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Indivisibility of Heegner points in the multiplicative case
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abstract
For certain elliptic curves $E$ over $\mathbb{Q}$ with multiplicative reduction at a prime $p\geq 5$, we prove the $p$-indivisibility of the derived Heegner classes defined with respect to an imaginary quadratic field $K$, as conjectured by Kolyvagin. The conditions on $E$ include that $E[p]$ be irreducible and not finite at $p$ and that $p$ split in the imaginary quadratic field $K$, along with certain $p$-indivisibility conditions on various Tamagawa factors. The proof extends the arguments of the second author for the case where $E$ has good ordinary reduction at~$p$.
Forward citations
Cited by 3 Pith papers
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$L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings
The central L-derivative of an elliptic curve over a real quadratic field is expressed as a χ-twisted sum of regularized theta lifts (automorphic Green's functions) on X0(N)×X0(N).
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Kolyvagin's conjecture for modular forms
Under ordinary p and p>k+1 with standard residual assumptions, the strong Kolyvagin conjecture for even-weight newforms holds: derived Heegner-cycle cohomology classes are nonzero.
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Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions
Central derivative values of Rankin–Selberg L-functions are written as arithmetic heights of Hirzebruch–Zagier divisors on X_0(N)×X_0(N) or as geodesic Green's-function sums, by reorganizing Bruinier–Yang and AGHMP theorems.
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