The S-unit equation over real quadratic fields
On Fermat’s equation over some quadratic imaginary number fields
... On the other hand, modularity of elliptic curves over number fields with complex embeddings is highly conjectural. For general number fields, ¸Sengün and Siksek [27] proved an asymptotic version of Fermat’s Last ...
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Torsion of elliptic curves over real quadratic fields of smallest discriminant
... of all K-rational points on E, together with the base point O, is an abelian group. He also conjectured that this group is finitely generated when K is the field Q of rational numbers. In 1922, Mordell proved this ...
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Representation of Algebraic Integers as Sum of Units over the Real Quadratic Fields
... the real quadratic fields 𝑄√𝑑 into the sets 𝑊 1 , 𝑊 2 , 𝑊 3 … Jacobsons results shows that 𝑄√2, 𝑄√5 ∈ 𝑊 1 and Sliwa confirm that 𝑄√2 and 𝑄√5 are the only real quadratic fields in ...
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The generalized Fermat equation over totally real number fields
... Chapter 1 Introduction In 1637 according to the legend, Pierre de Fermat wrote in a copy of Arith- metica that there are no integers a, b, c such that a n + b n = c n if n is greater or equal to 3. Fermat wrote that had ...
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On approximations over ideal bases in quadratic number fields
... on fields, valuations, ideals, and quadratic forms, that will be subsequently ...The fields to be considered are quadratic number fields, ...2 over Q* distinction is to be made ...
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Modularity of abelian surfaces over imaginary quadratic fields
... varieties over different fields and arbitrary ...curves over totally real fields [FLHS15]; GL 2 -type varieties over Q [ KW09 ]; Q-curves [ ES01] Calabi-Yau threefolds [GY11] and ...
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Deterministic equation solving over finite fields
... We now broaden the perspective and describe the consequences of this result. We begin by considering the performance of our methods when compared to probabilistic methods. After that, we list several applications to ...
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Explicit Form of Fundamental Units of Certain Real Quadratic Fields
... all real quadratic fields with the period k d of the continued fraction expansion of the quadratic irrational number ω d is higher than ...Fundamental Unit and the Class Numbers of ...
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On the real quadratic fields with certain continued fraction expansions and fundamental units
... the real quadratic number fields Q( √ d) which contain the specific form of the continued fractions expansions of integral basis element where d ≡ 2, 3(mod4) is a square free positive ...
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Affine Nash groups over real closed fields
... work over an arbitrary real closed field R, a special case being when R = ...polynomial equation P (¯ x, f (¯ x)) = 0 on ...manifold over R, of Nash maps between such Nash manifolds, and hence ...
FACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set
... Proof. Suppose a|b, say b = ac. Ideals are O K -modules, so ac ⊂ a. Let’s summarize the situation right now. We have replaced multiplication and divisibility among elements of O K with multiplication and divisibility ...
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Irreducible binary cubics and the generalised superelliptic equation over number fields
... Remark. The assumption that K is Galois is needed in order to prove that, for large l, a certain mod l Galois representation is irreducible. If the number field is totally real but not Galois, it will become clear ...
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Quadratic Functions Unit
... the quadratic function has undergone a reflection in the ...with real life applications that can be modeled using quadratic functions, it is sometimes necessary to find that maximum or minimum point ...
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Solving Systems of Multivariate Quadratic Equations over Finite Fields or: From Relinearization to MutantXL
... If some equations are sparse you might introduce more new monomials than equations by multiplying them by all monomials of a special degree. So in some case it might be better to multiply some equations only by some ...
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An Algorithm for Explicit Form of Fundamental Units of Certain Real Quadratic Fields and Period Eight
... certain real quadratic number ...such quadratic fields is first in the literature and it gives us a more practical way to calculate the fundamental ...the quadratic irrational number of ...
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Determination of Fundamental Units of Real Quadratic Number Fields Related with Specific Continued Fraction
... damental unit d of such real quadratic number fields Q( √ d) along with parametrized forms of d which are still missing in the open ...fundamental unit of such real ...
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Explicit quadratic Chabauty over number fields Balakrishnan, Jennifer S.; Besser, Amnon; Bianchi, Francesca; Müller, J. Steffen
... certain quadratic Chabauty techniques do not find all rational points but only those that avoid certain residue ...p-adic equation and this equation does not depend on the chosen branch of the ...
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On elliptic curves of prime power conductor over imaginary quadratic fields with class number one
... number fields of class number one, as the number of units modulo squares is always ...of real quadratic fields is of particular interest, requiring almost no modification except to allow for O ...
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Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$
... is irreducible and modular, and it follows from Lemma 8.3 that ρ ¯ E ,` ∼ ¯ ρ f ,λ for some Hilbert eigenform over K of parallel weight 2 that is new at level N ` , where N ` = 2 O K or 2p. Now a standard argument ...
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On congruences of modular forms over imaginary quadratic fields
... associated to some automorphic representation Π 0 of GSp(4, A Q ), such that R 0 | I p ∼ = τ p , for all p ∈ S. This is yet unproved, but as we discussed in Subsection 6.2.1, one hopes that Theorem 6.2.5 will be ...
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