User:JosephSilverman

This user has been on Wikipedia for 19 years, 5 months and 14 days.

PhDThis user has a Doctor of Philosophy degree in Mathematics.

To Do

Articles that need writing

• Divisibility sequence (now exists, could be expanded)

• Tate pairing (now exists, but only one sentence)

• Division polynomial (referenced in Point counting of elliptic curves, and should add a link from the article on elliptic curves) (now exists, I did some editing Jan 2012)

• p-adic dynamics (add a redirect from nonarchmedean dynamics)

• Regular affine automorphism (with link from Henon map)

Articles and stubs that need major edit/rewrite

• S_unit add a section on unit equations with more then 2 variables. Maybe also quantitative upper bounds for number of solutions.

• Arithmetic dynamics add a section on dynamical Andre-Oort conjecture

• Vojta's conjecture added material March 2014, but could still use work

• Faltings height current article has no details

• Weil pairing could use some rewriting

• Lehmer's conjecture not well written, should also mention elliptic analogue (did a rewrite Jan 2012, could still use some cleanup work)

• Fibonacci sequence (very poorly written, includes several proofs, not in Wiki style)

• 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

• canonical height Add result on nef canonical heights for abelian varieties. In dynamics section, add fact that may want the line bundle in $NS(X)\otimes \mathbb {R}$.

• 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)

• Elliptic divisibility sequence - add section on Generalizations

• 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)

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Major articles contributed to Wikipedia

• Elliptic divisibility sequence - January 28, 2009
• Arithmetic dynamics - February 20, 2009
• Neron-Tate height - June 22, 2009 (expanded to include non-symmetric line bundles 2014)

(wikipedia formating examples)

Arithmetic dynamics

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

AAA	BBB
aaa	bbb
ccc	ddd

Definitions

Let S be a set

$F^{(n)}=F\circ F\circ \cdots \circ F.$

S → S

S ∈ S

A … Z

External links

• The Arithmetic of Dynamical Systems home page

External links

• JosephSilverman at the Mathematics Genealogy Project
• Jill Pipher's home page