A note on rank one quadratic twists of elliptic curves and the non-degeneracy of 𝑝-adic regulators at Eisenstein primes

Abstract

We show that for certain non-CM elliptic curves E / Q E_{/\mathbb {Q}} such that 3 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E ψ E_{\psi } of E E have Mordell–Weil rank one and the 3 3 -adic height pairing on E ψ ( Q ) E_{\psi }(\mathbb {Q}) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p p -adic height. It is not known – though expected – that the archimedian height of these higher-codimensional cycles is non-zero.

p-ADIC L-FUNCTIONS AND RATIONAL POINTS ON CM ELLIPTIC CURVES AT INERT PRIMES

“…Remark 4.15. For non-trivial χ, an abelian variety A χ associated to g does not have semistable reduction at p. Accordingly, the proposition complements [7], [17] and Theorem B. 3.…”

“…We now return to Theorem B. (i) A recent progress towards Conjecture B.1 appears in [7], [17], [19]. The key tools are (variants of) the Beilinson-Flach element and the BDP formula.…”

Mazur's main conjecture at Eisenstein primes

“…To be able to use these classes as a bridge between the anticyclotomic Z p -extension K − ∞ /K and the cyclotomic Z p -extension Q ∞ /Q (or rather its translate by K), here we are led to consider a variant of their construction attached to the pair (f, g), where f is the weight 2 newform attached to E, and g is a suitable CM Hida family. Because our g specializes in weight 1 to the p-irregular Eisenstein series Eis 1,η , where η = η K/Q is the quadratic character associated to K/Q, in fact we use a refinement of the construction in [KLZ17] studied in [BST21].…”