Related papers: Kato's Euler system and the Mazur-Tate refined con…

We provide a new interpretation of the Mazur-Tate Conjecture and then use it to obtain the first (unconditional) theoretical evidence in support of the conjecture for elliptic curves of strictly positive rank.

In this paper we study a new conjecture concerning Kato's Euler system of zeta elements for elliptic curves $E$ over $\mathbb{Q}$. This conjecture, which we refer to as the `Generalized Perrin-Riou Conjecture', predicts a precise congruence…

We give an elementary proof of the Mazur-Tate-Teitelbaum conjecture for elliptic curves by using Kato's element.

Let $E/\mathbb{Q}$ be an elliptic curve of level $N$ and rank equal to $1$. Let $p$ be a prime of ordinary reduction. We experimentally study conjecture $4$ of B. Mazur and J. Tate in his article "Refined Conjectures of the Birch and…

We construct Stickelberger elements for Hilbert modular cusp forms of parallel weight 2 and use recent results of Dasgupta and Spiess to bound their order of vanishing from below. As a special case the vanishing part of Mazur and Tate's…

The second part of the Birch and Swinnerton-Dyer (BSD) conjecture gives a conjectural formula for the order of the Shafarevich-Tate group of an elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated…

Let $E/\mathbb{Q}$ be an elliptic curve which has split multiplicative reduction at a prime $p$ and whose analytic rank $r_{an}(E)$ equals one. The main goal of this article is to relate the second order derivative of the…

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…

In 1987, Mazur and Tate stated conjectures which, in some cases, resemble the classical Birch-Swinnerton-Dyer conjecture and its $p$-adic analog. We study experimentally three conjectures stated by Mazur and Tate using SageMath. Our…

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…

Let $ E $ be an elliptic curve defined over a number field, the conjecture of Birch and Swinnerton-Dyer (BSD, for short) asserts a deep relation between the group $ E(K) $ of rational points and the $ L-$function $ L(E/K, s)$ of $ E $ at $…

If E is an elliptic curve over Q and K is an imaginary quadratic field, there is an Iwasawa main conjecture predicting the behavior of the Selmer group of E over the anticyclotomic Z_p-extension of K. The main conjecture takes different…

We study the Iwasawa main conjecture for quadratic Hilbert modular forms over the p-cyclotomic tower. Using an Euler system in the cohomology of Siegel modular varieties, we prove the "Kato divisibility" of the Iwasawa main conjecture under…

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…

We present an heuristic argument for the prediction of expected Mordell-Weil rank of elliptic curves over number fields, using Birch and Swinnerton-Dyer's original conjecture and Sato-Tate conjectures. We do calculations in some cases and…

We prove an analogue of Kida's formula for the Iwasawa invariants of the Mazur-Tate elements attached to elliptic curves over $\mathbb{Q}$. Let $p$ be an odd prime and let $L/K$ be a Galois extension of abelian number fields with $p$-power…

Let $\mathbb{F}_q$ be a finite field of odd characteristic and $K= \mathbb{F}_q(t)$. For any integer $d\geq 2$ coprime to $q$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x(x^2+t^{2d} x-4t^{2d})$. We show that the rank of the…

We discuss abelian equivariant Iwasawa theory for elliptic curves over $\mathbb{Q}$ at good supersingular primes and non-anomalous good ordinary primes. Using Kobayashi's method, we construct equivariant Coleman maps, which send the…

We prove a substantial part of conjectures of Mazur and Tate that refine the conjecture of Birch and Swinnerton-Dyer. Our approach, which also leads to some results even finer than the predictions of Mazur and Tate, is via the `rank-zero…

The primary objective of this paper is the study of different instances of the elliptic Stark conjectures of Darmon, Lauder and Rotger, in a situation where the elliptic curve attached to the modular form $f$ has split multiplicative…