Schedule -------------------2009 Dec 14 --------------------------------------------------------------------------------------------------------------------------------------------- Speaker: Zhang Qifan 张起帆 (Si Chuan University) Title: Permutation Polynomials and Their Applications Time: 2009 Dec 14 3:10pm-4:10pm Abstract: I'll introduce some results on permutation polynomials in several variables and an an interesting application to cryptography. Moreover, I'll give a simple proof for a classic result on polynomial functions. Speaker: Xu Fei 徐飞 （CNU） Title: Brauer-Manin Obstruction for integral points. Time: 2009 Dec 14 4:20pm-5:20pm _____2009 Nov. 30_____________________________________________________________________ Speaker: Wei Dasheng (Academy of Math. And System Sciences, China) Title: The torus and sum of integral squares. Time: 2009 Nov. 30 3:00pm-4:00pm Speaker: Xu Fei (Capital Normal University) Title: Remark on torus and sum of integral squares. Time: 2009 Nov. 30 4:10pm-5:10pm Tea time: 5:10pm-6:00pm at Coffee Room of the fourth floor. Abstract: Let F be a number field and O_F the ring of integers. Let X be a scheme over O_F whose generic fiber is a principal homogeneous space of tori. Recently, Harari showed that the Brauer-Manin obstruction is the only obstruction for existence of the integral points of X.. However, the Brauer-Manin obstruction of tori given by Harari is not constructive. Fei Xu and Dasheng gave a constructive proof of the result. In this talk, we mainly discuss some application of our method for binary quadratic forms (for exaple, the sum of two integral squares ). In Preparation ---------------------2009 Dec28------------------------------------------------------------- Speaker: Hui Heung Shan Theodore (许向山) (The Chinese University of Hong Kong香港中文大学) Title: Upper Bound of the rank of elliptic curves with quadratic twists and a little conjecture. Abstract: For elliptic curves having complex multiplication and defined by y2 = x3 – Dx, Silverman has given a proof on the upper bound of the ranks of such curves which depends only on D. The proof depends heavily on the fact that E has a rational point of finite rank. Based on John Coates’s former results, we can show that the rank of any elliptic curve having a quadratic twist is always bounded above by O(logD) where the curve is given by: y2 = 4x3 – g 2D2x – g3D3. Also, a conjecture related to the greatest common divisors of certain polynomials would be discussed. Speaker: Liang Zhibing Title: The Tate-Shafarevich group for elliptic curves with complex multiplication. Abstract: In this lecture we will discuss some joint work with Coates and Sujatha in proving some theoretical results on BSD conjecture and carrying out numerical testing of conjectures which go far beyond what is possible to do by classical methods on elliptic curves with complex multiplication. We will also state some open questions. ----------------------------2010--------------------------------------------- Speaker: Li Bin 李彬 (Academy of Mathematics and Systems Science) Title: The Euler Characteristic of Arithmetic D-Modules on Curves Abstract: The theory of Arithmetic D-modules is a p-adic cohomology theorey which is developed by Berthelot. We will prove an Euler characteristic formula for middle extensions of arithmetic D-modules on curves. It is a generalization of Christol and Mebkhout's formula, and can be used to compute the Euler characteristics of holomonic arithmetic D-modules on curves. Speaker: Gong Ke 龚克 （Henan University） Title: Character sums over shifted primes Abstract: We obtain a new bound for sums of a multiplicative character modulo an integer $q$ at shifted primes $p+a$ over primes $p\le N$. Our bound is nontrivial starting with $N\ge q^{8/9 +\varepsilon}$ for any $\varepsilon>0$. This extends the range of the bound of Z.~Kh.~Rakhmonov that is nontrivial for $N\ge q^{1 + \varepsilon}$. This is the joint work with Professors John Friedlander and Igor Shparlinski.

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