Summary
We present a synthesis of Saltman’s work (Adv. Math. 43(3):250–283, 1982; J. Alg. 314(2):817–843, 2007) on the division algebras of prime-to-p degree over the function field K of a p-adic curve. Suppose Δ is a K-division algebra. We prove that (a) Δ’s degree divides the square of its period; (b) if Δ has prime degree (different from p), then it is cyclic; (c) Δ has prime index different from p if and only if Δ’s period is prime, and its ramification locus on a certain model for K has no “hot points”.
For Parimala on her 60th birthday
2010 Mathematics subject classification. Primary: 16K20. Secondary: 11S15, 13A20, 16H05, 16K50.
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References
Artin, E., Tate, J.: Class Field Theory, AMS Chelsea Publishing, Am. Math. Soc., Rhode Island, 2009.
Auslander, M., Goldman, O.: The Brauer group of a commutative ring, Trans. Am. Math. Soc., 97 (1960) 367–409.
Colliot-Thélène, J.-L.: Birational invariants, purity and the Gersten conjecture, in K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (B. Jacob and A. Rosenberg, eds.), Proceedings of Symposia in Pure Mathematics, vol. 58.1, Am. Math. Soc., Providence, RI, 1995.
Colliot-Thélène, J.-L.: Review of [S1], Zentralblatt MATH database 1931–2009, Zbl 0902.16021.
Garibaldi, S., Serre, J.-P., Merkurjev, A.: Cohomological Invariants in Galois Cohomology, (University Lecture Series Volume 28), American Mathematical Society, 2003.
Gille, P., Szamuely, T.: Central Simple Algebras and Galois Cohomology, Cambridge University Press, Cambridge, 2006.
Grothendieck, A.: Le groupe de Brauer I, II, III, in Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam/Masson, Paris, 1968, 46–188.
Hoobler, R.T.: A cohomological interpretation of Brauer groups of rings, Pact. J. Math. 86 no. 1, (1980) 89–92.
Lichtenbaum, S.: Curves over discrete valuation rings, Am. J. Math., 90 no. 2, (1968) 380–405.
Liu, Q.: Algebraic Geometry and Arithmetic Curves Oxford University Press, New York, 2002.
Matsumura, H.: Commutative Ring Theory, (Cambridge Studies in Advanced Mathematics 8), Cambridge University Press, Cambridge, 1989.
Milne, J. S.: Etale Cohomology. Princeton University Press, New Jersey, 1980.
Milne, J. S.: Division algebras over p-adic curves, J. Raman. Math. Soc., 12 (1997) 25–47.
Milne, J. S.: Correction to division algebras over p-adic curves, J. Raman. Math. Soc., 13 (1998) 125–129.
Milne, J. S.: Cyclic algebras over p-adic curves, J. Alg., 314 no. 2 (2007), 817–843.
Milne, J. S.: Division algebras over surfaces, J. Alg., 315, no. 4 (2008), 1543–1585.
Grothendieck, A., Raynaud, M.: Séminaire de géométrie algébrique du Bois Marie 1960–1961: Revêtements étales et groupe fondamental, Documents mathématiques, vol. 3, Société Mathématique de France, 2003.
Silverman, J. H.: Advanced Topics in the Arithmetic of Elliptic Curves, Springer, New York, 1994.
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Brussel, E. (2010). On Saltman’s p-Adic Curves Papers. In: Colliot-Thélène, JL., Garibaldi, S., Sujatha, R., Suresh, V. (eds) Quadratic Forms, Linear Algebraic Groups, and Cohomology. Developments in Mathematics, vol 18. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6211-9_2
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