On Tate and His Mathematics – a Glossary∗
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Class Numbers of CM Algebraic Tori, CM Abelian Varieties and Components of Unitary Shimura Varieties
CLASS NUMBERS OF CM ALGEBRAIC TORI, CM ABELIAN VARIETIES AND COMPONENTS OF UNITARY SHIMURA VARIETIES JIA-WEI GUO, NAI-HENG SHEU, AND CHIA-FU YU Abstract. We give a formula for the class number of an arbitrary CM algebraic torus over Q. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result. 1. Introduction An algebraic torus T over a number field k is a connected linear algebraic group over k such d that T k k¯ isomorphic to (Gm) k k¯ over the algebraic closure k¯ of k for some integer d 1. The class⊗ number, h(T ), of T is by⊗ definition, the cardinality of T (k) T (A )/U , where A≥ \ k,f T k,f is the finite adele ring of k and UT is the maximal open compact subgroup of T (Ak,f ). Asa natural generalization for the class number of a number field, Takashi Ono [18, 19] studied the class numbers of algebraic tori. Let K/k be a finite extension and let RK/k denote the Weil restriction of scalars form K to k, then we have the following exact sequence of tori defined over k 1 R(1) (G ) R (G ) G 1, −→ K/k m,K −→ K/k m,K −→ m,k −→ where R(1) (G ) is the kernel of the norm map N : R (G ) G . -
2012 Cole Prize in Algebra
2012 Cole Prize in Algebra Alexander S. Merkurjev received the 2012 AMS speaker at the International Congress of Mathema- Frank Nelson Cole Prize in Algebra at the 118th An- ticians (Berkeley, 1986). Twice he has delivered nual Meeting of the AMS in Boston in January 2012. an invited address at the European Congress of Mathematics (1992, 1996), and he was a plenary Citation speaker in 1996 (Budapest). The 2012 Frank Nelson Cole Prize in Algebra is awarded to Alexander S. Merkurjev of the Univer- Response from Alexander S. Merkurjev sity of California, Los Angeles, for his work on the It is a great honor and great pleasure for me to essential dimension of groups. receive the 2012 Frank Nelson Cole Prize in Alge- The essential dimension of a finite or of an alge- bra. I would like to thank the American braic group G is the smallest number of parameters Mathematical Society and the Selection needed to describe G-actions. For instance, if G Committee for awarding the prize to is the symmetric group on n letters, this invari- me. ant counts the number of parameters needed to I am very grateful to my teacher, specify a field extension of degree n, which is the Andrei Suslin (he was awarded the algebraic form of Hilbert’s thirteenth problem. Merkurjev’s papers (“Canonical p-dimension Frank Nelson Cole Prize in Algebra in of algebraic groups”, with N. Karpenko, Adv. 2000). I also want to thank my parents, Math. 205 (2006), no. 2, 410–433; and “Essential family, friends, and colleagues for dimension of finite p-groups”, with N. -
Harish-Chandra 1923–1983
NATIONAL ACADEMY OF SCIENCES HARISH- C HANDRA 1 9 2 3 – 1 9 8 3 A Biographical Memoir by R O G E R H O W E Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 2011 NATIONAL ACADEMY OF SCIENCES WASHINGTON, D.C. Photo by Herman Landshoff; Courtesy Archives of the Institute for Advanced Study. HARISH-CHANDRA October 11, 1923–October 12, 1983 BY ROGER HOWE He taught them the Kshatria code of honor: that a warrior may never refuse a challenge…. The Five Sons of Pandu, The Story of the Mahabharata Retold by Elizabeth Seeger ARISH-CHANDRA WAS, if not the exclusive architect, cer- Htainly the chief engineer of harmonic analysis on semisimple Lie groups. This subject, with roots deep in mathematical physics and analysis, is a synthesis of Fou- rier analysis, special functions and invariant theory, and it has become a basic tool in analytic number theory, via the theory of automorphic forms. It essentially did not ex- ist before World War II, but in very large part because of the labors of Harish-Chandra, it became one of the major mathematical edifices of the second half of the twentieth century. Harish-Chandra was born in 1923 in Uttar Pradesh, in northern India. His family belonged to the Kshatria (war- rior) caste. Kshatria traditionally were rulers, landowners, and military leaders, and more recently have commonly been businessmen or civil servants. Harish-Chandra’s father, Chandrakishore, was a civil engineer who monitored and maintained the dikes and irrigation canals that sustain agri- 3 B IOGRA P HICAL MEMOIRS culture on the North Indian plains. -
Henri Darmon
Henri Darmon Address: Dept of Math, McGill University, Burnside Hall, Montreal, PQ. E-mail: [email protected] Web Page: http://www.math.mcgill.ca/darmon Telephone: Work (514) 398-2263 Home: (514) 481-0174 Born: Oct. 22, 1965, in Paris, France. Citizenship: Canadian, French, and Swiss. Education: 1987. B.Sc. Mathematics and Computer Science, McGill University. 1991. Ph.D. Mathematics, Harvard University. Thesis: Refined class number formulas for derivatives of L-series. University Positions: 1991-1994. Princeton University, Instructor. 1994-1996. Princeton University, Assistant Professor. 1994-1997. McGill University, Assistant Professor. 1997-2000. McGill University, Associate Professor. 2000- . McGill University, Professor. 2005-2019. James McGill Professor, McGill University. Other positions: 1991-1994. Cercheur hors Qu´ebec, CICMA. 1994- . Chercheur Universitaire, CICMA. 1998- . Director, CICMA (Centre Interuniversitaire en Calcul Math´ematique Alg´ebrique). 1999- . Member, CRM (Centre de Recherches Math´ematiques). 2005-2014. External member, European network in Arithmetic Geometry. Visiting Positions: 1991. IHES, Paris. 1995. Universit´a di Pavia. 1996. Visiting member, MSRI, Berkeley. 1996. Visiting professor and guest lecturer, University of Barcelona. 1997. Visiting Professor, Universit´e Paris VI (Jussieu). 1997. Visitor, Institut Henri Poincar´e. 1998. Visiting Professor and NachDiplom lecturer, ETH, Zuric¨ h. 1999. Visiting professor, Universit`a di Pavia. 2001. Visiting professor, Universit`a di Padova. 2001. Korea Institute for Advanced Study. 2002. Visiting professor, RIMS and Saga University (Japan). 1 2003. Visiting Professor, Universit´e Paris VI, Paris. 2003. Visiting professor, Princeton University. 2004. Visiting Professor, Universit´e Paris VI, Paris. 2006. Visiting Professor, CRM, Barcelona, Spain. 2008. Visiting Professor, Universit´e Paris-Sud (Orsay). -
Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves Athanasios Angelakis
Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves Athanasios Angelakis To cite this version: Athanasios Angelakis. Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves. Group Theory [math.GR]. Université de Bordeaux; Universiteit Leiden (Leyde, Pays-Bas), 2015. English. NNT : 2015BORD0180. tel-01359692 HAL Id: tel-01359692 https://tel.archives-ouvertes.fr/tel-01359692 Submitted on 23 Sep 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves Proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden op gezag van Rector Magnificus prof. mr. C.J.J.M. Stolker, volgens besluit van het College voor Promoties te verdedigen op woensdag 2 september 2015 klokke 15:00 uur door Athanasios Angelakis geboren te Athene in 1979 Samenstelling van de promotiecommissie: Promotor: Prof. dr. Peter Stevenhagen (Universiteit Leiden) Promotor: Prof. dr. Karim Belabas (Universit´eBordeaux I) Overige leden: Prof. -
Prize Is Awarded Every Three Years at the Joint Mathematics Meetings
AMERICAN MATHEMATICAL SOCIETY LEVI L. CONANT PRIZE This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding fi ve years. Levi L. Conant (1857–1916) was a math- ematician who taught at Dakota School of Mines for three years and at Worcester Polytechnic Institute for twenty-fi ve years. His will included a bequest to the AMS effective upon his wife’s death, which occurred sixty years after his own demise. Citation Persi Diaconis The Levi L. Conant Prize for 2012 is awarded to Persi Diaconis for his article, “The Markov chain Monte Carlo revolution” (Bulletin Amer. Math. Soc. 46 (2009), no. 2, 179–205). This wonderful article is a lively and engaging overview of modern methods in probability and statistics, and their applications. It opens with a fascinating real- life example: a prison psychologist turns up at Stanford University with encoded messages written by prisoners, and Marc Coram uses the Metropolis algorithm to decrypt them. From there, the article gets even more compelling! After a highly accessible description of Markov chains from fi rst principles, Diaconis colorfully illustrates many of the applications and venues of these ideas. Along the way, he points to some very interesting mathematics and some fascinating open questions, especially about the running time in concrete situ- ations of the Metropolis algorithm, which is a specifi c Monte Carlo method for constructing Markov chains. The article also highlights the use of spectral methods to deduce estimates for the length of the chain needed to achieve mixing. -
LECTURES on HEIGHT ZETA FUNCTIONS of TORIC VARIETIES by Yuri Tschinkel
S´eminaires & Congr`es 6, 2002, p. 227–247 LECTURES ON HEIGHT ZETA FUNCTIONS OF TORIC VARIETIES by Yuri Tschinkel Abstract.— We explain the main ideas and techniques involved in recent proofs of asymptotics of rational points of bounded height on toric varieties. 1. Introduction Toric varieties are an ideal testing ground for conjectures: their theory is sufficiently rich to reflect general phenomena and sufficiently rigid to allow explicit combinato- rial computations. In these notes I explain a conjecture in arithmetic geometry and describe its proof for toric varieties. Acknowledgments. — Iwould like to thank the organizers of the Summer School for the invitation. The results concerning toric varieties were obtained in collaboration with V. Batyrev. It has been a great pleasure and privilege to work with A. Chambert- Loir, B. Hassett and M. Strauch — Iam very much indebted to them. My research was partially supported by the NSA. 1.1. Counting problems Example 1.1.1.—LetX ⊂ Pn be a smooth hypersurface given as the zero set of a homogeneous form f of degree d (with coefficients in Z). Let N(X, B)=#{x | f(x)=0, max(|xj |) B} n+1 (where x =(x0,...,xn) ∈ Z /(±1) with gcd(xj) = 1) be the number of Q-rational points on X of “height” B. Heuristically, the probability that f represents 0 is about B−d and the number of “events” about Bn+1. Thus we expect that lim N(X, B) ∼ Bn+1−d. B→∞ 2000 Mathematics Subject Classification.—14G05, 11D45, 14M25, 11D57. Key words and phrases.—Rational points, heights, toric varieties, zeta functions. -
On the History of Fermat's Last Theorem: a Down-To-Earth
On the History of Fermat’s Last Theorem: A Down-to-Earth Approach Leo Corry - Tel Aviv University DRAFT ov. 2007 – OT FOR QUOTATIO 1. Introduction .............................................................................................................. - 1 - 2. Origins – in the margins .......................................................................................... - 5 - 3. Early stages – in the margins as well ..................................................................... - 8 - 4. FLT between 1800 and 1855 – still in the margins .............................................. - 13 - 5. Marginality within diversification in number theory (1855-1908) ....................... - 22 - 6. The Wolfskehl Prize and its aftermath ................................................................. - 34 - 7. FLT in the twentieth century: developing old ideas, discovering new connections .................................................................................................... - 44 - 8. Concluding Remarks: the Fermat-to-Wiles drama revisited .............................. - 56 - 9. REFERENCES......................................................................................................... - 60 - 1. Introduction Andrew Wiles’ proof of Fermat’s Last Theorem (FLT), completed in 1994, was a landmark of late twentieth century mathematics. It also attracted a great deal of attention among both the media and the general public. Indeed, not every day a mathematical problem is solved more than 350 years after it was first -
Harish Chandra the Iisc and Set up the Molecular Biophysics Unit (MBU)
I NDIAN Ramachandran resigned from Madras in 1970 and then spent two years as a visiting professor at the Biophysics Department of the University of Chicago. During this visit he devised a new method to reconstruct three-dimensional N A images from two-dimensional data, thus laying the foundations of TIONAL computerized tomography. On his return from Chicago Ramachandran joined Harish Chandra the IISc and set up the Molecular Biophysics Unit (MBU). In 1977 he visited S the National Institute for Health in Bethesda, Maryland, USA as a Fogarty CIENCE (1923 – 1983) Scholar. In the same year he was elected a Fellow of the Royal Society, London. He retired from MBU in 1978 but continued as a Professor of Mathematical Philosophy at the IISc until 1989. A CADEMY From the early 1980’s he developed Parkinson’s disease and was cared for by his wife Rajam whom he married in 1945. In 1998, Rajam suddenly died of a Harish Chandra was an outstanding mathematician of his generation. He was heart attack and this was a grievous shock from which Ramachandran never a mathematician who transformed the peripheral topic of ‘representation theory’ recovered. In 1999, the International Union of Crystallography awarded him INSA into a major field which became central to contemporary mathematics. the 5th Ewald Prize for his outstanding contributions to crystallography. In PLA Harish was born on 11 October 1923 in Kanpur. His grandfather was a senior 1999 he had a cardiac arrest and since then remained in the hospital until his railroad clerk in Ajmer. He was deeply committed to give his son Chandrakishore death on 7, April 2001. -
ADELIC ANALYSIS and FUNCTIONAL ANALYSIS on the FINITE ADELE RING Ilwoo
Opuscula Math. 38, no. 2 (2018), 139–185 https://doi.org/10.7494/OpMath.2018.38.2.139 Opuscula Mathematica ADELIC ANALYSIS AND FUNCTIONAL ANALYSIS ON THE FINITE ADELE RING Ilwoo Cho Communicated by P.A. Cojuhari Abstract. In this paper, we study operator theory on the -algebra , consisting of all ∗ MP measurable functions on the finite Adele ring AQ, in extended free-probabilistic sense. Even though our -algebra is commutative, our Adelic-analytic data and properties on ∗ MP MP are understood as certain free-probabilistic results under enlarged sense of (noncommutative) free probability theory (well-covering commutative cases). From our free-probabilistic model on AQ, we construct the suitable Hilbert-space representation, and study a C∗-algebra M generated by under representation. In particular, we focus on operator-theoretic P MP properties of certain generating operators on M . P Keywords: representations, C∗-algebras, p-adic number fields, the Adele ring, the finite Adele ring. Mathematics Subject Classification: 05E15, 11G15, 11R47, 11R56, 46L10, 46L54, 47L30, 47L55. 1. INTRODUCTION The main purposes of this paper are: (i) to construct a free-probability model (under extended sense) of the -algebra ∗ consisting of all measurable functions on the finite Adele ring A , implying MP Q number-theoretic information from the Adelic analysis on AQ, (ii) to establish a suitable Hilbert-space representation of , reflecting our MP free-distributional data from (i) on , MP (iii) to construct-and-study a C∗-algebra M generated by under our representa- P MP tion of (ii), and (iv) to consider free distributions of the generating operators of M of (iii). -
From Simple Algebras to the Block-Kato Conjecture Alexander Merkurjev
Public lecture From simple algebras to the Block-Kato conjecture Alexander Merkurjev Professor at the University of California, Los Angeles Guest of the Atlantic Algebra Centre of MUN Distinguished Lecturer Atlantic Association for Research in the Mathematical Sciences Abstract. Hamilton’s quaternion algebra over the real numbers was the first nontrivial example of a simple algebra discovered in 1843. I will review the history of the study of simple algebras over arbitrary fields that culminated in the proof of the Bloch-Kato Conjecture by the Fields Medalist Vladimir Voevodsky. Time and Place The lecture will take place on June 15, 2016, on St. John’s campus of Memorial University of Newfound- land, in Room SN-2109 of Science Building. Beginning at 7 pm. Awards and Distinctions of the Speaker In 1986, Alexander Merkurjev was a plenary speaker at the International Congress of Mathematicians in Berkeley, California. His talk was entitled "Milnor K-theory and Galois cohomology". In 1994, he gave an invited plenary talk at the 2nd European Congress of Mathematics in Budapest, Hungary. In 1995, he won the Humboldt Prize, a prestigious international prize awarded to the renowned scholars. In 2012, he won the Cole Prize in Algebra, for his fundamental contributions to the theory of essential dimension. Short overview of scientific achievements The work of Merkurjev focuses on algebraic groups, quadratic forms, Galois cohomology, algebraic K- theory, and central simple algebras. In the early 1980s, he proved a fundamental result about the struc- ture of central simple algebras of period 2, which relates the 2-torsion of the Brauer group with Milnor K- theory. -
Introduction of Langlands
Peter Roquette 4. Nov. 2004 Mathematisches Institut, Universit¨at Heidelberg Introduction of Langlands at the Arf lecture, 11 November 2004.1 Ladies and Gentlemen ! It is my pleasure and a great honor to me to introduce Robert Langlands as the speaker of the Arf Lecture of this year 2004, here at the Middle East Technical University. Professor Langlands is a permanent member of the Institute for Advanced Study in Princeton, which is one of the most pres- tigious places for mathematical research studies. He has received a number of highly valued awards, among them the Cole Prize in Number Theory from the American Mathematical Society in 1982, the National Academy of Sciences reward in Mathema- tics in 1988, and the Wolf prize from the Wolf foundation in Israel 1996. He is elected fellow of the Royal Society of Canada and the Royal Society of London, and he has received honorary doctorates from at least 7 universities (to my knowledge). In the laudation for the Wolf prize it was said that he received the prize for his “ path-blazing work and extraordinary insights in the fields of number theory, automorphic forms and group representations”. 1Each year there is scheduled a special colloquium lecture at the Middle East Technical University in Ankara, commemorating Cahit Arf who had served as a professor for many years at METU. In the year 2004 the invited speaker was Robert Langlands, and I had been asked to chair his lecture. The following text was read to introduce the speaker. P.Rq. Introduction of Langlands 4.11.04 Seite 2 Ladies and gentlemen, while I was preparing the text for this introduction it soon became clear to me that in this short time I would not be able to describe, not even approximately, Lang- lands’ rich work, its underlying ideas, its enormous impact on the present mathematical research world wide, and its consequence for the future picture of Mathematics.