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THE CHIRAL SUPERSTRING SIEGEL FORM IN DEGREE TWO IS A LIFT
Poor, Cris,Yuen, David S. Korean Mathematical Society 2012 대한수학회지 Vol.49 No.2
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.
The chiral superstring Siegel form in degree two is a lift
Cris Poor,David S. Yuen 대한수학회 2012 대한수학회지 Vol.49 No.2
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 Γ1(1, 2) to Siegel modular cusp forms over certain subgroups Γpara(t, 1, 2) of paramodular groups. The theta group lift given here is a modication of the Gritsenko lift.
THE CUSP STRUCTURE OF THE PARAMODULAR GROUPS FOR DEGREE TWO
Poor, Cris,Yuen, David S. Korean Mathematical Society 2013 대한수학회지 Vol.50 No.2
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.
Nonlift weight two paramodular eigenform constructions
Cris Poor,Jerry Shurman,David S. Yuen 대한수학회 2020 대한수학회지 Vol.57 No.2
We complete the construction of the nonlift weight two cusp para\-modular Hecke eigenforms for prime levels $N<600$, which arise in conformance with the paramodular conjecture of Brumer and Kramer.
The cusp structure of the paramodular groups for degree two
Cris Poor,David S. Yuen 대한수학회 2013 대한수학회지 Vol.50 No.2
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.
Computations of spaces of paramodular forms of general level
Cris Poor,David S. Yuen,Jeffery Breeding II 대한수학회 2016 대한수학회지 Vol.53 No.3
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.
NONLIFT WEIGHT TWO PARAMODULAR EIGENFORM CONSTRUCTIONS
Poor, Cris,Shurman, Jerry,Yuen, David S. Korean Mathematical Society 2020 대한수학회지 Vol.57 No.2
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.
COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL
Breeding, Jeffery II,Poor, Cris,Yuen, David S. Korean Mathematical Society 2016 대한수학회지 Vol.53 No.3
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.