21 research outputs found

    Statistics for Iwasawa invariants of elliptic curves, II

    Get PDF
    We study the average behavior of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants

    λ\lambda-invariant stability in families of modular Galois representations

    Full text link
    Consider a family of modular forms of weight 2, all of whose residual (modp)\pmod{p} Galois representations are isomorphic. It is well-known that their corresponding Iwasawa λ\lambda-invariants may vary. In this paper, we study this variation from a quantitative perspective, providing lower bounds on the frequency with which these λ\lambda-invariants grow or remain stable.Comment: final version; to appear in Research in the Mathematical Science

    Statistics for Iwasawa Invariants of elliptic curves, II\rm{II}

    Full text link
    We study the average behaviour of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain unconditional lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.Comment: 23 pages, minor changes. Accepted for publication in the International Journal of number theor

    Heuristics for anti-cyclotomic Zp\mathbb{Z}_p-extensions

    Full text link
    This paper studies Iwasawa invariants in anti-cyclotomic towers. We do this by proposing two heuristics supported by computations. First we propose the Intersection Heuristics: these model `how often' the pp-Hilbert class field of an imaginary quadratic field intersects the anti-cyclotomic tower and to what extent. Second we propose the Invariants Heuristics: these predict that the Iwasawa invariants λ\lambda and μ\mu usually vanish for imaginary quadratic fields where pp is non-split.Comment: v2: Incorporated changes as per the referee reports; accepted for publication (Experimental Math

    Studying Hilbert's 10th problem via explicit elliptic curves

    Full text link
    N.Garc\'ia-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form Q(p3,q)\mathbb{Q}(\sqrt[3]{p},\sqrt{-q}) for positive proportions of primes pp and qq. We improve their proportions and extend their results to the case of number fields of the form Q(p3,Dq)\mathbb{Q}(\sqrt[3]{p},\sqrt{Dq}), where DD belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.Comment: Comments very welcome

    Structure of fine Selmer groups in abelian p-adic Lie extensions

    Get PDF
    This paper studies fine Selmer groups of elliptic curves in abelian p -adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic Z p -extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg\u27s Conjecture is clarified

    Derived pp-adic heights and the leading coefficient of the Bertolini--Darmon--Prasanna pp-adic LL-function

    Full text link
    Let E/QE/\mathbf{Q} be an elliptic curve and let pp be an odd prime of good reduction for EE. Let KK be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which pp splits. In a previous work, Agboola--Castella formulated an analogue of the Birch--Swinnerton-Dyer conjecture for the pp-adic LL-function LpBDPL_{\mathfrak{p}}^{\rm BDP} of Bertolini--Darmon--Prasanna attached to E/KE/K, assuming the prime pp to be ordinary for EE. The goal of this paper is two-fold: (1) We formulate a pp-adic BSD conjecture for LpBDPL_{\mathfrak{p}}^{\rm BDP} for all odd primes pp of good reduction. (2) For an algebraic analogue FpBDPF_{\overline{\mathfrak{p}}}^{\rm BDP} of LpBDPL_{\mathfrak{p}}^{\rm BDP}, we show that the ``leading coefficient'' part of our conjecture holds, and that the ``order of vanishing'' part follows from the expected ``maximal non-degeneracy'' of an anticyclotomic pp-adic height. In particular, when the Iwasawa--Greenberg Main Conjecture (FpBDP)=(LpBDP)(F_{\overline{\mathfrak{p}}}^{\rm BDP})=(L_{\mathfrak{p}}^{\rm BDP}) is known, our results determine the leading coefficient of LpBDPL_{\mathfrak{p}}^{\rm BDP} at T=0T=0 up to a pp-adic unit. Moreover, by adapting the approach of Burungale--Castella--Kim in the pp-ordinary case, we prove the main conjecture for supersingular primes pp under mild hypotheses.Comment: 34 page

    STRUCTURE OF FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

    Full text link
    This paper studies fine Selmer groups of elliptic curves in abelian p-adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic Zp-extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified

    Efficacy and safety during endoscopic retrograde cholangiopancreatography (ERCP) under total intravenous anesthesia – propofol alone versus propofol supplemented with dexketa, a comparative study in medical college, Kolkata

    Get PDF
    Background: Endoscopic retrograde cholangiopancreatography (ERCP) is an invasive procedure and, hence, is distressing for awake patients, requiring an adequate level of anesthesia. Recent advancements have encouraged the use of monitored anesthesia care, that allows the patient to tolerate unpleasant experiences during procedures while maintaining cardio-respiratory function. Usually, propofol-based anesthesia is given in ERCP. The main aim of this study is to compare the effect of propofol alone and propofol with ketamine and dexmedetomidine on the hemodynamics during ERCP, recovery profile, and side effects (if any). Aims and Objectives: (1) To compare efficacy in terms of hemodynamic stability and desaturation events. (2) Recovery and quality of recovery. (3) Pain score. (4) Incidence of post-operative nausea and vomiting. Materials and Methods: This is a comparative double-blinded study. Adult patients from the age group of 18–70 years belonging to the American Society of Anesthesiologists (ASA-I) and ASA-II who had undergone ERCP under total intravenous anesthesia were taken and randomly assigned to either of the two groups. Both groups received 0.01 mg/kg glycopyrrolate, 0.1 mg/kg ondansetron, 0.05 mg/kg midazolam, 50 mcg fentanyl, and 40 mg hyoscine. Group A patients received 30 mg propofol as a bolus dose and then repeated according to requirements. Group B patients received 0.5 mcg/kg dexmedetomidine as a loading dose and 0.3 mcg/kg/h as a maintenance infusion dose. 30 mg propofol was given before negotiating scope and then 1 mL (1:1) mixture of propofol and ketamine was repeated according to requirements. Total propofol consumption, hemodynamics, quality of recovery, and side effects (if any) were recorded at regular intervals. Results: The study showed significant cases in Group A had episodes of hypotension and apneic events, whereas there were very few hemodynamic instability and almost no apneic events in Group B patients. The requirement of propofol was much higher in Group A patients. Conclusion: Dexmedetomidine when used along with propofol and ketamine in ERCP patients reduced the dose requirement of propofol and maintained hemodynamic stability without causing any apneic events
    corecore