William G Mccallum

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Journals/Publications

• Daro, P., McCallum, W., & Zimba, J. (2010). Common U.S. math standards. Science, 328(5976), 285-.
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  PMID: 20395478;
• Maurer, S. B., & McCallum, W. (2006). Advising a precollege curriculum project. Notices of the American Mathematical Society, 53(9), 1018-1020.
• McCallum, W. G., & Sharifi, R. T. (2003). A cup product in the Galois cohomology of number fields. Duke Mathematical Journal, 120(2), 269-310.
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  Abstract: Let K be a number field containing the group μn of nth roots of unity, and let S be a set of primes of K including all those dividing n and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal S-ramified extension of K with coefficients in μn which yields a pairing on a subgroup of K× containing the S-units. In this general situation, we determine a formula for the cup product of two elements that pair trivially at all local places. Our primary focus is the case in which K = ℚ(μp) for n = p, an odd prime, and S consists of the unique prime above p in K. We describe a formula for this cup product in the case that one element is a pth root of unity. We explain a conjectural calculation of the restriction of the cup product to p-units for all p ≤ 10,000 and conjecture its surjectivity for all p satisfying Vandiver's conjecture. We prove this for the smallest irregular prime p = 37 via a computation related to the Galois module structure of p-units in the unramified extension of K of degree p. We describe a number of applications: to a product map in K-theory, to the structure of S-class groups in Kummer extensions of K, to relations in the Galois group of the maximal pro-p extension of ℚ(μp) unramified outside p, to relations in the graded ℤp-Lie algebra associated to the representation of the absolute Galois group of ℚ in the outer automorphism group of the pro-p fundamental group of P1 (ℚ̄) - {0,1, ∞}, and to Greenberg's pseudonullity conjecture.
• An, S. Y., Kim, S. Y., Marshall, D. C., Marshall, S. H., McCallum, W. G., & Perlis, A. R. (2001). Jacobians of genus one curves. Journal of Number Theory, 90(2), 304-315.
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  Abstract: Consider a curve of genus one over a field K in one of three explicit forms: a double cover of P1, a plane cubic, or a space quartic. For each form, a certain syzygy from classical invariant theory gives the curve's jacobian in Weierstrass form and the covering map to its jacobian induced by the K-rational divisor at infinity. We give a unified account of all three cases. © 2001 Academic Press.
• McCallum, W. G. (2001). Brauer points on fermat curves. Bulletin of the Australian Mathematical Society, 63(3), 393-406.
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  Abstract: If X is a variety over a number field K, the set of K-rational points on X is contained in the subset of the adelic points cut out by the Brauer group; we call this set the set of Brauer points on the variety. If S is a set of valuations of K, we also define S-Brauer points in a natural way. It is natural to ask how good a bound on the rational points is provided by the Brauer (or S-Brauer) points. Let p > 3 be a prime number, and let X be the Fermat curve of degree p, xp + yp = 1. Let K be the field of p-th roots of unity, and let r be the p-rank of the class group of K. In this paper we show that if r < (p + 3)/8, then the set of p-Brauer points on X has cardinality at most p. We construct elements of the Brauer group of X by relating it to the Weil-Chatelet group of the jacobian of X, then use the method of Coleman and Chabauty to bound the points cut out by these elements.
• McCallum, W. G. (2001). Greenberg's conjecture and units in multiple ℤ_p-extensions. American Journal of Mathematics, 123(5), 909-930.
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  Abstract: Let A be the inverse limit of the p-part of the ideal class groups in a ℤrp-extension K∞/K. Greenberg conjectures that if r is maximal, then A is pseudo-null as a module over the Iwasawa algebra A (that is, has codimension at least 2). We prove this conjecture in the case that K is the field of pth roots of unity, p has index of irregularity 1, satisfies Vandiver's conjecture, and satisfies a mild additional hypothesis on units. We also show that if K is the field of pth roots of unity and r is maximal, Greenberg's conjecture for K implies that the maximal p-ramified pro-p-extension of K cannot have a free pro-p quotient of rank r unless p is regular. Finally, we prove a generalization of a theorem of Iwasawa in the case r = 1 concerning the Kummer extension of K∞ generated by p-power roots of p-units. We show that the Galois group of this extension is torsion-free as a Λ-module if there is only one prime of K above p and K∞ contains all the p-power roots of unity.
• McCallum, W. G. (1994). On the method of Coleman and Chabauty. Mathematische Annalen, 299(1), 565-596.
• McCallum, W. G. (1992). The arithmetic of Fermat curves. Mathematische Annalen, 294(1), 503-511.
• McCallum, W. G. (1990). Tate duality and wild ramification. Mathematische Annalen, 288(1), 553-558.
• McCallum, W. G. (1988). On the Shafarevich-Tate group of the jacobian of a quotient of the Fermat curve. Inventiones Mathematicae, 93(3), 637-666.