On Selmer Ranks of Elliptic Curves With a Rational 2-Torsion
DOI:
https://doi.org/10.70474/sqw8ys05Keywords:
Elliptic Curves, Selmer groups, Galois CohomologyAbstract
This study investigates the asymptotic behavior of the ranks of Selmer groups associated with elliptic curves possessing a rational 2-torsion point defined over the integers. The Selmer group plays a central role in understanding the Mordell–Weil group and the Birch and Swinnerton-Dyer conjecture. The arithmetic of elliptic curves with torsion points has long attracted significant interest, with foundational results tracing back to the work of Mordell, Selmer, and later refinements by Cassels and others. In particular, the behavior of 2-Selmer groups provides insights into the distribution of ranks and the structure of rational points. Building upon previous methods developed for quadratic twists and leveraging tools from Galois cohomology, we demonstrate that the upper bounds on the size of these Selmer groups are unbounded within certain infinite families of elliptic curves. Our approach highlights the interplay between local conditions at primes and global properties of the curve, offering new perspectives on how torsion influences Selmer ranks.
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