Corpus ID: 237940378

On the $p$-converse to a theorem of Gross-Zagier and Kolyvagin

@inproceedings{Kim2021OnT,
  title={On the \$p\$-converse to a theorem of Gross-Zagier and Kolyvagin},
  author={Chan-Ho Kim},
  year={2021}
}
  • Chan-Ho Kim
  • Published 25 September 2021
  • Mathematics
We give a simple proof of a p-converse theorem to a theorem of Gross–Zagier and Kolyvagin for non-CM elliptic curves with good ordinary reduction at p > 3 under mild hypotheses. In particular, no condition on the conductor is imposed. Combining with the known results, we obtain the equivalence rkZE(Q) = 1,#X(E/Q) < ∞ ⇔ ords=1L(E, s) = 1 for every elliptic curve E over Q. 1. Statement of the main result The aim of this short article is to give a succinct proof of the following p-converse result… Expand

References

SHOWING 1-10 OF 29 REFERENCES
A converse to a theorem of Gross, Zagier, and Kolyvagin
Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two oddExpand
PERRIN-RIOU’S HEEGNER POINT MAIN CONJECTURE AND p-CONVERSE TO GROSS–ZAGIER–KOLYVAGIN FOR SUPERSINGULAR PRIMES
Let E/Q be a semistable elliptic curve with good supersingular reduction at a prime p > 3. In this paper, we prove the implication corankZpSelp∞ (E/Q) = 1 =⇒ ords=1L(E, s) = 1, which is a p-converseExpand
Exceptional zero formulae and a conjecture of Perrin-Riou
Let $$A/\mathbf{Q}$$A/Q be an elliptic curve with split multiplicative reduction at a prime p. We prove (an analogue of) a conjecture of Perrin-Riou, relating p-adic Beilinson–Kato elements toExpand
On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes
Let $E/\mathbb{Q}$ be an elliptic curve, and $p$ a prime where $E$ has good reduction, and assume that $E$ admits a rational $p$-isogeny. In this paper, we study the anticyclotomic Iwasawa theory ofExpand
Kolyvagin's Conjecture and patched Euler systems in anticyclotomic Iwasawa theory
Let E/Q be an elliptic curve of conductor N and let K be an imaginary quadratic field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology classes for E using K-CM points andExpand
On the p-converse of the Kolyvagin-Gross-Zagier theorem
Let $A/\mathbb{Q}$ be an elliptic curve having split multiplicative reduction at an odd prime $p$. Under some mild technical assumptions, we prove the statement: $$rank_{\mathbb{Z}}A(\mathbb{Q})=1 \Expand
Heegner Point Kolyvagin System and Iwasawa Main Conjecture
We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points for semi-stable elliptic curves E over a quadratic imaginary field K satisfying a certain generalizedExpand
Galois Representations in Arithmetic Algebraic Geometry: Euler systems and modular elliptic curves
This paper consists of two parts. In the first we present a general theory of Euler systems. The main results (see §§3 and 4) show that an Euler system for a p-adic representation T gives a bound onExpand
THE IWASAWA MAIN CONJECTURE FOR HILBERT MODULAR FORMS
  • X. Wan
  • Mathematics
  • Forum of Mathematics, Sigma
  • 2015
Following the ideas and methods of a recent work of Skinner and Urban, we prove the one divisibility of the Iwasawa main conjecture for nearly ordinary Hilbert modular forms under certain localExpand
$p$-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou
Let $p$ be an odd prime. Given an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-D_K})$ where $p$ splits with $D_K>3$, and a $p$-ordinary newform $f \in S_k(\Gamma_0(N))$ such that $N$ verifies theExpand
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