Jared Weinstein. Reciprocity laws and Galois representations: recent breakthroughs
Given a polynomial f(x) with integer coefficients, a reciprocity law is a rule which determines, for a prime p, whether f(x) modulo p is the product of distinct linear factors. We examine reciprocity laws through the ages, beginning with Fermat, Euler and Gauss, and continuing through the modern theory of modular forms and Galois representations. We conclude with an exposition of Peter Scholze's astonishing work on torsion classes in the cohomology of arithmetic manifolds.
Jared Weinstein. Reciprocity laws and Galois representations: recent breakthroughs
Given a polynomial f(x) with integer coefficients, a reciprocity law is a rule which determines, for a prime p, whether f(x) modulo p is the product of distinct linear factors. We examine reciprocity laws through the ages, beginning with Fermat, Euler and Gauss, and continuing through the modern theory of modular forms and Galois representations. We conclude with an exposition of Peter Scholze's astonishing work on torsion classes in the cohomology of arithmetic manifolds.
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The S&P 500 fell 1.3% to 4,204.36, and the Dow Jones Industrial Average was down 0.7% to 32,943.33. The Dow posted a fifth straight weekly loss — its longest losing streak since 2019. The Nasdaq Composite tumbled 2.2% to 12,843.81. Though all three indexes opened in the green, stocks took a turn after a new report showed U.S. consumer sentiment deteriorated more than expected in early March as consumers' inflation expectations soared to the highest since 1981. Again, in contrast to Facebook, Google and Twitter, Telegram's founder Pavel Durov runs his company in relative secrecy from Dubai. Ukrainian forces successfully attacked Russian vehicles in the capital city of Kyiv thanks to a public tip made through the encrypted messaging app Telegram, Ukraine's top law-enforcement agency said on Tuesday. At this point, however, Durov had already been working on Telegram with his brother, and further planned a mobile-first social network with an explicit focus on anti-censorship. Later in April, he told TechCrunch that he had left Russia and had “no plans to go back,” saying that the nation was currently “incompatible with internet business at the moment.” He added later that he was looking for a country that matched his libertarian ideals to base his next startup. "Like the bombing of the maternity ward in Mariupol," he said, "Even before it hits the news, you see the videos on the Telegram channels."
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