Abstract
Let be an elliptic curve over a number field , and fix a rational prime . Put , where is the group of -power torsion points of . Let be an irreducible self-dual complex representation of . With certain assumptions on and , we give explicit formulas for the root number . We use these root numbers to study the growth of the rank of in its own division tower and also to count the trivial zeros of the -function of . Moreover, our assumptions ensure that the -division tower of is nonabelian.
In the process of computing the root number, we also study the irreducible self-dual complex representations of , where is the ring of integers of a finite extension of , for an odd prime. Among all such representations, those that factor through have been analyzed in detail in existing literature. We give a complete description of those irreducible self-dual complex representations of that do not factor through .
Citation
Ryota Matsuura. "Twisted root numbers of elliptic curves semistable at primes above 2 and 3." Algebra Number Theory 4 (3) 255 - 295, 2010. https://doi.org/10.2140/ant.2010.4.255
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