Timeline for Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Oct 27, 2022 at 14:51 | comment | added | Jeremy Rouse | Yes. Chapter 9 of O'Meara's book "Introduction to Quadratic Forms" is titled "Integral Theory of Quadratic Forms over Local Fields", and Theorem 92.2 of that book (page 247 in the second printing) gives a classification (in the non-dyadic cases) in terms of the discriminants of the terms in the Jordan splitting. (Basically, a term in the Jordan splitting arises from diagonalising the quadratic form and then grouping together all terms where the $p$-adic valuation of the coefficients are the same.) | |
| Oct 27, 2022 at 8:20 | comment | added | a196884 | Thanks, great answer! Do you a know a reference for studying equivalence over the localisation? Do we have invariants that classify equivalence over $\mathbb{Z}_{(p)}$ in the same way we do for equivalence over $\mathbb{Z}_p$ (and genera of forms)? | |
| Oct 27, 2022 at 8:13 | vote | accept | a196884 | ||
| Nov 4, 2022 at 17:58 | |||||
| Oct 27, 2022 at 1:25 | history | answered | Jeremy Rouse | CC BY-SA 4.0 |