Arithmetic of Abelian Varieties

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The investigator will conduct research in Number Theory and Arithmetical Algebraic Goemetry, with a focus on elliptic curves. She will study the distribution of ranks of elliptic cuves in families of quadratic twists, investigating how often a given rank occurs. Such questions arise when one studies the Conjecture of Birch and Swinnerton-Dyer, and conjectures on the boundedness or unboundedness of ranks of elliptic curves.Elliptic curves have both mathematical importance, and powerful applications. They were crucial in achieving important and fundamental mathematical results about arithmetic and geometry, including Fermat's Last Theorem, the Shimura-Taniyama Conjecture, and the Mordell Conjecture. They can also be used to factor integers and to test for primality. They have important applications to cryptography, for use in wireless and/or mobile devices including smart cards, cell phones, pagers, and hand-held computers.