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Showing 1 to 20 of 20 for “"quadratic fields"”.
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The Fermat equation over quadratic fields
… Last Theorem over the rational numbers to quadratic fields. In particular, under certain congruence conditions it is shown that the Fermat equation of exponent p has no solution over Q(√m) when p is a m-regular prime. Completely analogous to the work of Kummer, it is shown that m-regular …
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P-Class Towers of Imaginary Quadratic Fields
We conclude with a preliminary description of some joint work with Nigel Boston on the asymptotic distributions of non-abelian G with respect to the discriminant.
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The Tarry -Escott Problem Over Quadratic Fields
… related diophantine systems over Z and over some quadratic fields. We give infinitely many solutions of the Tarry-Escott problem over Zi of degrees 2, 3, 4, 5 and 7, none of which can be directly derived from solutions over Z . We show that there are infinitely many solutions of the Tarry-Escott …
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Solutions of the Cubic Fermat Equation in Quadratic Fields
We will examine when there are nontrivial solutions to the equation $x^3 + y^3 = z^3$ in $\mathbb{Q}(\sqrt{d})$ for a squarefree integer $d$. In this variation of Fermat's Last Theorem, it is possible for nontrivial solutions to exist in $\mathbb{Q}(\sqrt{d})$ for some choices of $d$, but not for …
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Improving regulator verification and compact representations in real quadratic fields
… unconditionally verifies the regulator of a real quadratic number field and refinements to the concept of a compact representation of a quadratic integer, originally given by Buchmann, Thiel, and Williams [12]. In addition, we consider the well-known applications of this theory to principal ideal …
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The Structure of the Class Group of Imaginary Quadratic Fields
Let Q(√(-d)) be an imaginary quadratic field with discriminant Δ. We use the isomorphism between the ideal class groups of the field and the equivalence classes of binary quadratic forms to find the structure of the class group. We determine the structure by combining two of Shanks' algorithms [7, …
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An Arithmetic-geometric Reciprocity between Theta Functions Attached to Real and Imaginary Quadratic Fields
… modular forms associated to ideal classes in quadratic number fields. These modular forms are theta functions that were originally introduced by Hecke in the 1920s andhave been investigated by several authors since. Our framework allows us to prove old and new results concerning the periods of …
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On the Galois Group of the 2-Class Field Towers of Some Imaginary Quadratic Fields
… for $Gal(k^{nr,2}/k)$ for the imaginary quadratic fields $k=\mathbb{Q}(\sqrt{-2379}),\mathbb{Q}(\sqrt{-445}),Q(\sqrt{-1015})$, and $\mathbb{Q}(\sqrt{-1595})$. In the case that $k=\mathbb{Q}(\sqrt{-2379})$, we illustrate a method that reduces the number of Bush's possibilities for …
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Two-Party and Threshold ECDSA constructions with bandwidth efficient instantiations from Class Groups of Imaginary Quadratic Fields
… (CL) costruito dai gruppi di classi in campi quadratici immaginari. Tale instanziazione migliora la soluzione di Lindell per quanto riguarda la sicurezza a 192-bit o superiore in termini di efficienza per quanto riguarda la generazione delle chiavi, e l'efficienza dell'algoritmo di firma per …
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Numerical Tests of Two Conjectures in Fake Real Quadratic Orders
A fake real quadratic order is defined based on an imaginary quadratic field and a prime p but behaves similarly to real quadratic orders. Two conjectures regarding fake real quadratic orders are discussed in the thesis. The first one is the Cohen-Lenstra heuristic. Our computation showed that for …
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Antisymmetry in the theory of rigid meromorphic cocycles
… the theory of p-adic singu- lar moduli for real quadratic fields. The central object in their theory are rigid meromorphic cocycles. These are elements of the first cohomology group of Ihara’s group SL2(Z[1/p]) with values in the multiplicative group of non-zero rigid mero- morphic functions on …
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Arithmetic of Maass forms of half-integral weight
… connections to invariants of real and imaginary quadratic fields, expanding on the work of Zagier and Duke-Imamoglu-Toth. Next, we examine the deep relationship between sums of Kloosterman sums and Maass cusp forms, motivated by work of Kuznetsov and Sarnak-Tsimerman, among others. Finally, we …
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Topics in analytic number theory
… work on large gaps between primes to imaginary quadratic fields. Suppose K is an imaginary quadratic field, and let N_K denote the field norm on O_K. For x₀ in O_K and r > 0, let (x₀, r) = { x in O_K : |N_K(x − x₀)| < r }. Define G_K(X) = max { r > 0 : there exists x₀ in O_K such that |N_K(x₀)| …
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Irreducible elements in algebraic number fields
… irreducible elements in algebraic number fields. The first question is: Given an algebraic integer β in a field with class number greater than two, how many different lengths of factorizations into irreducibles exist? The distribution into ideal classes of the prime ideals whose product is …
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Distribution of sequences related to L-functions
… primes of all rings of integers of all imaginary quadratic fields. One would want to know if it is possible to walk to infinity stepping only on points in $P$ and such that the sequence of lengths of steps used in the process is bounded. However, the problem is surprisingly connected to some …
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On Selmer groups and factoring p-adic L-functions
… p-adic L-function associated to an imaginary quadratic field (constructed by Katz) into a product of two Kubota-Leopoldt p-adic L-functions. In 1982, Greenberg proved the corresponding result on the algebraic side involving classical Iwasawa modules, as predicted by the main conjectures for …
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Obstructions to rational and integral points
… is necessary and sufficient, over imaginary quadratic fields and totally real fields unconditionally, and over all number fields conditionally on the section conjecture. This is part of a joint project with Tomer Schlank.
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Bijective proofs of partition identities and covering systems
… 6, we first define covering systems in number fields, and extend those results to arbitrary number fields. In Chapter 7, we give an explicit version of their first theorem to provide a specific number for the least modulus of a covering system, where the reciprocal sum is strictly bigger than …