Counting number fields in fibers (with an Appendix by Jean Gillibert)
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DOI: 10.1007/s00209-017-1900-5
Abstract: Let X be a projective curve over $${\mathbb Q}$$Q and $${t\in {\mathbb Q}(X)}$$t∈Q(X) a non-constant rational function of degree $${n\ge 2}$$n≥2. For every $${\tau \in {\mathbb Z}}$$τ∈Z pick $${P_\tau \in X(\bar{\mathbb Q})}$$Pτ∈X(Q¯) such that $${t(P_\tau… read more here.
Keywords: fields fibers; tau; number fields; counting number ... See more keywords
Metric Mahler measures over number fields
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DOI: 10.1007/s10474-017-0770-y
Abstract: For an algebraic number α, the metric Mahler measure $${m_1(\alpha)}$$m1(α) was first studied by Dubickas and Smyth [4] and was later generalized to the t-metric Mahler measure $${m_t(\alpha)}$$mt(α) by the author [16]. The definition of… read more here.
Keywords: metric mahler; number fields; mahler measures; number ... See more keywords
Elements with prime and small indices in bicyclic biquadratic number fields
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DOI: 10.1007/s10998-017-0233-9
Abstract: We give necessary and sufficient conditions for the existence of primitive algebraic integers with index A in totally complex bicyclic biquadratic number fields where A is an odd prime or a positive rational integer at… read more here.
Keywords: biquadratic number; number fields; bicyclic biquadratic; prime small ... See more keywords
On the Waldspurger formula and the metaplectic Ramanujan conjecture over number fields
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DOI: 10.1016/j.jfa.2019.05.013
Abstract: In this paper, by inputting the Bessel identities over the complex field in previous work of the authors, the Waldspurger formula of Baruch and Mao is extended from totally real fields to arbitrary number fields.… read more here.
Keywords: number; ramanujan conjecture; number fields; formula metaplectic ... See more keywords
Corrigendum to “Some real quadratic number fields with their Hilbert 2-class field having cyclic 2-class group” [J. Number Theory 173 (2017) 529–546]
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DOI: 10.1016/j.jnt.2017.03.023
Abstract: Abstract This corrigendum describes a number of calculation and typographical errors, primarily in the examples, in the 2017 JNT paper Some Real Quadratic Number Fields with their Hilbert 2-Class Field Having Cyclic 2-Class Group, by… read more here.
Keywords: fields hilbert; class; number fields; number ... See more keywords
A Voronoi–Oppenheim summation formula for totally real number fields
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DOI: 10.1016/j.jnt.2018.04.008
Abstract: Abstract We obtain a Voronoi–Oppenheim summation formula for divisor functions of totally real number fields. This generalizes a formula proved by Oppenheim in 1927. We use a similar method to the one developed by Beineke… read more here.
Keywords: number; oppenheim summation; voronoi oppenheim; number fields ... See more keywords
On pro-p-extensions of number fields with restricted ramification over intermediate Zp-extensions
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DOI: 10.1016/j.jnt.2020.12.002
Abstract: Abstract We consider pro-p-extensions of a number field in which the ramification and decomposition are restricted over an intermediate Z p -extension. For such a maximal pro-p-extension under a certain restriction condition, we obtain a… read more here.
Keywords: number; ramification; fields restricted; pro extensions ... See more keywords
NONMONOGENITY OF NUMBER FIELDS DEFINED BY TRUNCATED EXPONENTIAL POLYNOMIALS
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DOI: 10.1017/s0004972724000819
Abstract: Abstract Let p be a prime number. Let $n\geq 2$ be an integer given by $n = p^{m_1} + p^{m_2} + \cdots + p^{m_r}$ , where $0\leq m_1 < m_2 < \cdots < m_r$ are… read more here.
Keywords: number; truncated exponential; nonmonogenity number; number fields ... See more keywords
On the distribution of norm groups in the intervals corresponding to odd degree extensions of algebraic number fields
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DOI: 10.1080/00927872.2018.1492586
Abstract: Abstract Let X be a subgroup of a group Y. The interval (X, Y) is the set of subgroups of Y that contain X including X and Y. Let K/k be a finite extension of… read more here.
Keywords: algebraic number; number fields; extensions algebraic; number ... See more keywords
Integral Bases and Monogenity of Composite Fields
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DOI: 10.1080/10586458.2017.1382404
Abstract: ABSTRACT We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly… read more here.
Keywords: monogenity composite; number fields; index form; monogenity ... See more keywords
Abel’s Problem, Gauss and Cartier Congruences Over Number Fields
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DOI: 10.1093/imrn/rnaf295
Abstract: Abel’s problem consists in identifying the conditions under which the differential equation $y^{\prime}=\eta y$, with $\eta $ an algebraic function in $\mathbb{C}(x)$, possesses a non-zero algebraic solution $y$. This problem has been algorithmically solved by… read more here.
Keywords: problem; eta; cartier congruences; abel problem ... See more keywords