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 proved by Diamond-Flach-Guo and Dimitrov and an explicit comparison of adjoint $L$-values. As applications, we discuss the comparison of various integral canonical periods, the $\mu$-part of the anticyclotomic main conjecture for modular forms, and the primitivity of Kato's Euler systems

Variation of anticyclotomic Iwasawa invariants in Hida families

Building on the construction of big Heegner points in the quaternionic setting, and their relation to special values of Rankin-Selberg $L$-functions, we obtain anticyclotomic analogues of the results of Emerton-Pollack-Weston on the variation of Iwasawa invariants in Hida families. In particular, combined with the known cases of the anticyclotomic Iwasawa main conjecture in weight $2$, our results yield a proof of the main conjecture for $p$-ordinary newforms of higher weights and trivial nebentypus.Comment: Essentially final version, to appear in Algebra & Number Theor

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.Comment: to appear in Transactions of AM

Effect of the Relocation Policy of Public Agencies in South Korea

In 2005, the government announced the relocation policy of public agencies to non-capital regions. The main purpose of the policy was to decentralize population in the SMA (Seoul Metropolitan Area) through building regional competitiveness. The transfer of 98% of the public institutions was completed in April 2017 (MOLIT 2017). However, unlike the blueprint for the initial plan for the relocation of the agencies, there may be an increase in social cost and insufficient contribution to the regional economic development. This study empirically examines effectiveness of the policy and its impact on regional economic growth and on how differently the policy affects regions depending on the distance from the city for relocation. To measure the policy effect, I use a Difference-in-Difference (DID) model. After analyzing the effects of the relocation of public institutions on the local economy through research models, there is a policy effect only on the Regional Gross Domestic Production (RGDP) in the target city where the innovation city is located. The model for analyzing the surrounding regions shows that there are not statistically significant effects of the policy on the real GDP and population change, indicating that the effects of public sector relocation policies are hardly visible in the surrounding area. As for the factors affecting the real GDP, the increase in the number of houses and the increase in the land price were found to affect the increase in the real GDP. However, analysis of the population change model for the target areas showed that the coefficient value of the policy effect is not statistically significant. This means that the public agency transfer policy has little impact on the population growth of the regions

Indivisibility of Kato's Euler systems and Kurihara numbers (Algebraic Number Theory and Related Topics 2018)

Algebraic Number Theory and Related Topics 2018. November 26-30, 2018. edited by Takao Yamazaki and Shuji Yamamoto. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.In this survey article, we discuss our recent work [KKS20], [KN20] on the numerical verification of the Iwasawa main conjecture for modular forms of weight two at good primes and elliptic curves with potentially good reduction. The criterion is based on the Euler system method and the equality of the main conjecture can be checked via the non-vanishing of Kurihara numbers. We also discuss further arithmetic applications of Kurihara numbers to study the structure of Selmer groups following the philosophy of refined Iwasawa theory `a la Kurihara. In the appendix by Alexandru Ghitza, the SageMath code for an effective computation of Kurihara numbers is illustrated