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Showing 1 to 20 of 22 for “"rational points"”.
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Rational Points on Lattice Varieties
… In this thesis we investigate the properties of rational points on lattice varieties over Witt vectors with algebraically closed residue field in prime characteristic. These varieties are defined over a finite field. For varieties over finite field, the local zeta function encapsulates valuable …
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RATIONAL POINTS ON SOME FAMILIES OF ELLIPTIC CURVES
… curve over Q(m). (Here Q represents the field of rational numbers.) Brown and Myers show that a certain quadratic polynomial m(t) has the property that E_m(t) contains an additional rational point that is independent from the two original generators. This implies that there are infinitely many …
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Computability of rational points on curves over function fields in characteristic p
… thesis is that of explicitly computing the K-rational points of a regular nonsmooth curve X over a αnitely generated αeld K of characteristic p. We start with an in-depth study of such curves in general and the tools exclusive to characteristic p geometry needed to compute their K-points. We …
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Effective Chabauty for symmetric powers of curves
… that curves of genus g > 2 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the upper bound on the number of rational points, XI, [paragraph]2, but this bound is too large to be used in any reasonable sense. In 1985, …
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Generalizations of Golod -Shafarevich and Applications
Let Nq(g) be the maximal number of rational points a global function field of genus g with constant field Fq can have. Let A(q) = lim sup g→infinity Nqg g . As an application of our results, we improve the best known lower bound for A(2) from A(2) > 0.255 to A(2) > 0.3.
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Good reduction of 1-motives
… to questions dealing with reductions of points in semiabelian varieties. We prove a version of the Néron-Ogg-Shafarevich criterion for 1-motives and show how the image of the Frobenius in the ℓ-adic Galois representation associated to a 1-motive determines the ℓ-part of its reduction …
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The Birch and Swinnerton-Dyer Conjecture for elliptic curves
… a characterisation for elliptic curves over rational numbers. We investigate the group structure of rational points on elliptic curves, and show that this group is finitely generated by the Mordell-Weil Theorem. The Shafarevich-Tate group is introduced by way of an example. Thereafter, with …
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Coleman integration for hyperelliptic curves : algorithms and applications
… application is the method of Chabatv to find rational points on curves of genus greater than 1. The second is Mlihyong Kim's recent nonabelian analogue of the Chabauty method for elliptic curves. The last two applications concern p-adic heights on Jacobians of hyperelliptic curves. necessary …
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A Contribution to Metric Diophantine Approximation : the Lebesgue and Hausdorff Theories
… set to understand better the distribution of rational points on algebraic planar curves, Chapter 1 provides an extension of the celebrated Theorem of Duffin and Schaeffer. This enables one to set a generalized version of the Duffin–Schaeffer conjecture. Chapter 2 deals with the topic of …
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Contributions to Trigonometric Sums and Fifth -Order Mock Theta Functions
… a more complex behavior on radial approach to rational points on the unit circle. We prove a special case of a general transformation formula for the fifth order mock theta functions, extending the work of G. E. Andrews on the third order functions. Finally, the last chapter is devoted to …
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A spatiotemporal analysis of Army base distributions in the contiguous United States from 1800 to 1900
… literature it was necessary to establish some rational points of reference to compare fort and personnel attributes of significant timeframes from 1800 to 1900. More specifically, once the distributional patterns were revealed, they were compared by using center of population, geographic means, …
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A spatiotemporal analysis of Army base distributions in the contiguous United States from 1800 to 1900
… literature it was necessary to establish some rational points of reference to compare fort and personnel attributes of significant timeframes from 1800 to 1900. More specifically, once the distributional patterns were revealed, they were compared by using center of population, geographic means, …
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Minimum distance of error correcting codes versus encoding complexity, symmetry, and pseudorandomness
… zeros of a codeword corresponds to the number of rational points.
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The Algorithmic Solution of Simultaneous Diophantine Equations
… for the free abelian part of the group of rational points on each of these curves are presented, including the case p= 16421 which has a canonical height of 137.2290. By combining the method with existing techniques, we also find a generator for the set of points of infinite order on the …
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Elliptic Curves
… of genus 1. Under a geometric operation, the rational points <em>E</em>(<strong><em>Q</em></strong>) of an elliptic curve <em>E</em> form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of …
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Relative Springer isomorphisms and the conjugacy classes in Sylow p-subgroups of Chevalley groups
… consisting of the \(\char{bbold10}{0x46}_q\)-rational points of \(G\). Let \(F : G \rightarrow G\) be the standard Frobenius morphism such that \(G^F\)= \(G(q)\). Let \(B\) be an \(F\)-stable Borel subgroup of \(G\); write \(U\) for the unipotent radical of \(B\) and \(\char{eufm10}{0x75}\) …
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Torsion Subgroups Of Rational Elliptic Curves Over Odd Degree Galois Fields
… and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of …
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Torsion Subgroups of Rational Elliptic Curves Over Odd Degree Galois Fields
… and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of …
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Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics
… between the arithmetic of these curves and the rational orbits of these representations. In this thesis, we complete Thorne's picture and show that $2$-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. …
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On Some Applications of a Generalized Dwork Trace Formula to L-functions associated to Exponential Sums over Galois Rings
… main ingredients in his celebrated proof of the rationality of the zeta function of an (affine or projective) algebraic variety over a finite field. In this thesis, we will prove a generalization of Dwork’s trace formula that applies to exponential sums over Galois rings and the associated …
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