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Uniform Boundedness Conjecture For Rational Points
In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field K and a positive integer g \geq 2 that there exists a number N(K,g) depending only on K and g such that for any algebraic curve C defined over K having genus equal to g has at most N(K,g) K- rational points. This is a refinement of Faltings's theorem, which asserts that the set of K-rational points C(K) is necessarily finite. Progress The first significant progress towards the conjecture was due to Caporaso, Harris, and Mazur. They proved that the conjecture holds if one assumes the Bombieri–Lang conjecture. Mazur's Conjecture B A variant of the conjecture, due to Mazur, asserts that there should be a number N(K,g,r) such that for any algebraic curve C defined over K having genus g and whose Jacobian variety J_C has Mordell–Weil rank over K equal to r, the number of K-rational points of C is at most N(K,g,r). This variant of the conjecture is known as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic variety, algebraic varieties. In more abstract terms, arithmetic geometry can be defined as the study of scheme (mathematics), schemes of Finite morphism#Morphisms of finite type, finite type over the spectrum of a ring, spectrum of the ring of integers. Overview The classical objects of interest in arithmetic geometry are rational points: solution set, sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or Algebraic function field, function fields, i.e. field (mathematics), fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The European Mathematical Society
'' Journal of the European Mathematical Society'' is a monthly peer-reviewed mathematical journal. Founded in 1999, the journal publishes articles on all areas of pure and applied mathematics. Most published articles are original research articles but the journal also publishes survey articles.Summary of the journal The journal has been published by until 2003. Since 2004, it is published by the . The first editor-in-chief was |
Bogomolov Conjecture
In mathematics, the Bogomolov conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin-Mumford conjecture in arithmetic geometry. The conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998. A further generalization to general abelian varieties was also proved by Zhang in 1998. Statement Let ''C'' be an algebraic curve of genus ''g'' at least two defined over a number field ''K'', let \overline K denote the algebraic closure of ''K'', fix an embedding of ''C'' into its Jacobian variety ''J'', and let \hat h denote the Néron-Tate height on ''J'' associated to an ample symmetric divisor. Then there exists an \epsilon > 0 such that the set : \ is finite. Since \hat h(P)=0 if and only if ''P'' is a torsion point, the Bogomolov conjecture generalises the Manin-Mumford conjecture. Proof The original Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998. Genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philipp Habegger
Philipp Habegger (born 23 July 1978) is a Swiss mathematician and a professor of mathematics at the University of Basel who works in Diophantine geometry. Early life and education Habegger was born on 23 July 1978. He received his Ph.D. under the supervision of David Masser at the University of Basel in 2007. Career From 2008 to 2010, Habegger was a ETH Fellow at ETH Zurich. He moved to the University of Zurich for a lectureship position in 2010. In 2013, he was a von Neumann Fellow at the Institute for Advanced Study. As of 2021, Habegger is a professor of mathematics at the University of Basel. Research Habegger's research focuses on height functions and their applications to unlikely intersections In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on .... Selected publicatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ziyang Gao
Ziyang () prefecture-level city in eastern Sichuan province, China. It is bordered by the provincial capital of Chengdu to the northwest, Deyang to the north, Suining to the northeast, Chongqing municipality to the east, and Neijiang to the west. Its development is going to be very important because of the proximity of Chengdu new Airport and economic zone. As of the 2020 Chinese census, Ziyang's total population was 2,308,631 inhabitants whom 867,119 lived in the built-up (''or metro'') area made of Yanjiang District. Subdivisions Government and infrastructure Sichuan Provincial Women's Prison is in Yangma Town (), Jianyang,Archive. Sichuan Provincial Administration of Prisons which was previously under the jurisdiction of Ziyang. Climate Demographics Ziyang's permanent population is approximately 2.503 million in 2019. Ziyang's population has been experiencing a steady decline during the 2010s, with its population in 2010 standing at approximately 3.665 mill ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chabauty's Method
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality. Diophantine geometry in general is the study of algebraic varieties ''V'' over fields ''K'' that are finitely generated over their prime fields—including as of special interest number fields and finite fields—and over local fields. Of those, only the complex numbers are algebraically closed; over any other ''K'' the existence of points of ''V'' with coordinates in ''K'' is something to be proved and studied as an extra topic, even knowing the geometry of ''V''. Arithmetic geometry can be more generally defined as the study of schemes of finite type over the spectrum of the ring of integers In mathematics, the ring of integers of an algebraic number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duke Mathematical Journal
''Duke Mathematical Journal'' is a peer-reviewed mathematics journal published by Duke University Press. It was established in 1935. The founding editors-in-chief were David Widder, Arthur Coble, and Joseph Miller Thomas Joseph Miller Thomas (16 January 1898 – 1979) was an American mathematician, known for the Thomas decomposition of algebraic and differential systems. Thomas received his Ph.D., supervised by Frederick Wahn Beal, from the University of Pennsylva .... The first issue included a paper by Solomon Lefschetz. Leonard Carlitz served on the editorial board for 35 years, from 1938 to 1973. The current managing editor is Richard Hain (Duke University). Impact According to the journal homepage, the journal has a 2018 impact factor of 2.194, ranking it in the top ten mathematics journals in the world. References External links * Mathematics journals Duke University, Mathematical Journal Publications established in 1935 Multilingual journals English-language jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Zureick-Brown
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". was, according to the Hebrew Bible, the third king of the United Kingdom of Israel. In the Books of Samuel, he is described as a young shepherd and harpist who gains fame by slaying Goliath, a champion of the Philistines, in southern Canaan. David becomes a favourite of Saul, the first king of Israel; he also forges a notably close friendship with Jonathan, a son of Saul. However, under the paranoia that David is seeking to usurp the throne, Saul attempts to kill David, forcing the latter to go into hiding and effectively operate as a fugitive for several years. After Saul and Jonathan are both killed in battle against the Philistines, a 30-year-old David is anointed king over all of Israel and Judah. Following his rise to power, Da ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Rabinoff
Joseph is a common male given name, derived from the Hebrew Yosef (יוֹסֵף). "Joseph" is used, along with "Josef", mostly in English, French and partially German languages. This spelling is also found as a variant in the languages of the modern-day Nordic countries. In Portuguese and Spanish, the name is "José". In Arabic, including in the Quran, the name is spelled ''Yūsuf''. In Persian, the name is "Yousef". The name has enjoyed significant popularity in its many forms in numerous countries, and ''Joseph'' was one of the two names, along with ''Robert'', to have remained in the top 10 boys' names list in the US from 1925 to 1972. It is especially common in contemporary Israel, as either "Yossi" or "Yossef", and in Italy, where the name "Giuseppe" was the most common male name in the 20th century. In the first century CE, Joseph was the second most popular male name for Palestine Jews. In the Book of Genesis Joseph is Jacob's eleventh son and Rachel's first son, and kn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric Katz
Eric Katz is a mathematician working in combinatorial algebraic geometry and arithmetic geometry. He is currently an associate professor in the Department of Mathematics at Ohio State University. In joint work with Karim Adiprasito and June Huh, he resolved the Heron–Rota–Welsh conjecture on the log-concavity of the characteristic polynomial of matroids. With Joseph Rabinoff and David Zureick-Brown, he has given bounds on rational and torsion points on curves. Education Katz went to Beachwood High School, in Beachwood, Ohio, a suburb of Cleveland. After earning a B.S. in Mathematics from Ohio State University in 1999, he pursued graduate studies at Stanford University, obtaining his Ph.D. in 2004 with a thesis written under the direction of Yakov Eliashberg Yakov Matveevich Eliashberg (also Yasha Eliashberg; russian: link=no, Яков Матвеевич Элиашберг; born 11 December 1946) is an American mathematician who was born in Leningrad, USSR. E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
