Wikipedia:Reference desk/Archives/Mathematics/2022 April 13

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April 13

Lemma?

"For all CM-points τ, the “singular moduli” j(τ) and the values of E₄(τ)/η⁸(τ) and E₆(τ)/η¹²(τ) are algebraic integers.", this is a lemma, like it is taught in college/university? Or any one tells you that? --ExclusiveEditor _{Notify Me!} 16:18, 13 April 2022 (UTC)

No idea on this specific statement, but in terms of form, it appears to qualify as a Lemma (mathematics), a short true statement that is part of a larger proof. Not sure if this one is true, or what its use is, but it is a lemma-like statement. --Jayron32 16:28, 13 April 2022 (UTC)

The lemma used here appears in "An efficient determination of the coefficients in the Chudnovskys’ series for 1/π" (on springer- An efficient determination of... on page 5(of pdf)/ 807 of papers.)--ExclusiveEditor --ExclusiveEditor _{Notify Me!} 17:50, 13 April 2022 (UTC)

This does not look like undergraduate material, and I don't think this is material routinely taught at the universities at the graduate level; if treated at all, it would be part of some specialized seminar. For one thing, the definitions of the Eisenstein series $E_{4}$ and $E_{6}$ are specific to this paper. If it was known material, it would have been published before. Instead of giving a proof themselves, the authors could then simply have referred the reader to the published proof, as they do for the statement that $j(\tau )$ is an algebraic integer for all $CM$-points $\tau .$  --Lambiam 22:56, 13 April 2022 (UTC)

I don't recognize the specific theorem or lemma, but the general "language" it uses is algebraic number theory. Particularly, it is talking about complex multiplication and the j-invariant. This is something that an undergraduate student could study if they wanted to pursue the topic, but it wouldn't be part of the regular curriculum for undergraduate math majors. A grad student specializing in number theory or related topics would have to study the general area (elliptic curves etc.) enough to read the paper, though the computation of pi specifically is a niche area that the Chudnovsky brothers (as well as the Borwein brothers, Jonathan and Peter) were leading figures. The Borweins published a book called "Pi and the AGM" with more material in this vein. 2601:648:8202:350:0:0:0:4671 (talk) 05:58, 15 April 2022 (UTC)