{"product_id":"elliptic-curves-mn40-9780691085593","title":"Elliptic Curves. (MN-40)","description":"\u003cp\u003eAn elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects--elliptic curves and modular forms--come together in Eichler-Shimura theory  which constructs elliptic curves out of modular forms of a special kind. The converse  that all rational elliptic curves arise this way  is called the Taniyama-Weil Conjecture and is known to imply Fermats Last Theorem.  Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions  and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is  therefore  a window into a broad expanse of mathematics--including class field theory  arithmetic algebraic geometry  and group representations--in which the concidence of L functions relates analysis and algebra in the most fundamental ways.  Developing  with many examples  the elementary theory of elliptic curves  the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern Eichler-Shimura theory  which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics  and no other explains Eichler-Shimura theory in such an accessible manner.\u003c\/p\u003e","brand":"My Store","offers":[{"title":"Default Title","offer_id":45651863502901,"sku":"ByrdShop_0691085595","price":132.28,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0627\/8139\/0901\/files\/9780691085593.jpg?v=1781842769","url":"https:\/\/atxbooks.com\/products\/elliptic-curves-mn40-9780691085593","provider":"ATX Books","version":"1.0","type":"link"}