Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's pp-rationality conjecture

In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements

Asymptotic growth of the signed Tate-Shafarevich groups for supersingular abelian varieties

Let $E$ be an elliptic curve over $\mathbb{Q}$ with supersingular reduction at $p$ with $a_p=0$. We study the asymptotic growth of the plus and minus Tate-Shafarevich groups defined by Lei along the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. In this paper, we work in the general framework of supersingular abelian varieties defined over $\mathbb{Q}$. Using Coleman maps constructed by Buyukboduk--Lei, we define the multi-signed Mordell-Weil groups for supersingular abelian varieties, provide an explicit structure of the dual of these groups as an Iwasawa module and prove a control theorem. Furthermore, we define the multi-signed Tate-Shafarevich groups and, by computing their Kobayashi rank, we provide an asymptotic growth formula along the cyclotomic tower of $\mathbb{Q}$

Evidence for strange stars from joint observation of harmonic absorption bands and of redshift

From recent reports on terrestrial heavy ion collision experiments it appears that one may not obtain information about the existence of asymptotic freedom (AF) and chiral symmetry restoration (CSR) for quarks of QCD at high density. This information may still be obtained from compact stars - if they are made up of strange quark matter. Very high gravitational redshift lines (GRL), seen from some compact stars, seem to suggest high ratios of mass and radius (M/R) for them. This is suggestive of strange stars (SS) and can in fact be fitted very well with SQM equation of state deduced with built in AF and CSR. In some other stars broad absorption bands appear at about ~ 0.3 keV and multiples thereof, that may fit in very well with resonance with harmonic compressional breathing mode frequencies of these SS. Emission at these frequencies are also observed in six stars. If these two features of large GRL and BAB were observed together in a single star, it would strengthen the possibility for the existence of SS in nature and would vindicate the current dogma of AF and CSR that we believe in QCD. Recently, in 4U 1700-24, both features appear to be detected, which may well be interpreted as observation of SS - although the group that analyzed the data did not observe this possibility. We predict that if the shifted lines, that has been observed, are from neon with GRL shift z = 0.4 - then the compact object emitting it is a SS of mass 1.2 M_sun and radius 7 km. In addition the fit to the spectrum leaves a residual with broad dips at 0.35 keV and multiples thereof, as in 1E1207-5209 which is again suggestive of SS.Comment: 5 pages, 4 figures, accepted for publication in the MNRA

Conjecture A and μ\mu-invariant for Selmer groups of supersingular elliptic curves

Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over $\mathbb{Z}_p^2$-extensions of an imaginary quadratic field where $p$ splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the $\mu$-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups (when $E$ has good supersingular reduction) which are completely analogous to the classical Selmer group (when $E$ has good ordinary reduction). In this survey paper, we do not present any proofs, however we have tried to give references of the discussed results for the interested reader.Comment: 31 pages. This is a survey pape

On the signed Selmer groups for motives at non-ordinary primes in Zp2\mathbb{Z}_p^2-extensions

Generalizing the work of Kobayashi and the second author for elliptic curves with supersingular reduction at the prime $p$, B\"uy\"ukboduk and Lei constructed multi-signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a number field $F$ for more general non-ordinary motives. In particular, their construction applies to abelian varieties over $F$ with good supersingular reduction at all the primes of $F$ above $p$. In this article, we scrutinize the case in which $F$ is imaginary quadratic, and prove a control theorem (that generalizes Kim's control theorem for elliptic curves) of multi-signed Selmer groups of non-ordinary motives over the maximal abelian pro-$p$ extension of $F$ that is unramified outside $p$, which is the $\mathbb{Z}_p^2$-extension of $F$. We apply it to derive a sufficient condition when these multi-signed Selmer groups are cotorsion over the corresponding two-variable Iwasawa algebra. Furthermore, we compare the Iwasawa $\mu$-invariants of multi-signed Selmer groups over the $\mathbb{Z}_p^2$-extension for two such representations which are congruent modulo $p$