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Physical Sciences and Mathematics Commons™
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Articles 1 - 2 of 2
Full-Text Articles in Physical Sciences and Mathematics
There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, Allison F. Arnold-Roksandich
There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, Allison F. Arnold-Roksandich
HMC Senior Theses
L-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions.
Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy
Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy
CMC Senior Theses
In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.