Three-Selmer groups for elliptic curves with 3-torsion |
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Authors: | Tony Feng Kevin James Carolyn Kim Eric Ramos Catherine Trentacoste Hui Xue |
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Institution: | 1. Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA, 02138, USA 2. Department of Mathematical Sciences, Clemson University, Box 340975, Clemson, SC, 29634-0975, USA 3. Department of Mathematical Sciences, Carnegie Mellon University, Wean Hall 6113, Pittsburgh, PA, 15213, USA
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Abstract: | We consider a specific family of elliptic curves with rational 3-torsion subgroup. We arithmetically define 3-Selmer groups through isogeny and 3-descent maps, then associate the image of the 3-descent maps to solutions of homogeneous cubic polynomials affiliated with the elliptic curve E and an isogenous curve E′. Thanks to the work of Cohen and Pazuki, we have solubility conditions for the homogeneous polynomials. Using these conditions, we give a graphical approach to computing the size of 3-Selmer groups. Finally, we translate the conditions on graphs into a question concerning ranks of matrices and give an upper bound for the rank of the elliptic curve E by calculating the size of the Selmer groups. |
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