A graph-theoretic approach to computing Selmer groups of elliptic curves y² = x³ + bx over Q(i) · arXivDesk
2410.22714Oct 30, 2024Updated title. 26 pages, 1 figure. Improved exposition and corrected typos throughout. Major additions include the mod 2 reduction of the graph $G_b$, an explicit calculation of the $\varphi$-Selmer group when $b$ is a product of inert primes, and two new infinite families of elliptic curves over $\mathbb{Q}(i)$ with trivial rank
A graph-theoretic approach to computing Selmer groups of elliptic curves y2=x3+bx over Q(i)
We develop a graph-theoretic algorithm to compute the φ-Selmer group of the elliptic curve Eb:y2=x3+bx
Nearby in the stack
over
Q(i)
, where
b∈Z[i]
and
φ
is a degree 2 isogeny of
Eb
. We associate to
Eb
a weighted graph
Gb
, whose vertices are the odd Gaussian primes dividing
b
, and whose edge weights are determined by the quartic residue symbol between pairs of these primes. By applying our algorithm, we explicitly compute the
φ
-Selmer group of
Eb
when
b
is a product of inert primes, and we construct several infinite families of elliptic curves over
Q(i)
with trivial Mordell-Weil rank.
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