Potential good reduction

In mathematics, potential good reduction is a property of the reduction modulo a prime or, more generally, prime ideal, of an algebraic variety.

Definitions

Good reduction refers to the reduced variety having the same properties as the original, for example, an algebraic curve having the same genus, or a smooth variety remaining smooth. Potential good reduction refers to the situation over a sufficiently large finite extension of the field of definition.

Equivalent formulations

For elliptic curves, potential good reduction is equivalent to the j-invariant being an algebraic integer.

gollark: Pretty much all of the algorithms reduced size by ~50% or so and the difference is maybe 5% or so between them all, so this is definitely premature optimization, but bees?
gollark: I tested four different compression algorithms and brotli did fairly well; I would have used zstandard but the node bindings for it are awful, and brotli actually did do better on small inputs.
gollark: For example, it stores created/updated timestamps in a way which allows them to be looked up more quickly, makes it faster to look up the latest revision of stuff, allows me to do compression (I implemented brotli compression to reduce storage requirements a lot), and allows revisions to have data and represent stuff other than "the page content changed".
gollark: The new version *is* better, even if it involves something like 70 lines more code.
gollark: I've reworked minoteaur's design a bit again because productivity is BEES and happens to other people.```sqlCREATE TABLE pages ( id INTEGER PRIMARY KEY, name TEXT NOT NULL, updated INTEGER NOT NULL, content TEXT NOT NULL);``` I went from that small and thus uncool database thingy to this:```sqlCREATE TABLE versions ( vuuid TEXT PRIMARY KEY COLLATE BINARY, rawSize INTEGER NOT NULL, encoding TEXT, data BLOB NOT NULL);CREATE TABLE pages ( title TEXT PRIMARY KEY, created INTEGER NOT NULL, updated INTEGER NOT NULL, latestVersion TEXT NOT NULL REFERENCES versions(vuuid));CREATE TABLE revisions ( ruuid TEXT PRIMARY KEY COLLATE BINARY, page TEXT NOT NULL REFERENCES pages(title), timestamp INTEGER NOT NULL, type TEXT NOT NULL, data TEXT NOT NULL, -- JSON version TEXT NOT NULL REFERENCES versions(vuuid));CREATE INDEX revisions_page_ix ON revisions(page);```

See also

References
  • Serre, Jean-Pierre; Tate, John (1968). "Good reduction of abelian varieties". The Annals of Mathematics. Second. 88 (3): 492–517. doi:10.2307/1970722. JSTOR 1970722. Zbl 0172.46101.
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