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Chan-Ho Kim

Publications and source records attributed to Chan-Ho Kim.

At least 19 recordsLinked to original sources
arXiv2025

The refined Tamagawa number conjectures for $\mathrm{GL}_2$

Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that the associated $p$-adic Galois representation has large image. We establish a new and refined "Birch and Swinnerton-Dyer type" formula for Bloch-Kato Selmer groups of the central critical twist of $f$ via Kolyvagin derivatives of $L$-values instead of complex analytic or $p$-adic variation of $L$-values only under the Iwasawa main conjecture localized at the augmentation ideal. Our formula determines the exact rank and module structure of the Selmer groups and is insensitive to weight, the local behavior of $f$ at $p$, and analytic rank. As consequences, we prove the non-vanishing of Kato's Kolyvagin system and complete a "discrete" analogue of the Beilinson-Bloch-Kato conjecture for modular forms at ordinary primes. We also obtain the higher weight analogue of the $p$-converse to the theorem of Gross-Zagier and Kolyvagin, the $p$-parity conjecture, and a new computational upper bound of Selmer ranks. We also discuss how to formulate the refined conjecture on the non-vanishing of Kato's Kolyvagin system for modular forms of general weight. In the appendix with Robert Pollack, we compute several numerical examples on the structure of Selmer groups of elliptic curves and modular forms of higher weight. Sometimes our computation provides a deeper understanding of Selmer groups than what is predicted by Birch and Swinnerton-Dyer conjecture.

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arXiv2025

On the Fitting ideals of anticyclotomic Selmer groups of elliptic curves with good ordinary reduction

We give a short proof of the anticyclotomic analogue of the "strong" main conjecture of Kurihara on Fitting ideals of Selmer groups for elliptic curves with good ordinary reduction under mild hypotheses. More precisely, we completely determine the initial Fitting ideal of Selmer groups over finite subextensions of an imaginary quadratic field in its anticyclotomic $\mathbb{Z}_p$-extension in terms of Bertolini--Darmon's theta elements.

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arXiv2024

A user's guide to Beilinson-Kato's zeta elements

In his ground-breaking work, K. Kato constructed the Euler system of Beilinson--Kato's zeta elements and proved spectacular results on the Iwasawa main conjecture for elliptic curves and the classical and $p$-adic Birch and Swinnerton-Dyer conjectures by using these elements. The goal of this expository lecture note is to explain how Kato's Euler systems fit into the framework of the arithmetic of elliptic curves and their Iwasawa theory, and we hope that this approach eventually helps the reader to read Kato's original paper more easily and with less pain.

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arXiv2024

On the adjoint Selmer groups of semi-stable elliptic curves and Flach's zeta elements

We explicitly construct the rank one primitive Stark (equivalently, Kolyvagin) system extending a constant multiple of Flach's zeta elements for semi-stable elliptic curves. As its arithmetic applications, we obtain the equivalence between a specific behavior of the Stark system and the minimal modularity lifting theorem, and we also discuss the cyclicity of the adjoint Selmer groups. This Stark system construction yields a more refined interpretation of the collection of Flach's zeta elements than the "geometric Euler system" approach due to Flach, Wiles, Mazur, and Weston.

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arXiv2022

An explicit comparison of anticyclotomic $p$-adic $L$-functions for Hida families

The aim of this note is to compare several anticyclotomic $p$-adic $L$-functions for modular forms and $p$-adic families of ordinary modular forms, which have been defined and studied from different perspectives by Skinner-Urban, Hida, Perin-Riou, Bertolini-Darmon, Vatsal, Chida-Hsieh, Longo-Vigni, Castella-Longo and Castella-Kim-Longo. The main result of this paper is a comparison between the central critical twist of the two-variable anticyclotomic $p$-adic $L$-function obtained as specialisation of the three-variable $p$-adic $L$-function of Skinner-Urban and the two-variable $p$-adic $L$-function introduced by one of the authors on collaboration with Vigni by means of $p$-adic families of Gross points.

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arXiv2022

Anticyclotomic main conjecture and the non-triviality of Rankin-Selberg $L$-values in Hida families

The aim of this paper is to prove the two-variable anticyclotomic Iwasawa main conjecture for Hida families and a definite version of the horizontal non-vanishing conjecture, which are formulated in Longo-Vigni. Our approach is based on the two-variable anticyclotomic control theorem for Selmer groups for Hida families and the relation between the two-variable anticyclotomic $L$-function for Hida families built out of $p$-adic families of Gross points on definite Shimura curves studied in Castella-Longo and Castella-Kim-Longo and the self-dual twist of the specialisation to the anticyclotomic line of the three-variable $p$-adic $L$-function of Skinner-Urban.

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arXiv2022

Refined applications of Kato's Euler systems for modular forms

We discuss refined applications of Kato's Euler systems for modular forms of higher weight at good primes (with more emphasis on the non-ordinary ones) beyond the one-sided divisibility of the main conjecture and the finiteness of Selmer groups. These include a proof of the Mazur--Tate conjecture on Fitting ideals of Selmer groups over $p$-cyclotomic extensions and a new interpretation of the Iwasawa main conjecture via the non-triviality of Kato's Kolyvagin systems with structural applications. Some applications to Birch and Swinnerton-Dyer conjecture are also discussed.

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arXiv2022

The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves

We reveal a new and refined application of (a weaker statement than) the Iwasawa main conjecture for elliptic curves to the structure of Selmer groups of elliptic curves of arbitrary rank. For a large class of elliptic curves, we obtain the following arithmetic consequences. 1. Kato's Kolyvagin systems is non-trivial. It is the cyclotomic analogue of the Kolyvagin conjecture. 2. The structure of Selmer groups of elliptic curves over the rationals is completely determined in terms of certain modular symbols. It is a structural refinement of Birch and Swinnerton-Dyer conjecture. 3. The rank zero $p$-converse, the $p$-parity conjecture, and a new upper bound of the ranks of elliptic curves are obtained. 4. The conjecture of Kurihara on the semi-local description of mod $p$ Selmer groups is confirmed. 5. An application of the $p$-adic Birch and Swinnerton-Dyer conjecture to the structure of Iwasawa modules is discussed.

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arXiv2022

A higher Gross-Zagier formula and the structure of Selmer groups

We describe a Kolyvagin system-theoretic refinement of Gross--Zagier formula by comparing Heegner point Kolyvagin systems with Kurihara numbers when the root number of a rational elliptic curve $E$ over an imaginary quadratic field $K$ is $-1$. When the root number of $E$ over $K$ is 1, we first establish the structure theorem of the $p^\infty$-Selmer group of $E$ over $K$. The description is given by the values of certain families of quaternionic automorphic forms, which is a part of bipartite Euler systems. By comparing bipartite Euler systems with Kurihara numbers, we also obtain an analogous refinement of Waldspurger formula. No low analytic rank assumption is imposed in both refinements. We also prove the equivalence between the non-triviality of various ``Kolyvagin systems" and the corresponding main conjecture localized at the augmentation ideal. As consequences, we obtain new applications of (weaker versions of) the Heegner point main conjecture and the anticyclotomic main conjecture to the structure of $p^\infty$-Selmer groups of elliptic curves of arbitrary rank. In particular, the Heegner point main conjecture localized at the augmentation ideal implies the strong rank one $p$-converse to the theorem of Gross-Zagier and Kolyvagin.

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arXiv2021

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

We give a proof of a soft version of the $p$-converse to a theorem of Gross--Zagier and Kolyvagin for non-CM elliptic curves with good ordinary reduction at $p >3$ under the irreducibility assumption on the residual representation. In particular, no condition on the conductor is imposed. Combining with the known results, we obtain that the Mordell-Weil rank is one and the Tate-Shafarevich group is finite if and only if the analytic rank is one for every elliptic curve over the rationals.

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arXiv2020

On the indivisibility of derived Kato's Euler systems and the main conjecture for modular forms

We provide a simple and efficient numerical criterion to verify the Iwasawa main conjecture and the indivisibility of derived Kato's Euler systems for modular forms of weight two at any good prime under mild assumptions. In the ordinary case, the criterion works for all members of a Hida family once and for all. The key ingredient is the explicit computation of the integral image of the derived Kato's Euler systems under the dual exponential map. We provide explicit new examples at the end. This work does not appeal to the Eisenstein congruence method at all.

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arXiv2019

On the Iwasawa invariants of Kato's zeta elements for modular forms

We study the behavior of the Iwasawa invariants of the Iwasawa modules which appear in Kato's main conjecture without $p$-adic $L$-functions under congruences. It generalizes the work of Greenberg-Vatsal, Emerton-Pollack-Weston, B.D. Kim, Greenberg-Iovita-Pollack, and one of us simultaneously. As a consequence, we establish the propagation of Kato's main conjecture for modular forms of higher weight at arbitrary good prime under the assumption on the mod $p$ non-vanishing of Kato's zeta elements. The application to the $\pm$ and $\sharp/\flat$-Iwasawa theory for modular forms is also discussed.

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arXiv2019

A proof of Perrin-Riou's Heegner point main conjecture

Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $p>3$ be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate-Shafarevich group of $E$ over the anticyclotomic $\mathbf{Z}_p$-extension of $K$ in terms of Heegner points. In this paper, we give a proof of Perrin-Riou's conjecture under mild hypotheses. Our proof builds on Howard's theory of bipartite Euler systems and Wei Zhang's work on Kolyvagin's conjecture. In the case when $p$ splits in $K$, we also obtain a proof of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna.

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arXiv2019

On the refined conjectures on Fitting ideals of Selmer groups of elliptic curves with supersingular reduction

In this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the "weak main conjecture" à la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime $p$. We also prove the "strong main conjecture" suggested by the second named author under the validity of the $\pm$-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among "finite layer objects", "$\pm$-objects", and "fine objects" in Iwasawa theory. The case of good ordinary reduction is also treated.

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arXiv2019

On the quantitative variation of congruence ideals and integral periods of modular forms

We prove the conjecture of Pollack and Weston on the quantitative analysis of the level lowering congruence \`{a} la Ribet for modular forms of higher weight. It was formulated and studied in the context of the integral Jacquet-Langlands correspondence and anticyclotomic Iwasawa theory for modular forms of weight two and square-free level for the first time. We use a completely different method based on the $R=\mathbb{T}$ theorem established by Diamond-Flach-Guo and Dimitrov and an explicit comparison of adjoint $L$-values. We briefly discuss arithmetic applications of our main result at the end.

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arXiv2019

Remarks on Kato's Euler systems for elliptic curves with additive reduction

Extending the former work for the good reduction case, we provide a numerical criterion to verify a large portion of the "Iwasawa main conjecture without $p$-adic $L$-functions" for elliptic curves with additive reduction at an odd prime $p$ over the cyclotomic $\mathbb{Z}_p$-extension. We also deduce the corresponding $p$-part of the Birch and Swinnerton-Dyer formula for elliptic curves of rank zero from the same numerical criterion. We give explicit examples at the end and specify our choice of Kato's Euler system in the appendix.

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