Algebra and Number Theory Seminar: Zichao Lin

Title: On Iwasawa 𝜇-invariants of Selmer groups for elliptic curves

Abstract: Let 𝐸 be an elliptic curve over ℚ with good ordinary reduction at 𝑝, 𝐸⁡[𝑝] the 𝑝-torsion points of 𝐸 and ℚ∞/ℚ the cyclotomic ℤ𝑝-extension. In 1999, Greenberg offered sufficient conditions for the Pontryagin dual of the Selmer group over ℚ∞ to have 𝜇-invariant 0, and obtained a lower bound when the 𝜇-invariant is positive. In this talk, we will focus on providing upper bounds for the 𝜇-invariant. Combining with Greenberg's result, we are able to prove Greenberg's Conjecture for elliptic curves when 𝐸⁡[𝑝] is reducible. As a corollary, when 𝑝 =2, we are able to classify all the elliptic curves with 𝜇-invariant 0 or 1 under the additional assumptions that 𝐸⁡[4] is reducible. This is a joint work with Mulun Yin.