2017/07/19 by Matteo Longo, Longo, Matteo, Maria Rosaria Pati +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1707.06019
openalex publication_date 2017/07/19 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Iwasawa theory of modular forms over anticyclotomic \ℤp-extensions\nof imaginary quadratic fields has been studied by several authors, starting\nfrom the works of Bertolini-Darmon and Iovita-Spiess, under the crucial\nassumption that the prime p is unramified in K. We start in this article\nthe systematic study of anticyclotomic p-adic L-functions when p is\nramified in K. In particular, when f is a weight 2 modular form attached\nto an elliptic curve E/\ℚ having multiplicative reduction at p, and\np is ramified in K, we show an analogue of the exceptional zeroes\nphenomenon investigated by Bertolini-Darmon in the setting when p is inert in\nK. More precisely, we consider situations in which the p-adic L-function\nLp(E/K) of E over the anticyclotomic \ℤp-extension of K does\nnot vanish identically but, by sign reasons, has a zero at certain characters\n\χ of the Hilbert class field of K. In this case we show that the value\nat \χ of the first derivative of Lp(E/K) is equal to the formal group\nlogarithm of the specialization at p of a global point on the elliptic curve\n(actually, this global point is a twisted sum of Heegner points). This\ngeneralizes similar results of Bertolini-Darmon, available when p is inert in\nK and \χ is the trivial character.\n