This proposal is to support US participants to "An international conference on Galois representations, automorphic forms and Shimura varieties" which will take place during June 20-23, 2011, at the National Center for Theoretical Sciences (NCTS), Hsinchu, Taiwan. The purpose of this conference is to showcase the multi-facets of Galois representations and their inter-connections. The invited speakers include experts working on noncongruence modular forms, transcendence theory over function fields, p-adic Hodge theory, modularity of Galois representations, automorphic representations, L-functions, and arithmetic of Shimura varieties. Recent advances in and applications of Galois representations, automorphic forms and arithmetic of Shimura varieties, from theoretical and computational aspects, will be addressed. Preceding this conference, Henri Darmon will offer a short course on "Algebraic cycles and p-adic deformations" to prepare the students. Two accompanying special day activities on noncongruence modular forms and arithmetic geometry will be held after the conference to facilitate further research oriented discussions on more focused subjects. The slides of talks during the conference will be posted on the website of NCTS.

The research topics on Galois representations has generated a tremendous attention after Wiles' proof of Fermat's last theorem. Wiles' proof opened up new connections among several important areas of mathematics, from analytic and combinatorial number theory to arithmetic geometry. The topics of the conference focuses on the interaction among these topics. It is hoped that international collaborations will be fostered through this kind of activities leading to further and deeper connections among these research fields. More information about the conference can be found at the website

Project Report

was held June 20-23, 2011 in the National Center for Theoretical Sciences (NCTS), Hsinchu, Taiwan. It was cosponsored by NCTS and NSF. Two short courses preceded the conference to provide background materials and a one day workshop followed the conference to allow for more in-depth discussion on a specialized topic. Four graduate students from the US were invited to participate in these activities with full support. The conference, short courses, and the workshop were all very well-received. Intellectual Merit The Langlands philosophy predicts a connection between Galois representations and automorphic forms, which vastly generalizes the class field theory. Most of the progress has been from automorphic forms/ representations to Galois representations, and Shimura varieties played an important role in establishing such correspondences. Little was known about the reverse direction until the remarkable breakthrough by Wiles and Taylor-Wiles. Since then, tremendous progress has been made in this area in recent years. The purpose of this conference was to show recent advances in and applications of Galois representations, automorphic forms and arithmetic of Shimura varieties, from theoretical and computational aspects. The 17 invited speakers from 6 countries were: Chieh-Yu Chang (NCTS, Taiwan), Ching-Li Chai (U. Penn), Kuok-Fai Chao (NSYSU, Taiwan), Suh Hyu Choi (KAIST, Korea), Henri Darmon (McGill, Canada), Ulrich Goertz (Duisburg-Essen, Germany), Jerome Hoffman (Louisiana State), Kai-Wen Lan (Princeton), Tong Liu (Purdue), Ling Long (Iowa State), Ye-Tien (Chinese Academy of Sciences, China), John Voight (Vermont), Fu-Tsun Wei (NTHU, Taiwan), Tong-Hai Yang (Madison), Yifan Yang (NCTU, Taiwan), Chia-Fu Yu (Academia Sinica, Taiwan), Jiu-Kang Yu (Purdue). Their talks covered the following areas: 1. Values of L-functions (Darmon, Yang, Tien, Wei) 2. Shimura curves and varieties (Lan, T. Yang, Tien, Voight, Wei, Y. Yang, Goertz, C. Yu, Chai) 3. Deformation of Galois representations (Darmon, Choi, Liu) 4. Galois representations and noncongruence modular forms (Long, Hoffman) 5. Trace formula (Tien, J. Yu) 6. Galois representations and transcendence (Chang) 7. Differential equations and modular forms (Y. Yang, Hoffman) 8. Geometric structure of representations (Chao) The site provides information about this conference, including program and poster, on which NSF is acknowledged. The award of this grant is $14116, which is used to support the following people: Jerome William Hoffman (Louisiana State), Tong Liu (Purdue), Jiu-Kang Yu (Purdue), Ryan Flynn (Penn State, graduate student), Nguyen Ngoc Dong Quan (U. Arizona, graduate student), Peiyu Tsai (Harvard, graduate student, female), Jonas Kibelbek (Penn State, graduate student), and Wen-Ching Li (Penn State). Other speakers from US, including Ching-Li Chai, Jerome Hoffman, Kai-Wen Lan, Ling Long, Ravi Ramakrishna, John Voight, and Tong-Hai Yang, were supported by different sources. Broader Impacts Before the conference, Henri Darmon (McGill, Canada) offered a 6-hour short course on "Algebraic Cycles and p-adic Deformations", June 15-17, 2011, and Ravi Ramakrishna (Cornell) offered a 6-hour course on "A Survey on Serre’s Conjecture and Its Generalizations", June 16-17, 2011. Please access the site for details of these two short courses. The purpose of these courses was to survey recent developments in special values of L-functions and Galois representations to prepare the audience for the conference. After the conference, a one day workshop on "Noncongruence Modular Forms" was held on June 27, 2011 for people interested in this topic to have more indepth discussions. It can also be accessed from The slides of the talks given in the conference are posted in available to the general public. This conference was attended by 47 number theorists from Canada, China, Croatia, Germany, Korea, Taiwan, and United States. In particular, four graduate students from US (one from Arizona, two from Penn State, and one from Harvard) were invited to participate with full support. One student gave a talk in the one-day workshop following the conference. Among the 17 invited speakers of the conference, there were 3 postdoctoral fellows, 4 junior faculty members, and 1 female associate professor. Among the 5 speakers in the workshop, Kibelbek was a graduating student, Kazalicki, who graduated from U. Wisconsin, Madison in 2010, was a postdoctoral fellow from Zagreb, Hoffman and Long both spoke in the conference, and Ng from Iowa State was a visitor at NCTS. This event was well-attended and well-received by all participants. Darmon’s course provided the essential background materials for his conference talk. Ramakrishna summarized recent developments on Serre’s conjecture, a very exciting achievement in Galois representations. The conference talks showed very interesting connections among various topics, and also among different speakers. The one-day workshop revealed a very interesting new development on congruences of Atkin-Swinnerton-Dyer type, extending the ASD congruences from holomorphic noncongruence forms to weakly holomorphic ones numerically. This topic is taken up by Kazalicki and Scholl in their recent collaboration to provide a rigorous proof. The new discovery could lead to a satisfactory explanation of the case where ASD congruences fail on holomorphic forms.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tie Luo
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Pennsylvania State University
University Park
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