–Shafarevich group of its Jacobian. As a special case, we show how the methods developed yield better bounds on the size of the genus over function fields of elliptic curves and demonstrate how they can be computed directly using arithmetic data of the elliptic curve with a number of examples.

Étant donné un corps muni d’un ensemble de valuations discrètes $V$, nous montrons comment le genre d’une algèbre à division sur ce corps est lié aux genres des algèbres résiduelles par rapport à $V$ et à des propriétés de ramification. Nous démontrons la trivialité du genre pour les algèbres de quaternions sur une large classe de corps, y compris les corps locaux multidimensionnels et les corps de fonctions des courbes définies sur un corps local multidimensionnel ou sur un corps réel clos. En outre, nous considérons le cas du corps de fonctions d’une courbe ayant un point rationnel sur un corps global et établissons un lien entre le problème de genre sur ce corps et la $2$-torsion du groupe de Tate–Shafarevich de sa jacobienne. En particulier, nous montrons que sur les corps de fonctions des courbes elliptiques, les méthodes développées donnent de meilleurs bornes pour le genre et expliquons comment elles peuvent être calculées directement en utilisant des informations arithmétiques, en donnant des exemples.

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DOI : 10.5802/jtnb.1331
Classification : 14F22, 11G05, 16K50, 12G05
Keywords: Brauer groups, genus of division algebras, quaternions, elliptic curves, Tate–Shafarevich group, discrete valued fields, higher local fields, semi-global fields

Srinivasan Srimathy 1

1 School of Mathematics, Tata Institute of Fundamental Research, Mumbai- 400005, India
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Srinivasan Srimathy. The genus of division algebras over discrete valued fields. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 535-567. doi: 10.5802/jtnb.1331

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