Relative $p$-class groups and $p$-Selmer groups
Debajyoti De, Dipramit Majumdar, Sudipa Mondal

TL;DR
This paper explores the relationship between relative p-class groups, p-Selmer groups, and root numbers for elliptic curves with specific j-invariants, revealing new insights into their structure and applications to rank and number theory problems.
Contribution
It establishes a link between relative p-class groups, root numbers, and p-Selmer groups for certain elliptic curves, providing a method to construct large rank p-class groups.
Findings
Relative p-class groups determine p-Selmer group dimensions.
Construction of families with large p-class group rank.
Connection between congruent number problem and relative p-class groups.
Abstract
Let be an elliptic curve with -invariant or and let be a twist of . We show that for any prime of good reduction of , a degree relative -class group and the root number of determines the dimension of the -Selmer group of . As a consequence, we construct families of large rank -class group. We also relate congruent number and cube sum problem with relative -class group.
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Taxonomy
TopicsFinite Group Theory Research
