Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Optimal norm form integer space-time codes for two antenna MIMO systems

Abstract: In this paper, we introduce the definitions of a full diversity integer generating matrix and the corresponding norm form space-time code for MIMO systems. Subject to a power constraint, we characterize all full diversity integer generating matrices with the first three largest gains in the Gaussian integer ring and the Eisenstein integer ring for two transmitter antennas. Using this generating matrix family to separately design space-time codes layer by layer for two transmitter antenna and two receiver antenna MIMO systems, we obtain the optimal norm form integer space-time codes both in the Gaussian integer ring and the Eisenstein integer ring in the sense of maximizing the minimum determinant of codeword matrices. As a consequence, we prove that the golden code constructed by Dayal and Varanasi, Belfiore, Rekaya and Viterbo is optimal in the Gaussian integer ring. Also, we find the optimal code in the Eisentein integer ring, the coding gain of which is greater than that of the golden code.

Ring-LWE Cryptography for the Number Theorist.

Abstract: In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [EHL, ELOS] gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q.

Realizable Galois module classes for tetrahedral extensions

Abstract: Let k be a number field with ring of integers $\mathfrak{O}_k$ , and let $\Gamma=A_4$ be the tetrahedral group. For each tame Galois extension N / k with group isomorphic to $\Gamma$ , the ring of integers $\mathfrak{O}_{N}$ of N determines a class in the locally free class group $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ . We show that the set of classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ realized in this way is the kernel of the augmentation homomorphism from $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ to the ideal class group $\mathrm{Cl}(\mathfrak{O}_k)$ . This refines a result of Godin and Sodaigui ( J. Number Theory 98 (2003), 320–328) on Galois module structure over a maximal order in $k[\Gamma]$ . To the best of our knowledge, our result gives the first case where the set of realizable classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ has been determined for a nonabelian group $\Gamma$ and an arbitrary number field k .

Automorphic products on unitary groups

Abstract: The dissertation provides a construction for Borcherds products on unitary groups of signature (1,q). The starting point for this is the multiplicative lifting due to R. E. Borcherds. He employs the singular theta-correspondence to construct a lifting, which takes as inputs weakly holomorphic vector valued modular forms, transforming under the Weil-representation of SL(2,Z) for a quadratic lattice, and lifts these to meromorphic automorphic forms for an arithmetic subgroup of O(2,n). The resulting functions have expansions as infinite products and take their zeros and poles along Heegner divisors. In order to transfer this result to unitary groups, we construct an embedding between the symmetric domain of the unitary group and that of an orthogonal group, respectively. This embedding is compatible with the complex structures of either symmetric domain and a suitable choice of cusps. The main result is the construction of Borcherds products, on unitary groups of signature (1,q). In this setting we prove a result which is analogous to that of Borcherds. As in the case of orthogonal groups, the infinite products thus constructed have their zeros and poles on Heegner divisors. Here, the role of the quadratic lattice is taken by a hermitian lattice, which we assume to have as multiplier system the ring of integers of an imaginary quadratic number field. Further, we study the behavior of these automorphic products on the boundary of the symmetric domain. It turns out that the values taken on the boundary points can be interpreted as CM-values of generalized eta-products. In the finial chapter, we construct examples for the unitary group SU(1,1) and unimodular lattices, which in this case are simply hyperbolic planes over the rings of integers of imaginary quadratic number fields. In this case, the resulting products can be viewed as meromorphic elliptic modular forms on the (classical) complex upper half-plane.