Wikipedia

JosephSilverman

To Do

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Articles that need writing

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Articles and stubs that need major edit/rewrite

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  • S_unit add a section on unit equations with more then 2 variables. Maybe also quantitative upper bounds for number of solutions.
  • Lehmer's conjecture not well written, should also mention elliptic analogue (did a rewrite Jan 2012, could still use some cleanup work)
  • Elliptic curve (at least, the number theory section should be expanded. For example, it should include the statement of Mazur's theorem (with mention of Merel's generalization). After doing this, delete the very short page entitled Mazur's torsion theorem. The French language article on elliptic curves is actually much better than the English one, so it could be used as a guide.
  • Weil-Châtelet_group It's actually an article that discusses both WC and SHA, which is fine. But it should include a section describing the Tate local pairing and the Cassels-Tate global pairing
  • Rosati_involution Added two paragraphs defining the Rosati involution and how its used to characterize NS(A). Should add a link from the abelian variety page?
  • Zsigmondy theorem Added generalizations (Lucas, Lehmer seqs, EDS), could use more work, e.g. add dynamics results, including use of abc by Gratton, Nguyen, Tucker)

Articles that need minor edit

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  • canonical height Add result on nef canonical heights for abelian varieties. In dynamics section, add fact that may want the line bundle in  .
  • Henon map - describe generalization to regular affine automorphisms, describe some arithmetic properties, i.e., finitely many rational periodic points
  • Sato-Tate_conjecture - needs reformatting, the Lang-Trotter part should be made more precise. Also, Koblitz value for constant is not correct(?), see paper(s) by Zywina
  • Mahler measure - add references for (1) Mahler measure equals value of L-function; (2) Fact that higher dim'l Mahler measure converges (Lawton paper)
  • Faltings'_theorem - needs a generalization section describing rational points on subvarieties of abelian varieties and integral points on affine subvarieties of abelian vareieties. will need to add references.
  • Smooth number - needs a section explaining (or at least stating) that smooth numbers and their distribution underlie all modern factorization algorithms, including the quadratic and number field sieves and Lenstra's elliptic curve factorization algorithm.
  • Newton-Raphson - in the generalizations section, add a brief discussion of its use in the p-adic setting.
  • Moufang_loop - add a reference that the points on a cubic surface form a Moufang loop (more or less)

(work in progress)

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Major articles contributed to Wikipedia

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(wikipedia formating examples)

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Arithmetic dynamics[1]


The following table describes a rough correspondence between Diophantine equations, especially abelian varieties, and dynamical systems:

AAA BBB
aaa bbb
ccc ddd

Definitions

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Let S be a set

 

S → S

S ∈ S

A … Z

References

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  1. ^ J.H. Silverman (2007). The Arithmetic of Dynamical Systems. Springer. ISBN 978-0-387-69903-5. {{cite book}}: External link in |title= (help)
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