The subject of counting positive lattice points in n-dimensional simplexes
has interested mathematicians for decades due to its applications in singularity the-
ory and number theory. Enumerating the lattice points in a right-angled simplex is
equivalent to determining the geometric genus of an isolated singularity of a weighted
homogeneous complex polynomial. It is also a method to shed insight into large gaps
in the sequence of prime numbers. Seeking to contribute to these applications, in this
paper, we prove the Yau Geometric Conjecture in six dimensions, a sharp upper bound
for the number of positive lattice points in a six-dimensional tetrahedron. The main
method of proof is summing existing sharp upper bounds for the number of points in
5-dimensional simplexes over the cross sections of the six-dimensional simplex. Our new
results pave the way for the proof of a fully general sharp upper bound for the number
of lattice points in a simplex. It also sheds new light on proving the Yau Geometric and
Yau Number-Theoretic Conjectures in full generality.
We study the structures of Fourier coefficients of automorphic forms on symplectic groups based on their local and global structures related to Arthur parameters. This is a first step towards the general conjecture on the relation between the structure of Fourier coefficients and Arthur parameters for automorphic forms occurring in the discrete spectrum, given by the first named author.
In , J. Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets. We continue with our investigation of Fourier coefficients and their implication to the structure of the cuspidal spectrum for symplectic groups ( and ). As result, we obtain certain characterization and construction of small cuspidal automorphic representations and gain a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, which has impacts to the generalized Ramanujan problem as posted by P. Sarnak in .