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In mathematics, arithmetic geometry is 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 varieties. In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers; the classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. 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 non-algebraically-closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fields, étale cohomology provides topological invariants associated to algebraic varieties.

P-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields. In the early 19th century, Carl Friedrich Gauss observed that non-zero integer solutions to homogeneous polynomial equations with rational coefficients exist if non-zero rational solutions exist. In the 1850s, Leopold Kronecker formulated the Kronecker–Weber theorem, introduced the theory of divisors, made numerous other connections between number theory and algebra, he conjectured his "liebster Jugendtraum", a generalization, put forward by Hilbert in a modified form as his twelfth problem, which outlines a goal to have number theory operate only with rings that are quotients of polynomial rings over the integers. In the late 1920s, André Weil demonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to the Mordell–Weil theorem which demonstrates that the set of rational points of an abelian variety is a finitely generated abelian group.

Modern foundations of algebraic geometry were developed based on contemporary commutative algebra, including valuation theory and the theory of ideals by Oscar Zariski and others in the 1930s and 1940s. In 1949, André Weil posed the landmark Weil conjectures about the local zeta-functions of algebraic varieties over finite fields; these conjectures offered a framework between algebraic geometry and number theory that propelled Alexander Grothendieck to recast the foundations making use of sheaf theory, scheme theory, in the 1950s and 1960s. Bernard Dwork proved one of the four Weil conjectures in 1960. Grothendieck developed étale cohomology theory to prove two of the Weil conjectures by 1965; the last of the Weil conjectures would be proven in 1974 by Pierre Deligne. Between 1956 and 1957, Yutaka Taniyama and Goro Shimura posed the Taniyama–Shimura conjecture relating elliptic curves to modular forms; this connection would lead to the first proof of Fermat's Last Theorem in number theory through algebraic geometry techniques of modularity lifting developed by Andrew Wiles in 1995.

In the 1960s, Goro Shimura introduced Shimura varieties as generalizations of modular curves. Since the 1979, Shimura varieties have played a crucial role in the Langlands program as a natural realm of examples for testing conjectures. In papers in 1977 and 1978, Barry Mazur proved the torsion conjecture giving a complete list of the possible torsion subgroups of elliptic curves over the rational numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves. In 1996, the proof of the torsion conjecture was extended to all number fields by Loïc Merel. In 1983, Gerd Faltings proved the Mordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points. In 2001, the proof of the local Langlands conjectures for GLn was based on the geometry of certain Shimura varieties. In the 2010s, Peter Scholze developed perfectoid spaces and new cohomology theories in arithmetic geometry over p-adic fields with application to Galois representations and certain cases of the weight-monodromy conjecture.

Arithmetic dynamics Arithmetic of abelian varieties Birch and Swinnerton-Dyer conjecture Moduli of algebraic curves Siegel modular variety Siegel's theorem on integral points

Corunastylis leptochila is a small terrestrial orchid endemic to Victoria. It has a single thin leaf fused to the flowering stem and up to twenty small reddish-brown to dark purplish flowers, it is known from one population with only six plants in forest near a swamp. Corunastylis leptochila is a terrestrial, deciduous, herb with an underground tuber and a single thin leaf 100–150 mm long. Between five and twenty reddish-brown to dark purplish flowers are densely crowded along a flowering stem 8–15 mm tall; the flowers are 4 -- 5 mm wide. The flowers and are inverted; the dorsal sepal is egg-shaped, 2–2.5 mm long and about 1.5 mm wide. The lateral sepals are linear to lance-shaped, 3.5–4.5 mm long, 1 mm wide with a small white gland on the tip. The petals are egg-shaped, about 2.5 mm long and 1 mm wide and have a small gland on the tip. The labellum is about 2 mm long and 1 mm wide. There is an oblong callus in the centre of the labellum and extending nearly to its tip. Flowering occurs in December.

Corunastylis leptochila was first formally described in 2017 by David Jones from a specimen collected near Lavers Hill and the description was published in Australian Orchid Research. The specific epithet is derived from the Ancient Greek words leptos meaning "fine" or "small" and cheilos meaning "lip" or "rim". Corunastylis leptochila grows in forest near a swamp near Lavers Hill where only six plants are known. Data related to Corunastylis leptochila at Wikispecies

Poottu is a Malayalam movie directed by Rajeevnath. The story is based on U. K. Kumaran’s short story and depicts the life in an apartment complex when the lock system malfunctions; the movie had its world premiere in Dubai on Saturday, 28 January 2017. Shibu Gangadharan is the associate director of this movie; the film, with around 30 Indian expatriate actors, was shot in Dubai and Fujeirah and was completed within 13 days. The movie launched its official trailer on 8 March 2017 and announced its theatrical release in April 2017, it was premiered on television on 3 September 2017. Poottu on IMDb "Poottu Malayalam Movie Photos". "Rajeev Nath Is All Set For His Next Venture! - Nettv4u.com". "പൂട്ടുമായി രാജീവ് നാഥ്!"

Bloody Romance is a 2018 Chinese television series based on the novel of the same name by Banming Banmei. It airs on Youku on July 2018 which stars Li Yitong and Qu Chuxiao as the leads; the series air internationally in 13 foreign countries via Dramafever. The series was a commercial success and passed 600 million views online by August 2018, it was praised for its tight plot and high production quality despite a low budget, as well as for featuring a strong female lead. A story about a young woman, exploited in the past, having gone through hell and back to become an assassin. During the chaotic period towards the end of the Tang Dynasty, Qi Xue accidentally enters a mysterious city and was given the name Wan Mei. Tasked with dangerous missions, she puts herself in danger for each task but gains the protection of Chang An, a man mysterious as a shadow; the two become embroiled in greater conspiracy involving a deadly struggle for power. The series is produced by established director. Xu Jizhou, as well as Zhang Wei.

Guo Yong is the martial arts director. Filming took place in Hengdian from September 2017 to January 2018

The Travel Show is an EP by British rapper Braintax, was released through Low Life Records in 1999. The EP was never released on CD. "Tools" — 3:21 "Rational Geographics" — 3:29 "Go There" — 3:33 "Making Moves" — 3:56 A Side: "Tools" — 3:21 "Rational Geographics" — 3:29 "Tools" — 3:21B Side: "Go There" — 3:33 "Making Moves" — 3:56 "Go There" — 3:33 Braintax produced all songs on this EP, expect "Go There," which Ben Grymm produced. Scratches on "Go There" are by Giacomo and scratches on "Making Moves" are by Lewis Parker; the Travel Show was the first album recorded after DJ T. E. S. T. Left the Braintax duo. Joseph Christie thus became the sole member of Braintax and started using Braintax as his recording name