Book Review: Arithmetic of algebraic curves

Holography Principle and Arithmetic of Algebraic Curves

According to the holography principle (due to G. ‘t Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which ...

BOOK REVIEW: Quantum Logic in Algebraic Approach

Since the invention of quantum logic by Birkhoff & von Neumann there has been a steady interest in the topic. Quantum logic owes much of its current attention to the rise of experimental foundational physics. Many researchers interested in the foundation of physics begin to view this field as a way to understand the quantum better; or at least from a new, hitherto unknown viewpoint. Yet much is...

Arithmetic of Hyperelliptic Curves

Arithmetic of Elliptic Curves

Here are the notes I am taking for Wei Zhang’s ongoing course on the arithmetic of elliptic curves offered at Columbia University in Fall 2014 (MATH G6761: Topics in Arithmetic Geometry). As the course progresses, these notes will be revised. I recommend that you visit my website from time to time for the most updated version. Due to my own lack of understanding of the materials, I have inevita...

Arithmetic of Elliptic Curves

Our research focuses on 9 specific elliptic curves E over Q, each with complex multiplication by the maximal order in an imaginary quadratic field. Viewed over C, each E gives rise to tori, defined by the generators ω1, ω2 ∈ C of the period lattice. Using SAGE, information and characteristics about the curves and their tori were calculated and compiled. Additionally, the tori were virtually con...