| 1. |
- Ahlgren, Scott, et al.
(författare)
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SCARCITY OF CONGRUENCES FOR THE PARTITION FUNCTION
- 2023
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Ingår i: American Journal of Mathematics. - 0002-9327 .- 1080-6377. ; 145:5, s. 1509-1548
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Tidskriftsartikel (refereegranskat)abstract
- The arithmetic properties of the ordinary partition function p(n) have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form p(ℓn + β) ≡ 0 (mod ℓ) for the primes ℓ = 5, 7, 11, and it is known that there are no others of this form. On the other hand, for every prime ℓ ≥ 5 there are infinitely many examples of congruences of the form p(ℓQm n + β) ≡ 0 (mod ℓ) where Q ≥ 5 is prime and m ≥ 3. This leaves open the question of the existence of such congruences when m = 1 or m = 2 (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve X(ℓQ), Galois representations and the arithmetic large sieve.
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| 2. |
- Beckwith, Olivia, et al.
(författare)
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Congruences of Hurwitz class numbers on square classes
- 2022
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 1090-2082 .- 0001-8708. ; 409
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Tidskriftsartikel (refereegranskat)abstract
- We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish Ramanujan-type congruences for Hurwitz class numbers on square classes, where the holomorphic case parallels previous work by Radu on partition congruences. We offer two applications. The first application demonstrates common divisibility features of Ramanujan-type congruences for Hurwitz class numbers. The second application provides a dichotomy between congruences for class numbers of imaginary quadratic fields and Ramanujan-type congruences for Hurwitz class numbers.
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| 3. |
- Beckwith, Olivia, et al.
(författare)
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Imaginary Quadratic Fields With ℓ-Torsion-Free Class Groups and Specified Split Primes
- 2024
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Ingår i: International Mathematics Research Notices. - 1073-7928 .- 1687-0247. ; In Press
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Tidskriftsartikel (refereegranskat)abstract
- Given an odd prime $\ell $ and finite set of odd primes $S_{+}$ , we prove the existence of an imaginary quadratic field whose class number is indivisible by $\ell $ and which splits at every prime in $S_{+}$ . Notably, we do not require that $p \not \equiv -1 \,\;(\mathrm{mod}\, \ell )$ for any of the split primes $p$ that we impose. Our theorem is in the spirit of a result by Wiles, but we introduce a new method. It relies on a significant improvement of our earlier work on the classification of non-holomorphic Ramanujan-type congruences for Hurwitz class numbers.
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