Arthur classified the discrete automorphic representations of symplectic and orthogonal groups over a number field by that of the general linear groups. In this classification, those that are not from endoscopic lifting correspond to pairs (ϕ,b), where ϕ is an irreducible unitary cuspidal automorphic representation of some general linear group and b is an integer. In this paper, we study the local components of these automorphic representations at a nonarchimedean place, and we give a complete description of them in terms of their Langlands parameters.

Okumura proposed a candidate of post-quantum cryptosystem based on Diophantine equations of degree increasing type (DEC). Sizes of public keys in DEC are small, e.g., 1,200 bits for 128 bit security, and it is a strongly desired property in post-quantum erea. In this paper, we propose a polynomial time attack against DEC. We show that the one-wayness of DEC is reduced to finding special (relatively) short vectors in some lattices. The usual LLL algorithm does not work well for finding the most important target vector in our attack. The most technical point of our method is to heuristically find a special norm called a weighted norm to find the most important target vector. We call this method “weighted LLL algorithm” in this paper. Our experimental results suggest that our attack can break the one-wayness of DEC for 128 bit security with sufficiently high probability.

Characterization of homogeneous polynomials with isolated critical point at the origin follows from a study of complex geometry. Yau previously proposed a Numerical Characterization Conjecture. A step forward in solving this Conjecture, the Granville-Lin-Yau Conjecture was formulated, with a sharp estimate that counts the number of positive integral points in ndimensional (n≥3) real right-angled simplices with vertices whose distance to the origin are at least n-1. The estimate was proven for n≤6 but has a counterexample for n = 7. In this project we come up with an idea of forming a new sharp estimate conjecture where we need the distances of the vertices to be n. We have proved this new sharp estimate conjecture for n≤7 and are in the process of proving the general n case.

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