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Creators/Authors contains: "Morrow, Jackson S"

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  1. ; ; (, Mathematische Zeitschrift)
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  2. ; (, International Mathematics Research Notices)
    Abstract We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $$F$$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $$\varphi \colon Y \to X$$ of degree at least $$2\dim (X)^{2}$$, there exists an ample line bundle $$\mathscr{L}$$ on $$Y$$ such that for a general member $$D$$ of the complete linear system $$|\mathscr{L}|$$, $$D$$ is geometrically irreducible and any set of $$\varphi (D)$$-integral points on $$X$$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable. 
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  3. ; ; ; ; (, Algebra and number theory)
    Let K be a number field, and let E/K be an elliptic curve over K. The Mordell–Weil theorem asserts that the K-rational points E(K) of E form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of E ( K ) for K a cubic number field. To do so, we determine the cubic points on the modular curves X1(N) for N = 21,22,24,25,26,28,30,32,33,35,36,39,45,65,121. As part of our analysis, we determine the complete lists of N for which J0(N), J1(N), and J1(2,2N) have rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on J1(N)(Q) is generated by Galois-orbits of cusps of X1(N) for N ≤55, N ̸=54. 
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