TY - JOUR
T1 - p∞ -Selmer groups and rational points on CM elliptic curves
AU - Burungale, Ashay
AU - Castella, Francesc
AU - Skinner, Christopher
AU - Tian, Ye
N1 - Publisher Copyright:
© 2022, The Author(s).
PY - 2022/10
Y1 - 2022/10
N2 - Let E/ Q be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if Selp∞(E/Q) has Zp-corank one, then E(Q) has a point of infinite order. The non-torsion point arises from a Heegner point, and thus ords=1L(E,s)=1, yielding a p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For p> 3 , this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real fields [4].
AB - Let E/ Q be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if Selp∞(E/Q) has Zp-corank one, then E(Q) has a point of infinite order. The non-torsion point arises from a Heegner point, and thus ords=1L(E,s)=1, yielding a p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For p> 3 , this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real fields [4].
KW - Elliptic waves
KW - Heegen points
KW - L-functions
KW - P-adic
KW - Selmen groups
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U2 - 10.1007/s40316-022-00203-y
DO - 10.1007/s40316-022-00203-y
M3 - Article
AN - SCOPUS:85128979211
SN - 2195-4755
VL - 46
SP - 325
EP - 346
JO - Annales Mathematiques du Quebec
JF - Annales Mathematiques du Quebec
IS - 2
ER -