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Superstring Theory is a study using supersymmetric strings to give an explanation for fundamental elements in the nature. One of its main focuses is an algebraic geometric object called Calabi- Yau manifold. A 6-dimensional Calabi-Yau manifold leads to the idea of mirror symmetry. The article “From Polygons to String Theory” suggests that we can stud the mirror symmetry by working on reflexive polygons. Each polygon provides us a family of curves. In this project, we will study representative polygons that produce elliptic curves.
Hasse Theorem says that the trace of Frobenius endomorphism of an elliptic curve over finite field Fp, denoted ap, satisfies the absolute value of ap is less than or equal to 2√p. One of well-known results regarding to this theorem is the Sato Tate conjecture. Motivated by the article “Finding meaning in error terms”, this project attempts to investigate this conjecture on particular chosen elliptic curves. Our method is using data collected by programing PARI/GP and MAGMA in order to make observations. We shall also have a closer look on the torsion subgroup of Mordell-Weil group of our elliptic curves and a relation between these curves over rational field Q and their reductions over finite field Fp.
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Pham, Linh Thi Dieu, "From String Theory to Elliptic Curves over finite field Fp" (2014). Senior Projects Spring 2014. 72.