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By Nicholas M. Katz

For 1000's of years, the examine of elliptic curves has performed a critical function in arithmetic. The earlier century particularly has noticeable large growth during this research, from Mordell's theorem in 1922 to the paintings of Wiles and Taylor-Wiles in 1994. still, there stay many basic questions the place we don't even be aware of what kind of solutions to anticipate. This ebook explores of them: what's the commonplace rank of elliptic curves, and the way does the rank range in different types of households of elliptic curves?

Nicholas Katz solutions those questions for households of ''big'' twists of elliptic curves within the functionality box case (with a growing to be consistent field). The monodromy-theoretic tools he develops prove to use, nonetheless within the functionality box case, both good to households of massive twists of gadgets of every kind, not only to elliptic curves.

The leisurely, lucid creation offers the reader a transparent photo of what's identified and what's unknown at this time, and situates the issues solved during this ebook in the broader context of the final learn of elliptic curves. The book's technical center uses, and explains, a number of complicated themes starting from fresh ends up in finite workforce idea to the equipment of l-adic cohomology and monodromy. Twisted L-Functions and Monodromy is key examining for a person attracted to quantity concept and algebraic geometry.

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