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Rational solutions of certain Diophantine equations involving norms

Maciej Ulas (2014)

Acta Arithmetica

We present some results concerning the unirationality of the algebraic variety f given by the equation N K / k ( X ₁ + α X ₂ + α ² X ₃ ) = f ( t ) , where k is a number field, K=k(α), α is a root of an irreducible polynomial h(x) = x³ + ax + b ∈ k[x] and f ∈ k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a = 0 and b ∈ k∖k³. We prove that if deg f = 4 and f contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then f is k-unirational. A similar result is proved for a broad family of quintic polynomials f satisfying...

Rational torsion points on Jacobians of modular curves

Hwajong Yoo (2016)

Acta Arithmetica

Let p be a prime greater than 3. Consider the modular curve X₀(3p) over ℚ and its Jacobian variety J₀(3p) over ℚ. Let (3p) and (3p) be the group of rational torsion points on J₀(3p) and the cuspidal group of J₀(3p), respectively. We prove that the 3-primary subgroups of (3p) and (3p) coincide unless p ≡ 1 (mod 9) and 3 ( p - 1 ) / 3 ≡ 1 ( m o d p ) .

Rational values of the arccosine function

Juan Varona (2006)

Open Mathematics

We give a short proof to characterize the cases when arccos(√r), the arccosine of the squareroot of a rational number r ∈ [0, 1], is a rational multiple of π: This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind.

Reading along arithmetic progressions

T. Downarowicz (1999)

Colloquium Mathematicae

Given a 0-1 sequence x in which both letters occur with density 1/2, do there exist arbitrarily long arithmetic progressions along which x reads 010101...? We answer the above negatively by showing that a certain regular triadic Toeplitz sequence does not have this property. On the other hand, we prove that if x is a generalized binary Morse sequence then each block can be read in x along some arithmetic progression.

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