Srimathy Srinivasan
TIFR, Mumbai
February 1, 2024
On the genus problem over discrete valued fields: Given a field with a set of discrete valuations, we show how the genus of any division algebra over certain fields is related to the genus of some residue algebras at various valuations and the ramification data. Applications include showing triviality of the genus of quaternions over many fields such as higher local fields, function fields of curves over higher local fields and function fields of curves over real closed fields. When the base field is a function field of a curve over a global field with a rational point, the genus of any quaternion is related to the $2$-torsion of the Tate-Shafarevich group of the Jacobian. As a consequence, when the curve is elliptic, the size of the genus can be computed directly using arithmetic data of the elliptic curve.
TIFR, Mumbai
February 1, 2024
On the genus problem over discrete valued fields: Given a field with a set of discrete valuations, we show how the genus of any division algebra over certain fields is related to the genus of some residue algebras at various valuations and the ramification data. Applications include showing triviality of the genus of quaternions over many fields such as higher local fields, function fields of curves over higher local fields and function fields of curves over real closed fields. When the base field is a function field of a curve over a global field with a rational point, the genus of any quaternion is related to the $2$-torsion of the Tate-Shafarevich group of the Jacobian. As a consequence, when the curve is elliptic, the size of the genus can be computed directly using arithmetic data of the elliptic curve.