Hyperelliptic Curves, Continued Fractions and Somos Sequences

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The surprising integral Z X 6t dt p t4 + 4t3 − 6t2 + 4t + 1 “ = log X 6 + 12X 5 + 45X 4 + 44X 3 − 33X 2 + 43 + (X 4 + 10X 3 + 30X 2 p + 22X − 11) X 4 + 4X 3 − 6X 2 ” + 4X + 1 is a nice example of a class of pseudo-elliptic integrals Z X f (t)dt p = log D(t) ` q a(X) + b(X) D(X) ´ . (∗) Here we take D to be a monic polynomial defined over Q, of even degree 2g + 2, and not the square of a polynomial; f , a, and b denote appropriate polynomials. We suppose a to be nonzero, say of degree m at least g + 1. We will see that necessarily deg f = g , that deg b = m − g − 1, and that f has leading coefficient m. In the example, m = 6 and g = 1. 5
The surprising integral Z X 6t dt p t4 + 4t3 − 6t2 + 4t + 1 “ = log X 6 + 12X 5 + 45X 4 + 44X 3 − 33X 2 + 43 + (X 4 + 10X 3 + 30X 2 p + 22X − 11) X 4 + 4X 3 − 6X 2 ” + 4X + 1 is a nice example of a class of pseudo-elliptic integrals Z X f (t)dt p = log D(t) ` q a(X) + b(X) D(X) ´ . (∗) Here we take D to be a monic polynomial defined over Q, of even degree 2g + 2, and not the square of a polynomial; f , a, and b denote appropriate polynomials. We suppose a to be nonzero, say of degree m at least g + 1. We will see that necessarily deg f = g , that deg b = m − g − 1, and that f has leading coefficient m. In the example, m = 6 and g = 1. 5
Page 1 and 2: Hyperelliptic Curves, Continued Fra
Page 3 and 4: A pseudo-elliptic integral Z x 6t p
Page 9 and 10: Michael Somos’s Sequences The sto
Page 17 and 18: Concerning (Bh) — thus 5-Somos
Page 25: The surprising integral Z X 6t dt p
Page 29 and 30: The surprising integral Z X 6t dt p
Page 31 and 32: Plainly, the equations detailing ps
Page 43 and 44: Moreover, u ′ = a ′ + b ′√
Page 53 and 54: Units and Torsion The notion unit e
Page 61 and 62: Units in Quadratic Fields and Conti
Page 71 and 72: Continued Fraction of the Square Ro
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Continued Fraction of the Square Ro
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To detail the continued fraction ex
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There is a minor miracle. Because t
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I should point out that any actual
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Second, if we replace D by D + 1 th
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Second, if we replace D by D + 1 th
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In the course of studying continued
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In the course of studying continued
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What the Continued Fraction Does No
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The elliptic case When g = 1 we hav
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The −dh are in fact the abscissæ
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As it happens, a combinatorial/alge
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Elliptic Divisibility Sequences Now
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Remarkably, Ward introduces his seq
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Elliptic Division Polynomials I ins
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4-Somos Suppose (Ch) = (. . . , 2,
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Higher Genus There surely are analo
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Others can do worse, and better. If
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Others can do worse, and better. If
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A cute example à la Somos Whatever
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A cute example à la Somos Whatever
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References Rather than attempt a li
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Hyperelliptic Curves, Continued Fractions and Somos Sequences

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