Explicit non-Gorenstein $$R={\mathbb {T}}$$ via rank bounds II: Computational aspects

Hsu, Catherine; Wake, Preston; Wang-Erickson, Carl

doi:10.1007/s40993-022-00401-1

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2023

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“…They require a numerical condition which turns out to force the rank of R and T over Z p to be 3 (cf. [17,Theorem 1.3.3]).…”

Section: Overview Of Q4mentioning

confidence: 99%

On Eisenstein congruences and beyond

Lecouturier

2023

Notices of the International Consortium of Chinese Mathematicians

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Eisenstein congruences play an important role in modern number theory. We survey some topics related to these congruences, starting from the example of Ramanujan's Delta function modulo 691. This paper does not contain any new results, except Theorem 2.4. This paper is a survey about (higher) Eisenstein congruences and their applications in number theory. As far as the author is aware, the first example of Eisenstein congruences was found by Ramanujan in [46]. Consider the formal series ∆ = q ∏ n≥1 (1 − q n ) 24 = ∑ n≥1 τ(n)q n (for τ(n) ∈ Z). Ramanujan proved that for all prime p we have τ(p) ≡ 1 + p 11 (modulo 691).

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“…They require a numerical condition which turns out to force the rank of R and T over Z p to be 3 (cf. [17,Theorem 1.3.3]).…”

Section: Overview Of Q4mentioning

confidence: 99%