2020/07/17 by Talia Blum, Caroline Choi, Blum, Talia +9
Mathematics · #11G05 #11R29 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2007.08756
openalex publication_date 2020/07/17 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
The Mordell-Weil groups E(\ℚ) of elliptic curves influence the\nstructures of their quadratic twists E-D(\ℚ) and the ideal class\ngroups \CL(-D) of imaginary quadratic fields. For appropriate (u,v)\n\∈ \ℤ2, we define a family of homomorphisms \Φu,v:\nE(\ℚ) \→ \CL(-D) for particular negative fundamental\ndiscriminants -D:=-DE(u,v), which we use to simultaneously address questions\nrelated to lower bounds for class numbers, the structures of class groups, and\nranks of quadratic twists. Specifically, given an elliptic curve E of rank\nr, let \ΨE be the set of suitable fundamental discriminants -D<0\nsatisfying the following three conditions: the quadratic twist E-D has\nrank at least 1; E\tor(\ℚ) is a subgroup of\n\CL(-D); and h(-D) satisfies an effective lower bound which grows\nasymptotically like c(E) \log (D)\(r)/(2) as D \→ \∞. Then for\nany \ε > 0, we show that as X \→ \∞, we have\n
#
,
left
-X lt; -D lt; 0: -D
in
PsiE
right
,
gg
varepsilon\nX^
frac12-
varepsilon. In particular, if \ℓ \∈ 3,5,7 and \ℓ\n\| |E\tor(\ℚ)|, then the number of such discriminants\n-D for which \ℓ \| h(-D) is \≫\ε\nX\(1)/(2)-\ε. Moreover, assuming the Parity Conjecture, our\nresults hold with the additional condition that the quadratic twist E-D\nhas rank at least 2.\n