Regular formal modules in local fields and irregularly degree

In this paper we investigate the irregular degree of finite not ramified local field extantions with respect to a polynomial formal group and in the multiplicative case. There was found necessary and sufficient conditions for the existence of primitive roots of ps power from 1 and (endomorphism [ps]Fm) in L-th unramified extension of the local field K (for all positive integer s). These conditions depend only on the ramification index of the maximal abelian subextension of the field K Ka/Qp.

Ramification of the Eigencurve at Classical RM Points

J. Bellaïche and M. Dimitrov showed that the $p$-adic eigencurve is smooth but not étale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the overconvergent cuspidal Eisenstein points, being the base change lift for $\text{GL}(2)_{/F}$ of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.

Enumerating extensions of (π)-adic fields with given invariants

We give an algorithm that constructs a minimal set of polynomials defining all extensions of a [Formula: see text]-adic field with given inertia degree, ramification index, discriminant, ramification polygon, and residual polynomials of the segments of the ramification polygon.

LIFTING TORSION GALOIS REPRESENTATIONS

Let $p\geqslant 5$ be a prime, and let ${\mathcal{O}}$ be the ring of integers of a finite extension $K$ of $\mathbb{Q}_{p}$ with uniformizer ${\it\pi}$. Let ${\it\rho}_{n}:G_{\mathbb{Q}}\rightarrow \mathit{GL}_{2}\left({\mathcal{O}}/({\it\pi}^{n})\right)$ have modular mod-${\it\pi}$ reduction $\bar{{\it\rho}}$, be ordinary at $p$, and satisfy some mild technical conditions. We show that ${\it\rho}_{n}$ can be lifted to an ${\mathcal{O}}$-valued characteristic-zero geometric representation which arises from a newform. This is new in the case when $K$ is a ramified extension of $\mathbb{Q}_{p}$. We also show that a prescribed ramified complete discrete valuation ring ${\mathcal{O}}$ is the weight-$2$ deformation ring for $\bar{{\it\rho}}$ for a suitable choice of auxiliary level. This implies that the field of Fourier coefficients of newforms of weight 2, square-free level, and trivial nebentype that give rise to semistable $\bar{{\it\rho}}$ of weight 2 can have arbitrarily large ramification index at $p$.

Gracilaria spp. morphology cultured in brackish water pond Pantai Sederhana Village, Muara Gembong

<p class="NoParagraphStyle" align="center"><strong>ABSTRACT</strong></p><p class="NoParagraphStyle" align="center"><strong> </strong></p><p><em>Gracilaria </em>spp. is a euryhaline species of seaweed which can live in the marine and brackish water. Development of <em>Gracilaria </em>spp. culture in Bekasi is potential because this seaweed can be cultured in ex shrimp pond by polyculture system. The objective of this research was to evaluate the phenotype morphological characteristic of <em>Gracilaria </em>spp. based on and its relationship with shrimp pond water quality. <em>Sampling </em>was done at three shrimp ponds with a salinity range at 13.7–19.2 g/L. Phenotypical characteristics of <em>Gracilaria </em>spp. consisted of colour and thallus morfometrics, while measurement of water quality consisted of physical and chemical charactersof shrimp pond. The result showed that <em>Gracilaria </em>spp. generally had light brown colour. At salinity higher than 13.7 g/kg, the number of secondary thalli increased, the distance among internode tertiary thalli declined, and the number of ramification index increased. Salinity showed a positive correlation with remification index which was 0.571.</p><p> </p><p>Keywords: <em>Gracilaria </em>spp., remification index, phenotype, salinity, brackishwater culture</p><br /><p class="NoParagraphStyle"> </p><p class="NoParagraphStyle"> </p><p class="NoParagraphStyle" align="center"><strong>ABSTRAK</strong><strong></strong></p><p class="NoParagraphStyle"> </p><p><em>Gracilaria </em>spp. merupakan spesies rumput laut eurihalin yang dapat hidup di laut dan di perairan payau. Pengembangan budidaya <em>Gracilaria </em>spp. di Bekasi potensial dilakukan karena memanfaatkan tambak bekas budidaya udang dengan sistem polikultur. Penelitian ini bertujuan untuk mengevaluasi karakteristik fenotipe morfologi <em>Gracilaria </em>spp. dan hubungannya dengan kualitas air di tambak budidaya. <em>Sampling </em>dilakukan pada tiga tambak dengan kisaran salinitas 13,7–19,2 g/L. Karakterisasi fenotipe meliputi warna dan morfometrik talus <em>Gracilaria </em>spp., sedangkan parameter kualitas air meliputi karakter fisika dan kimia air tambak. Hasil menunjukkan talus <em>Gracilaria</em>s spp. umumnya berwarna coklat muda dan pada salinitas di atas 13,7 g/L menunjukkan jumlah talus sekunder meningkat, jarak internode talus tersier menurun, dan indeks percabangan meningkat (P&lt;0,05). Salinitas berkorelasi positif dengan indeks percabangan sebesar 0,571.</p><p> </p><p>Kata kunci: <em>Gracilaria </em>spp., indeks percabangan, fenotipe, salinitas, budidaya air payau</p>

AN ELEMENTARY PROOF OF THE CONDUCTOR-DISCRIMINANT FORMULA

For a finite abelian extension K/ℚ, the conductor-discriminant formula establishes that the absolute value of the discriminant of K is equal to the product of the conductors of the elements of the group of Dirichlet characters associated to K. The simplest proof uses the functional equation for the Dedekind zeta function of K and its expression as the product of the L-series attached to the various Dirichlet characters associated to K. In this paper, we present an elementary proof of this formula considering first K contained in a cyclotomic number field of pn-roots of unity, where p is a prime number, and in the general case, using the ramification index of p given by the group of Dirichlet characters.

Local Bounds for Torsion Points on Abelian Varieties

We say that an abelian variety over a p-adic field K has anisotropic reduction (AR) if the special fiber of its Néronminimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the K-rational torsion subgroup of a g-dimensional AR variety depending only on g and the numerical invariants of K (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of g, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

p-ADIC ORDER BOUNDED GROUP VALUATIONS ON ABELIAN GROUPS

For a fixed integer e and prime p we construct the p-adic order bounded group valuations for a given abelian group G. These valuations give Hopf orders inside the group ring KG where K is an extension of $\mathbb{Q} _{p}$ with ramification index e. The orders are given explicitly when G is a p-group of order p or p2. An example is given when G is not abelian.