2015/03/26 by Matteo Longo, Longo, Matteo, Stefano Vigni +1
Mathematics · #11G05 #11R23 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1503.07812
openalex publication_date 2015/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E be an elliptic curve over mathbb Q and let p\≥5 be a prime of\ngood supersingular reduction for E. Let K be an imaginary quadratic field\nsatisfying a modified "Heegner hypothesis" in which p splits, write\nK_\∞ for the anticyclotomic mathbb Zp-extension of K and let\n\Λ denote the Iwasawa algebra of K_\∞/K. By extending to the\nsupersingular case the \Λ-adic Kolyvagin method originally developed by\nBertolini in the ordinary setting, we prove that Kobayashi's plus/minus\np-primary Selmer groups of E over K_\∞ have corank 1 over\n\Λ. As an application, when all the primes dividing the conductor of E\nsplit in K, we combine our main theorem with results of cCiperiani and of\nIovita-Pollack and obtain a "big O" formula for the mathbb Zp-corank of the\np-primary Selmer groups of E over the finite layers of K_\∞/K that\nrepresents the supersingular counterpart of a well-known result for ordinary\nprimes.\n