Quadratic Diophantine Equations and Orders in Quaternion Algebras

2016 ◽  
pp. 49-86
Author(s):  
Goro Shimura
Keyword(s):  
Diophantine Equations ◽  
Quaternion Algebras

The Gross-Zagier Formula on Shimura Curves

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-wu Zhang ◽  
Wei Zhang
Keyword(s):  
Diophantine Equations ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
Heegner Points ◽  
Quaternion Algebras ◽  
Complete Proof ◽  
Weil Representations ◽  
Arakelov Theory ◽  
Generating Series ◽  
New Applications

This comprehensive account of the Gross–Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross–Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross–Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross–Zagier formula is reduced to local formulas. This book will be of great use to students wishing to enter this area and to those already working in it.

Secure Watermarking using Diophantine Equations for Authentication and Recovery

2015 ◽  
Vol 3 (2) ◽  
Author(s):  
Jayashree Nair ◽  
T. Padma
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Jpeg Compression ◽  
Authentication Scheme ◽  
Original Image ◽  
Invariant Features ◽  
Jpeg2000 Compression ◽  
Frequency Properties ◽  
Visual Authentication

This paper describes an authentication scheme that uses Diophantine equations based generation of the secret locations to embed the authentication and recovery watermark in the DWT sub-bands. The security lies in the difficulty of finding a solution to the Diophantine equation. The scheme uses the content invariant features of the image as a self-authenticating watermark and a quantized down sampled approximation of the original image as a recovery watermark for visual authentication, both embedded securely using secret locations generated from solution of the Diophantine equations formed from the PQ sequences. The scheme is mildly robust to Jpeg compression and highly robust to Jpeg2000 compression. The scheme also ensures highly imperceptible watermarked images as the spatio –frequency properties of DWT are utilized to embed the dual watermarks.

scholarly journals Quaternion Algebras

2021 ◽  
Author(s):  
John Voight
Keyword(s):  
Quaternion Algebras

scholarly journals On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis

2021 ◽  
Vol 19 (1) ◽  
pp. 583-599
Author(s):  
Beata Bajorska-Harapińska ◽  
Jakub Jan Ludew ◽  
Barbara Smoleń-Duda ◽  
Roman Wituła
Keyword(s):  
Generating Functions ◽  
Split Quaternion ◽  
Quaternion Algebras

Abstract In this paper, we introduce generalizations of Quaternacci sequences (Quaternaccis), called Split Quaternacci sequences, which arose on a base of split quaternion algebras. Explicit and recurrent formulae for Split Quaternacci sequences are given, as well as generating functions. Also, matrices related to Split Quaternaccis sequences are investigated. Moreover, new identities connecting Horadam sequences with other known sequences are generated. Analogous identities for Horadam quaternions and split Horadam quaternions are proved.

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