Explicit isogenies of prime degree over quadratic fields

Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This builds on work of David, Larson-Vaintrob, and Momose. Combining this algorithm with work of Bruin-Najman, \"{O}zman-Siksek, and most recently Box, we determine the above set of primes for the four quadratic fields $\mathbb{Q}(\sqrt{7})$, $\mathbb{Q}(\sqrt{-5})$, $\mathbb{Q}(\sqrt{-10})$, and $\mathbb{Q}(\sqrt{5})$, providing the first such examples after Mazur's 1978 determination for $K = \mathbb{Q}$. The termination of the algorithm relies on the Generalised Riemann Hypothesis.Comment: 34 pages, with an Appendix written jointly with Maarten Derick

Towards strong uniformity for isogenies of prime degree

Let $E$ be an elliptic curve over a number field $k$ of degree $d$ that admits a $k$-rational isogeny of prime degree $p$. We study the question of finding a uniform bound on $p$ that depends only on $d$, and obtain, under a certain condition on the signature of the isogeny, such a uniform bound by explicitly constructing nonzero integers that $p$ must divide. As a corollary we find a uniform bound on torsion points defined over unramified extensions of the base field, generalising Merel's Uniform Boundedness result for torsion.Comment: 19 pages, comments welcom

Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville

We study the problem of determining the groups that can arise as the torsion subgroup of an elliptic curve over a fixed quadratic field, building on work of Kamienny-Najman, Krumm, and Trbovi\'c. By employing techniques to study rational points on curves developed by Bruin and Stoll, we determine the possible torsion subgroups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$ for all squarefree $d$ with $|d| < 800$, improving on the previously known range of $-5 < d < 26$. We use our computations to study the validity of a conjecture of Granville concerning how many twists of a given hyperelliptic curve admit a nontrivial rational point.Comment: 19 pages, 3 figures, comments welcom

IMECE2010-38092 AN INVESTIGATION OF ENTERPRISE RESOURCE PLANNING IMPLEMENTATION: EMPIRICAL EVIDENCE FROM INDIAN COMPANIES

ABSTRACT Due to globalization, there is tremendous demand on Indian companies to lower costs, enlarge product assortment, improve product quality, and provide reliable delivery dates through effective and efficient coordination of production and distribution activities. To achieve these conflicting goals, companies must constantly re-engineer or change their business practices and employ information systems like Enterprise Resource Planning (ERP). However, implementing ERP system is a difficult and high cost proposition. There are mixed results of success and failure in different companies. With a survey of manufacturing companies in India and subsequent detailed case study in one ERP implemented company, this study analyzes various parameters of ERP implementations with factor analysis and logit regression analysis. It is revealed that with clear goals of ERP implementation and proper ERP software selection, the companies are benefited in reducing inventory, improving customer service and other intangible benefits. The top management support, strong and meaningful training program are found the enabling factors of its success. It is argued that this study and ERP implementation model proposed in this paper is valuable to researchers and practitioners interested in implementing ERP system. The findings will also be helpful in extracting the better results from ERP implementation

Molecular progression to cervical precancer, epigenetic switch or sequential model?

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