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- W4361268813 abstract "Abstract We link Ramanujan-type congruences, which emerge abundantly in combinatorics, to the Galois- and geometric theory of modular forms. Specifically, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra, and discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The latter provide congruences among algebraic parts of twisted central <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>L</m:mi> </m:math> {mathrm{L}} -values. We specialize our results to integer partitions, for which we investigate the landmark proofs of partition congruences by Atkin and by Ono. Based on a modulo <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>ℓ</m:mi> </m:math> {ell} analogue of the Maeda conjecture for certain partition generating functions, we conclude that their approach by Hecke operators acting diagonally modulo <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>ℓ</m:mi> </m:math> {ell} on modular forms is indeed close to optimal. This work is enabled by several structure results for Ramanujan-type congruences that we establish. In an extended example, we showcase how to employ them to also benefit experimental work." @default.
- W4361268813 created "2023-03-31" @default.
- W4361268813 creator A5001242979 @default.
- W4361268813 date "2023-03-31" @default.
- W4361268813 modified "2023-10-14" @default.
- W4361268813 title "Relations among Ramanujan-type congruences II: Ramanujan-type congruences in half-integral weights" @default.
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- W4361268813 doi "https://doi.org/10.1515/forum-2022-0041" @default.
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