| 1. |
- Ahlgren, Scott, et al.
(författare)
-
SCARCITY OF CONGRUENCES FOR THE PARTITION FUNCTION
- 2023
-
Ingår i: American Journal of Mathematics. - 0002-9327 .- 1080-6377. ; 145:5, s. 1509-1548
-
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.
|
|
| 2. |
- Beckwith, Olivia, et al.
(författare)
-
Congruences of Hurwitz class numbers on square classes
- 2022
-
Ingår i: Advances in Mathematics. - : Elsevier BV. - 1090-2082 .- 0001-8708. ; 409
-
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.
|
|
| 3. |
- Beckwith, Olivia, et al.
(författare)
-
Imaginary Quadratic Fields With ℓ-Torsion-Free Class Groups and Specified Split Primes
- 2024
-
Ingår i: International Mathematics Research Notices. - 1073-7928 .- 1687-0247. ; In Press
-
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.
|
|
| 4. |
- Beckwith, O., et al.
(författare)
-
Nonholomorphic Ramanujan-type congruences for Hurwitz class numbers
- 2020
-
Ingår i: Proceedings of the National Academy of Sciences of the United States of America. - : Proceedings of the National Academy of Sciences. - 0027-8424 .- 1091-6490. ; 117:36, s. 21953-21961
-
Tidskriftsartikel (refereegranskat)abstract
- In contrast to all other known Ramanujan-type congruences, we discover that Ramanujan-type congruences for Hurwitz class numbers can be supported on nonholomorphic generating series. We establish a divisibility result for such nonholomorphic congruences of Hurwitz class numbers. The two key tools in our proof are the holomorphic projection of products of theta series with a Hurwitz class number generating series and a theorem by Serre, which allows us to rule out certain congruences.
|
|
| 5. |
- Bringmann, K., et al.
(författare)
-
Almost holomorphic Poincaré series corresponding to products of harmonic Siegel-Maass forms
- 2016
-
Ingår i: Research in the Mathematical Sciences. - : Springer Science and Business Media LLC. - 2197-9847 .- 2522-0144. ; 3
-
Tidskriftsartikel (refereegranskat)abstract
- We investigate Poincare series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincare series are almost holomorphic as well. In general, this is not the case. The main point of this paper is the study of Siegel-Poincare series of degree 2 attached to products of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. We establish conditions on the convergence and nonvanishing of such Siegel-Poincare series. We surprisingly discover that these Poincare series are almost holomorphic Siegel modular forms, although the product of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms (in contrast to the elliptic case) is not almost holomorphic. Our proof employs tools from representation theory. In particular, we determine some constituents of the tensor product of Harish-Chandra modules with walls.
|
|
| 6. |
- Bringmann, Kathrin, et al.
(författare)
-
Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition
- 2015
-
Ingår i: Transactions of the American Mathematical Society. - 0002-9947 .- 1088-6850. ; 367:9, s. 6647-6670
-
Tidskriftsartikel (refereegranskat)abstract
- Real-analytic Jacobi forms play key roles in different areas of mathematics and physics, but a satisfactory theory of such Jacobi forms has been lacking. In this paper, we fill this gap by introducing a space of harmonic Maass-Jacobi forms with singularities which includes the real-analytic Jacobi forms from Zwegers’s PhD thesis. We provide several structure results for the space of such Jacobi forms, and we employ Zwegers’s μ-functions to establish a theta-like decomposition.
|
|
| 7. |
- Conley, C. H., et al.
(författare)
-
Harmonic Maass-Jacobi forms of degree 1 with higher rank indices
- 2016
-
Ingår i: International Journal of Number Theory. - 1793-0421. ; 12:7, s. 1871-1897
-
Tidskriftsartikel (refereegranskat)abstract
- We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.
|
|
| 8. |
- Conley, C. H., et al.
(författare)
-
Harmonic Maass-Jacobi forms of degree 1 with higher rank indices
- 2016
-
Ingår i: International Journal of Number Theory. - : World Scientific Pub Co Pte Lt. - 1793-0421 .- 1793-7310. ; 12:7, s. 1871-1897
-
Tidskriftsartikel (refereegranskat)abstract
- We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.
|
|
| 9. |
- Magnusson, Tobias, 1990, et al.
(författare)
-
Scalar-valued depth two Eichler–Shimura integrals of cusp forms
- 2023
-
Ingår i: Transactions of the London Mathematical Society. - 2052-4986. ; 10:1, s. 156-174
-
Tidskriftsartikel (refereegranskat)abstract
- Given cusp forms (Formula presented.) and (Formula presented.) of integral weight (Formula presented.), the depth two holomorphic iterated Eichler–Shimura integral (Formula presented.) is defined by (Formula presented.), where (Formula presented.) is the Eichler integral of (Formula presented.) and (Formula presented.) are formal variables. We provide an explicit vector-valued modular form whose top components are given by (Formula presented.). We show that this vector-valued modular form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted by (Formula presented.), that can be seen as a higher depth generalization of the scalar-valued Eichler integral (Formula presented.) of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol–Popa. We show that (Formula presented.) can be expressed in terms of sums of products of components of vector-valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form (Formula presented.). This allows for effective computation of (Formula presented.).
|
|
| 10. |
- Raum, Martin, 1985
(författare)
-
Formal Fourier Jacobi expansions and special cycles of codimension two
- 2015
-
Ingår i: Compositio Mathematica. - 0010-437X .- 1570-5846. ; 151:12, s. 2187-2211
-
Tidskriftsartikel (refereegranskat)abstract
- We prove that formal Fourier Jacobi expansions of degree two are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension two, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree two. Combining both results enables us to determine relations of special cycles in the second Chow group.
|
|
