A Proof of the Andre-Oort Conjecture For
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The André-Oort Conjecture for Ag
Annals of Mathematics 187 (2018), 379{390 The Andr´e-Oortconjecture for Ag By Jacob Tsimerman Abstract We give a proof of the Andr´e-Oortconjecture for Ag | the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven \averaged" version of the Colmez conjecture yields lower bounds for Galois orbits of CM points. The Andr´e-Oortconjecture then follows from previous work of Pila and the author. 1. Introduction Recall the statement of the Andr´e-Oortconjecture: Conjecture 1.1. Let S be a Shimura variety, and let V be an irreducible closed algebraic subvariety of S. Then V contains only finitely many maximal special subvarieties. For the definition of the terms \Shimura variety" and \special subvariety," as well as a brief history of the origins of the conjecture, see [21]. In the past few decades, there has been an enormous amount of work on the Andr´e-Oort conjecture. Notably, a full proof of the conjecture under the assumption of the Generalized Riemann Hypothesis (GRH) for CM fields has been given by Klingler, Ullmo, and Yafaev [11], [21]. Following a strategy introduced by Pila and Zannier, in previous work [16] it was shown that the Andr´e-Oortconjecture for the coarse moduli space of principally polarized abelian varieties Ag follows once one establishes that the sizes of the Galois orbits of special points are \large." This is what we prove in this paper. More specifically, we prove the following: Theorem 1.2. There exists δg > 0 such that if Φ is a primitive CM type for a CM field E, and if A is any g-dimensional polarized abelian variety over Q with endomorphism ring equal to the full ring of integers OE and CM type Φ, Keywords: Andr´e-Oort,Complex Multiplication, Faltings Height, Colmez Conjecture AMS Classification: Primary: 11G15, 11G18. -
Point Counting for Foliations Over Number Fields 3
POINT COUNTING FOR FOLIATIONS OVER NUMBER FIELDS GAL BINYAMINI Abstract. Let M be an affine variety equipped with a foliation, both defined over a number field K. For an algebraic V ⊂ M over K write δV for the maximum of the degree and log-height of V . Write ΣV for the points where the leafs intersect V improperly. Fix a compact subset B of a leaf L. We prove effective bounds on the geometry of the intersection B∩V . In particular when codim V = dim L we prove that #(B ∩ V ) is bounded by a polynomial −1 in δV and log dist (B, ΣV ). Using these bounds we prove a result on the interpolation of algebraic points in images of B ∩ V by an algebraic map Φ. For instance under suitable conditions we show that Φ(B∩V ) contains at most poly(g,h) algebraic points of log-height h and degree g. We deduce several results in Diophantine geometry. i) Following Masser- Zannier, we prove that given a pair of sections P,Q of a non-isotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever P,Q are simultaneously torsion their order of torsion is bounded effectively by a polynomial in δP , δQ. In particular the set of such simultaneous torsion points is effectively computable in polynomial time. ii) Following Pila, we prove that n given V ⊂ C there is an (ineffective) upper bound, polynomial in δV , for the degrees and discriminants of maximal special subvarieties. In particular it follows that Andr´e-Oort for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). -
2015 SASTRA Ramanujan Prize Awarded to Jacob Tsimerman - 11-19-2015 by Manjil Saikia - Gonit Sora
2015 SASTRA Ramanujan Prize awarded to Jacob Tsimerman - 11-19-2015 by Manjil Saikia - Gonit Sora - http://gonitsora.com 2015 SASTRA Ramanujan Prize awarded to Jacob Tsimerman by Manjil Saikia - Thursday, November 19, 2015 http://gonitsora.com/2015-sastra-raacob-tsimerman/ The 2015 SASTRA Ramanujan Prize has been awarded to Jacob Tsimerman of the University of Toronto, Candan. The official prize announcement can be found here. The prize is awarded every year since 2005 to a young mathematician under the age of 32 in areas which are influenced by the work of Ramanujan. Previous winners include Terence Tao, Manjul Bhargava, James Maynard, etc. Tsimerman has made deep and highly original contributions to diverse parts of number theory, and most notably to the famous Andre-Oort Conjecture. More information about his research can be found here. Jacob Tsimerman was born in Kazan, Russia on April 26, 1988. In 1990 his family first moved to Israel and then in 1996 to Canada, where he participated in various mathematical competitions from the age of 9. In 2003 and 2004 he represented Canada in the International Mathematical Olympiad (IMO) and won gold medals both years, with a perfect score in 2004. During 2004-06, in just two years, he finished his Bachelors degree courses in Toronto. During 2006-11, he was a doctoral student at Princeton under the supervision of Professor Peter Sarnak; there he was supported by an American Mathematical Society (AMS) Centennial Fellowship. Following his PhD, he had a post-doctoral position at Harvard University as a Junior Fellow of the Harvard Society of Fellows. -
The SASTRA Ramanujan Prize — Its Origins and Its Winners
Asia Pacific Mathematics Newsletter 1 The SastraThe SASTRA Ramanujan Ramanujan Prize — Its Prize Origins — Its Originsand Its Winnersand Its ∗Winners Krishnaswami Alladi Krishnaswami Alladi The SASTRA Ramanujan Prize is a $10,000 an- many events and programs were held regularly nual award given to mathematicians not exceed- in Ramanujan’s memory, including the grand ing the age of 32 for path-breaking contributions Ramanujan Centenary celebrations in December in areas influenced by the Indian mathemati- 1987, nothing was done for the renovation of this cal genius Srinivasa Ramanujan. The prize has historic home. One of the most significant devel- been unusually effective in recognizing extremely opments in the worldwide effort to preserve and gifted mathematicians at an early stage in their honor the legacy of Ramanujan is the purchase in careers, who have gone on to accomplish even 2003 of Ramanujan’s home in Kumbakonam by greater things in mathematics and be awarded SASTRA University to maintain it as a museum. prizes with hallowed traditions such as the Fields This purchase had far reaching consequences be- Medal. This is due to the enthusiastic support cause it led to the involvement of a university from leading mathematicians around the world in the preservation of Ramanujan’s legacy for and the calibre of the winners. The age limit of 32 posterity. is because Ramanujan lived only for 32 years, and The Shanmugha Arts, Science, Technology and in that brief life span made revolutionary contri- Research Academy (SASTRA), is a private uni- butions; so the challenge for the prize candidates versity in the town of Tanjore after which the is to show what they have achieved in that same district is named. -
Jacob Tsimerman to Receive 2015 Sastra Ramanujan Prize
JACOB TSIMERMAN TO RECEIVE 2015 SASTRA RAMANUJAN PRIZE The 2015 SASTRA Ramanujan Prize will be awarded to Dr. Jacob Tsimerman of the University of Toronto, Canada. The SASTRA Ramanujan Prize was established in 2005 and is awarded annually for outstanding contributions by young mathematicians to areas influenced by the genius Srinivasa Ramanujan. The age limit for the prize has been set at 32 because Ramanujan achieved so much in his brief life of 32 years. The prize will be awarded during December 21-22, 2015 at the International Conference on Number Theory at SASTRA University in Kumbakonam (Ramanujan's hometown) where the prize has been given annually. Jacob Tsimerman is an extraordinary young mathematician who has made deep and highly original contributions to diverse parts of number theory, and most notably to the famous Andre-Oort Conjecture. He is one of the few mathematicians to have complete mastery over two very different areas of mathematics - analytic number theory and algebraic geometry. This has enabled him to achieve significant progress on a number of fundamental problems lying at the interface of the two subjects. Much of Tsimerman's research stems from the spectacular PhD thesis entitled \To- wards an unconditional proof of the Andre-Oort conjecture and surrounding problems" that he wrote at Princeton University in 2010 under the direction of Professor Peter Sar- nak. The thesis concerns arithmetical questions around the Andre-Oort conjecture and makes substantial progress towards it. The Andre-Oort Conjecture states that special subsets of Shimura varieties that are obtained as Zariski closures of special points, are finite unions of Shimura varieties. -
Functional Transcendence and Arithmetic Applications
P. I. C. M. – 2018 Rio de Janeiro, Vol. 2 (453–472) FUNCTIONAL TRANSCENDENCE AND ARITHMETIC APPLICATIONS J T Abstract We survey recent results in functional transcendence theory, and give arithmetic applications to the André-Oort conjecture and other unlikely-intersection problems. 1 Introduction The purpose of these notes is to give an introduction to the O-minimality approach to arithmetic geometric questions in the field now referred to as “Unlikely Intersections”, as well as give a brief survey of recent results in the literature. We emphasize the functional transcendence results, which are both necessary for many interesting arithmetic applica- tions, and which we hope are of independent interest. We include some recent results on variations of hodge structures, especially as we believe these are still in their infancy and could have fantastic developments in the near future. In §2, we give a survey of the types of functional transcendence statements that have stemmed from generalizing the Ax-Schanuel theorem. In §3 we introduce o-minimality and give a sort of users manual of the main results that have been useful for these sorts of applications. We then sketch the proofs of some functional transcendence results in §4. §5 is devoted to arithmetic appli- cations, where we explain the recent developments in the André-Oort conjecture and the Zilber-Pink conjecture. We give some sketches of proofs, but we only try to convey the main ideas, rather then give complete arguments. 2 Transcendence Theory 2.1 Classical Results. Classical transcendental number theory is largely concerned with the algebraic properties of special values of special functions. -
Mathematics People
Mathematics People problems’ that he wrote at Princeton University in 2010 Tsimerman Awarded 2015 under the direction of Professor Peter Sarnak. The thesis SASTRA Ramanujan Prize concerns arithmetical questions around the Andre-Oort conjecture and makes substantial progress towards it. Jacob Tsimerman of “The Andre-Oort conjecture states that special subsets the University of To- of Shimura varieties that are obtained as Zariski closures ronto has been awarded of special points are finite unions of Shimura varieties. the 2015 SASTRA Ra- Shimura varieties are special algebraic varieties (such manujan Prize, awarded as moduli spaces of Abelian varieties) which arise as annually for outstand- quotients of suitable complex domains by arithmetic ing contributions by groups. Thus Shimura varieties lie at the heart of arithme- young mathematicians tic geometry and automorphic forms. Yves Andre initially to areas influenced by stated this conjecture for one-dimensional subvarieties, the work of Srinivasa and subsequently Frans Oort proposed that it should hold Ramanujan. The age more generally. The conjecture lies at the confluence of limit for the prize has Diophantine problems and the arithmetic of modular Jacob Tsimerman, winner been set at thirty-two forms. By assuming the Generalized Riemann Hypothesis of the 2015 SASTRA because Ramanujan (GRH), the conjecture was proved in 2006 by Klinger, Ramanujan Prize, was born achieved so much in Ullmo, and Yafaev, but as of 2008 only the very simplest in Russia, lived in Israel, and his brief life of thirty- cases had been proved unconditionally. One of the tech- moved to Canada when he two years. The prize niques to attack the Andre-Oort conjecture is to obtain was eight.