On the maximality of the λ-invariants of Mazur--Tate elements

2025/11/29 by Antonio Lei, Lei, Antonio, Robert Pollack +3
Mathematics · #Analytic Number Theory Research #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2512.00525

Abstract

Let E be an elliptic curve with good ordinary reduction at an odd prime p. Assuming that Greenberg's μ=0 conjecture holds, we show that the λ-invariants of the Mazur--Tate elements attached to E either stabilise to the λ-invariant of the p-adic L-function or they attain the largest possible value at all finite levels. We characterise the latter phenomenon: it occurs if and only if \ord_p(\fracL(E',1)Ω_E') is negative for some E' that is isogenous to E. Furthermore, we relate this condition to congruences with boundary symbols coming from Eisenstein series. We also study the extension of these results to Hecke eigenforms of weight two.

Citations

• On the Iwasawa Invariants of Mazur--Tate elements of elliptic curves at additive primes
• Lambda-invariants of Mazur--Tate elements attached to Ramanujan's tau function and congruences with Eisenstein series
• Arithmetic conjectures suggested by the statistical behavior of modular symbols
• An algebraic version of a theorem of Kurihara

Related

• On the Iwasawa Invariants of Mazur--Tate elements of elliptic curves at additive primes
• An analogue of Kida's formula for Mazur-Tate elements
• On Nekovář's heights, exceptional zeros and a conjecture of Mazur-Tate-Teitelbaum
• The Iwasawa μ-invariants of Elliptic Curves over ℚ
• The Iwasawa μ-invariant of certain elliptic curves of analytic rank zero