| 1. |
- Beckwith, O., et al.
(författare)
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Nonholomorphic Ramanujan-type congruences for Hurwitz class numbers
- 2020
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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
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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.
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| 2. |
- Bringmann, K., et al.
(författare)
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Almost holomorphic Poincaré series corresponding to products of harmonic Siegel-Maass forms
- 2016
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Ingår i: Research in the Mathematical Sciences. - : Springer Science and Business Media LLC. - 2197-9847 .- 2522-0144. ; 3
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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.
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| 3. |
- Conley, C. H., et al.
(författare)
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Harmonic Maass-Jacobi forms of degree 1 with higher rank indices
- 2016
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Ingår i: International Journal of Number Theory. - 1793-0421. ; 12:7, s. 1871-1897
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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.
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| 4. |
- Westerholt-Raum, Martin, 1985, et al.
(författare)
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All modular forms of weight 2 can be expressed by Eisenstein series
- 2020
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Ingår i: Research in Number Theory. - : Springer Science and Business Media LLC. - 2522-0160 .- 2363-9555. ; 6:3
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Tidskriftsartikel (refereegranskat)abstract
- We show that every elliptic modular form of integral weight greater than 1 can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central L-values present in all previous work. For weights greater than 2, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.
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| 5. |
- Westerholt-Raum, Martin, 1985
(författare)
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Computing genus 1 Jacobi forms
- 2016
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 85:298, s. 931-960
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Tidskriftsartikel (refereegranskat)abstract
- We develop an algorithm to compute Fourier expansions of vector valued modular forms for Weil representations. As an application, we compute explicit linear equivalences of special divisors on modular varieties of orthogonal type. We define three families of Hecke operators for Jacobi forms, and analyze the induced action on vector valued modular forms. The newspaces attached to one of these families are used to give a more memory efficient version of our algorithm. - See more at: http://www.ams.org/journals/mcom/2016-85-298/S0025-5718-2015-02992-5/#sthash.bv7cxz8N.dpuf
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| 6. |
- Westerholt-Raum, Martin, 1985
(författare)
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H-harmonic Maaß-Jacobi forms of degree 1
- 2015
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Ingår i: Research in the Mathematical Sciences. - : Springer Science and Business Media LLC. - 2197-9847. ; 2
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Tidskriftsartikel (refereegranskat)
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| 7. |
- Westerholt-Raum, Martin, 1985
(författare)
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Harmonic Weak Siegel–Maaß Forms I: Preimages of Non-Holomorphic Saito-Kurokawa Lifts
- 2018
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Ingår i: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; :5, s. 1442-1472
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Tidskriftsartikel (refereegranskat)abstract
- Given a non-holomorphic Saito-Kurokawa lift we construct a preimage under the vector-valued lowering operator. In analogy with the case of harmonic weak elliptic Maaß forms, this preimage allows for a natural decomposition into a meromorphic and a non-holomorphic part. In this way every harmonic weak Siegel–Maaß form gives rise to a Siegel mock modular form.
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| 8. |
- Westerholt-Raum, Martin, 1985, et al.
(författare)
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Kudla's modularity conjecture and formal Fourier-Jacobi series
- 2015
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Ingår i: Forum of Mathematics. Pi. - : Cambridge University Press (CUP). - 2050-5086. ; 3
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Tidskriftsartikel (refereegranskat)abstract
- We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogs of Fourier–Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.
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| 9. |
- Westerholt-Raum, Martin, 1985
(författare)
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Products of vector valued Eisenstein series
- 2017
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Ingår i: Forum Mathematicum. - : Walter de Gruyter GmbH. - 0933-7741 .- 1435-5337. ; 29:1, s. 157-186
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Tidskriftsartikel (refereegranskat)abstract
- We prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin-Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context their previously mentioned result.
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| 10. |
- Westerholt-Raum, Martin, 1985, et al.
(författare)
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Sturm bounds for Siegel modular forms
- 2015
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Ingår i: Research in Number Theory. - : Springer Science and Business Media LLC. - 2363-9555. ; 1:1
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Tidskriftsartikel (refereegranskat)abstract
- We establish Sturm bounds for degree g Siegel modular forms modulo a prime p, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properties of specializations of Jacobi forms to torsion points. In particular, our approach is completely different from the proofs of the previously known cases g=1,2, which do not extend to the case of general g.
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