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  1. arXiv:2011.02058 (Published 2020-11-03)

    Local and Global Analysis

    Garth Warner

    The objective of this article is to give an introduction to p-adic analysis along the lines of Tate's thesis, as well as incorporating material of a more recent vintage, for example Weil groups.

  2. arXiv:1712.02897 (Published 2017-12-08)

    Bounding periods of subvarieties of (P^1)^n

    Jason P. Bell, Dragos Ghioca, Thomas J. Tucker

    Using methods of p-adic analysis, along with the powerful result of Medvedev-Scanlon (Annals of Mathematics, 2014) for the classification of periodic subvarieties of (P^1)^n, we bound the length of the orbit of a periodic subvariety Y of (P^1)^n under the action of a dominant endomorphism.

  3. arXiv:1404.4723 (Published 2014-04-18, updated 2016-03-16)

    On a conjecture of Kimoto and Wakayama

    Ling Long, Robert Osburn, Holly Swisher
    Comments: 8 pages, to appear in Proceedings of the AMS
    Categories: math.NT, math.CO
    Subjects: 33C20, 11B65, 11M41

    We prove a conjecture due to Kimoto and Wakayama from 2006 concerning Apery-like numbers associated to a special value of a spectral zeta function. Our proof uses hypergeometric series and p-adic analysis.

  4. arXiv:1310.5775 (Published 2013-10-22, updated 2013-11-11)

    Applications of p-adic analysis for bounding periods of subvarieties under etale maps

    Jason P. Bell, Dragos Ghioca, Thomas J. Tucker

    Using methods of p-adic analysis we give a different proof of Burnside's problem for automorphisms of quasiprojective varieties X defined over a field of characteristic 0. More precisely, we show that any finitely generated torsion subgroup of Aut(X) is finite. In particular this yields effective bounds for the size of torsion of any semiabelian variety over a finitely generated field of characteristic 0. More precisely, we obtain effective bounds for the length of the orbit of a preperiodic subvariety under the action of an etale map.

  5. arXiv:1305.0216 (Published 2013-05-01)

    Benedetto's trick and existence of rational preperiodic structures for quadratic polynomials

    Xander Faber

    We refine a result of R. Benedetto in p-adic analysis in order to exhibit infinitely many quadratic polynomials with rational coefficients having a specified graph of rational preperiodic points.

  6. arXiv:1302.0061 (Published 2013-02-01, updated 2014-06-15)

    Most odd degree hyperelliptic curves have only one rational point

    Bjorn Poonen, Michael Stoll
    Comments: 24 pages; to appear in Annals of Math
    Categories: math.NT
    Subjects: 11G30, 14G25, 14G40, 14K15, 14K20

    Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g tends to infinity. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using p-adic analysis and an idea of McCallum, we develop a reformulation of Chabauty's method that shows that certain computable conditions imply #C(Q)=1; on the other hand, using further p-adic analysis, the theory of arithmetic surfaces, a new result on torsion points on hyperelliptic curves, and crucially the Bhargava-Gross equidistribution theorem for nonzero 2-Selmer group elements, we prove that these conditions are often satisfied for p=2.

  7. arXiv:0905.0022 (Published 2009-04-30, updated 2009-05-04)

    Orientations and p-Adic Analysis

    Barry John Walker
    Comments: Updated references. Fixed two typos. Changed document class to a potentially more readable style
    Categories: math.AT, math.NT
    Subjects: 55P43, 55N22, 14L05

    Matthew Ando produced power operations in the Lubin-Tate cohomology theories and was able to classify which complex orientations were compatible with these operations. The methods used by Ando, Hopkins and Rezk to classify orientations of topological modular forms can be applied to complex K-Theory. Using techniques from local analytic number theory, we construct a theory of integration on formal groups of finite height. This calculational device allows us to show the equivalence of the two descriptions for complex K-Theory. As an application we show that the $p$-completion of the Todd genus is an $E_\infty$ map.