On the Field Extension by Complex Multiplication

Rational Conformal Field Theories and Complex Multiplication

We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFT’s corresponding to T 2 target and identify the Cardy branes with geometric branes. The T ’s leading to RCFT’s admit “complex multiplication” which characterizes Cardy branes as specific D0-branes. We propose a cond...

Complex Multiplication and Explicit Class Field Theory

Notes on class field theory and complex multiplication

2 Class field theory 2 2.1 Number fields and their completions . . . . . . . . . . . . . . . 2 2.1.1 Number fields, prime ideals . . . . . . . . . . . . . . . . 2 2.1.2 Fractional Ideals . . . . . . . . . . . . . . . . . . . . . . 3 2.1.3 Completions . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.4 Adeles and ideles . . . . . . . . . . . . . . . . . . . . . . 4 2.1.5 Cycles . . . . . . ....

Binary field multiplication on ARMv8

In this paper, we show efficient implementations of binary field multiplication over ARMv8. We exploit an advanced 64-bit polynomial multiplication (PMULL) supported by ARMv8 and conduct multiple levels of asymptotically faster Karatsuba multiplication. Finally, our method conducts binary field multiplication within 57 clock cycles for B-251. Our proposed method on ARMv8 improves the performanc...

Introductory Remarks on Complex Multiplication

Complex multiplication in its simplest form is a geometric tiling property. In its advanced form it is a unifying motivation of classical mathematics from elliptic integrals to number theory; and it is still of active interest. This interrelation is explored in an introductory expository fashion with emphasis on a central historical problem, the modular equation between j(z) and j(2z).