Mazur's Growth Number Conjecture in the Rank One Case
Debanjana Kundu, Antonio Lei
TL;DR
The paper addresses Mazur's Growth Number Conjecture for non-CM elliptic curves in the rank-one setting over a $ ext{Z}_p^2$-extension of an imaginary quadratic field $K$ with $p$ splitting. It develops an analytic framework based on a two-variable $p$-adic $L$-function $L_p(X,Y)$, its cyclotomic derivative, and Disegni's height-derivative formula to connect $p$-adic heights of Heegner points to the growth of Selmer coranks, under a one-sided IMC inclusion. Under the generalized Heegner hypothesis and ordinary reduction at $p$, the non-vanishing of $ rac{ ext{∂}L_p}{ ext{∂}Y}(X,0)$ (equivalently a nonzero $p$-adic height of the Heegner point) forces nontrivial specializations $oldsymbol{ extL}_{a,b}$ and, via the IMC, torsion structure and bounded coranks for Selmer groups in the corresponding extensions. Consequently, the growth of Selmer coranks matches Mazur's predictions in the rank-one context, and a pathway is provided to extend the approach to abelian varieties of $ ext{GL}_2$-type. The work highlights the power of $p$-adic L-functions and height pairings in translating algebraic rank questions into analytic non-vanishing phenomena.
Abstract
Let $p\geq 5$ be a prime number. Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. Let $K$ be an imaginary quadratic field where $p$ splits, and such that the generalized Heegner hypothesis holds. Under mild hypotheses, we show that if the $p$-adic height of the Heegner point of $\mathsf{E}$ over $K$ is non-zero, then Mazur's conjecture on the growth of Selmer coranks in the $\mathbb{Z}_p^2$-extension of $K$ holds.