• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.645-689
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• 2016

This article gives upper bounds on the number of Fourier-Jacobi coefficients that determine a paramodular cusp form in degree two. The level N of the paramodular group is completely general throughout. Additionally, spaces of Jacobi cusp forms are spanned by using the theory of theta blocks due to Gritsenko, Skoruppa and Zagier. We combine these two techniques to rigorously compute spaces of paramodular cusp forms and to verify the Paramodular Conjecture of Brumer and Kramer in many cases of low level. The proofs rely on a detailed description of the zero dimensional cusps for the subgroup of integral elements in each paramodular group.

• Journal of the Korean Mathematical Society
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• v.57 no.2
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• pp.507-522
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• 2020

We complete the construction of the nonlift weight two cusp paramodular Hecke eigenforms for prime levels N < 600, which arise in conformance with the paramodular conjecture of Brumer and Kramer.

• Journal of the Korean Mathematical Society
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• v.50 no.2
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• pp.445-464
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• 2013

We describe the one-dimensional and zero-dimensional cusps of the Satake compactification for the paramodular groups in degree two for arbitrary levels. We determine the crossings of the one-dimensional cusps. Applications to computing the dimensions of Siegel modular forms are given.

• Journal of the Korean Mathematical Society
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• v.61 no.5
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• pp.997-1033
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• 2024

We use Borcherds products to give a new construction of the weight 3 paramodular nonlift eigenform f_N for levels N = 61, 73, 79. We classify the congruences of f_N to Gritsenko lifts. We provide techniques that compute eigenvalues to support future modularity applications. Our method does not compute Hecke eigenvalues from Fourier coefficients but instead uses elliptic modular forms, specifically the restrictions of Gritsenko lifts and their images under the slash operator to modular curves.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.293-314
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• 2012

We prove that the Siegel modular form of D'Hoker and Phong that gives the chiral superstring measure in degree two is a lift. This gives a fast algorithm for computing its Fourier coefficients. We prove a general lifting from Jacobi cusp forms of half integral index t/2 over the theta group ${\Gamma}_1$(1, 2) to Siegel modular cusp forms over certain subgroups ${\Gamma}^{para}$(t; 1, 2) of paramodular groups. The theta group lift given here is a modification of the Gritsenko lift.