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Showing 1 to 18 of 18 for “"function fields"”.
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Constructing and Tabulating Dihedral Function Fields
… prime degree l dihedral extensions of a rational function field k_0(x), where k_0 is a perfect field with characteristic not dividing 2l. We begin with a class field theoretic construction algorithm when k0 is a finite field. We also describe modifications to this algorithm to improve the run time …
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Higher Siegel-Weil formulae over function fields
In their seminal work, Feng-Yun-Zhang introduced function field analogues of Kudla-Rapoport cycles for moduli spaces of unitary shtukas, and initiated the study of their intersection theory. They proved a higher Siegel-Weil formula in the case of non-degenerate Fourier coefficients, relating the …
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Finite Groups of Automorphisms of Algebraic Function Fields
Made available in DSpace on 2014-12-05T21:50:11Z (GMT). No. of bitstreams: 1 5805487.pdf: 1929314 bytes, checksum: 9227ba3bbd12be598885441a06850952 (MD5) Previous issue date: 1958
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On congruence function fields with many rational places
In this thesis, we study congruence function fields, in particular those with many rational places. This thesis consists of three parts, the first two parts present our results in two different aspects of function fields with many rational places, namely maximal function fields and asymptotically …
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L-functions of twisted elliptic curves over function fields
… and computationally, elliptic curves and their L-functions over number fields, in particular over the rational numbers. Much less work has been done over function fields, especially computationally, where the underlying geometry of the function field plays an intimate role in the arithmetic of …
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Primes of the form X² + nY² in function fields
Let n be a square-free polynomial over F_q, where q is an odd prime power. In this work, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the …
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Algorithmic Aspects of Biquadratic, Cubic and Radical Function Fields
… and an integral basis of a cyclic biquadratic function field are simple, explicit and efficient. The computation of the unit group on a global bicyclic biquadratic function field generalizes and simplifies Kubota's work [Kub56) to function fields. It thus settles this question for all global …
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Solving norm equations over global function fields using compact representations
… efficiency of solving norm equations over global function fields F and providing theoretical and empirical evidence of the improvement. We present two new algorithms for solving norm equations over F; one is an exhaustive search algorithm inspired by the sole existing method in [21], and the other …
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Toward a deformation theory for Galois representations of function fields
… a finite field of the absolute Galois group of a function field K. In this case, the characterization of abelian p-power extensions of fields of characteristic p can be extended and refined to allow only restricted ramification at the places of K, and can be a tool for analyzing one-dimensional …
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Computability of rational points on curves over function fields in characteristic p
The motivating problem of this 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 …
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Comparison of and Improvements to Degree Zero Divisor Class Group Arithmetic in Algebraic Function Fields
… algorithms for Jacobian arithmetic in global function fields and compares the improvements to previous works. We present two independent improvements to Jacobian arithmetic based on the unique representation described in [13]. The first improvement requires the function field to have a degree …
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Infrastructure, Arithmetic, and Class Number Computations in Purely Cubic Function Fields of Characteristic at Least 5
… upper and lower bounds on h, for a given cubic function field, than those given by the Hasse-Weil Theorem. We then employ Shanks' Baby Step-Giant Step algorithm [Sha71] and Pollard's Kangaroo method [Pol78], to search this interval and compute the desired invariants for purely cubic function …
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The Arithmetic of Elliptic and Hyperelliptic Curves with Applications to Pairing-Based Cryptography
… and their connection to the theory of algebraic function fields of one variable, the thesis concentrates on pairings (such as the Tate pairing) defined on groups related to a given absolutely irreducible non-singular curve over a finite field. Those pairings are being studied in terms of their …
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MUTATION INVARIANT FUNCTIONS ON CLUSTER ENSEMBLES ASSOCIATED WITH SURFACES
We define the notion of an invariant function on a cluster ensemble with respect to a group action of the cluster modular group on its associated function fields. We realize many examples of previously studied functions as elements of this type of invariant ring and give many new examples. We …
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Observational study of the monsoonal dynamics and eddy shedding phenomena
… the potential vorticity and Montgomery stream-function fields to document the characteristics of eddy shedding events. In most cases, the daughter cell propagates westward. In the vertical, eddy shedding is confined to the upper troposphere, between 300 and 100 mb. The relationship between …
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Invariant embeddings of the Deligne-Lusztig curves with applications
… of certain perfect hash families from towers of function fields. The fourth part is the study of the performance of binary concatenated algebraic geometry codes as polar codes. Let $C$ be a Deligne-Lusztig curve associated to a simple group $G$, i.e., $C$ is either the Hermitian, Suzuki, or Ree …
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Improved Bounds for Codes and Secret Sharing Schemes from Algebraic Curves
… V. Goppa established a remarkable connection: function fields of algebraic curves can be used to construct a large class of error-correcting codes. Such codes are called algebraic geometric (AG) codes. AG codes from divisors supported in only one point on the Hermitian curve produce long codes …
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Cohomological invariants for infinite groups
… an important result for arithmetic groups over function fields, due to Bux and Wortman. The first Grigorchuk group G was introduced in 1980 and has been extensively studied since due to its extraordinary properties. The class HF of hierarchically decomposable groups was introduced by Kropholler …