Pete L. Clark's papers

Pete L. Clark

Return to Pete's home page

Research papers:

Work in Progress:

Galois theory of arbitrary field extensions.
Draft, 8/11 (pdf)
Period-index problems in WC-groups III: biconic curves.
Draft, 11/08 (pdf)
Ramanujan graphs and Shimura curves.
Draft, Summer 2006 (pdf)

Preprints/Submitted for Publication:

[67] On the stratification conjecture for CM torsion subgroups (pdf)
[66] Functional degrees and arithmetic applications II: the group-theoretic prime Ax-Katz therem (with U. Schauz). (pdf)
[65] CM elliptic curves: volcanoes, reality and applications, Part II (with F. Saia) (pdf)
[64] CM elliptic curves: volcanoes, reality and applications, Part I (pdf)
[63] A Mean Value Inequality for all functions (pdf)
[62] Restricted Variable Chevalley-Warning Theorems (with A. Bishnoi) (pdf)
[61] On base size sets (pdf)
[60] Solution to the inverse Mordell-Weil problem for elliptic curves (pdf)
[59] Abstract geometry of numbers: linear forms. (pdf)
[58] Curves over global fields violating the Hasse Principle. (pdf).
[57] Period-index problems in WC-groups II: abelian varieties. (pdf)

Accepted and/or Published:

[56] Ax's lemma in the Aichinger-Moosbauer calculus (with N. Triantafillou). J. Algebra 659 (2024), 542-580. (pdf)
[55] Densities of integer sets represented by quadratic forms (with P. Pollack, J. Rouse and K. Thompson). J. Number Theory 256 (2024), 290-328. (pdf)
[54] Functional degrees and arithmetic applications I: the set of functional degrees. (with U. Schauz) J. Algebra 608 (2022), 691–718. (pdf)
[53] The least degree of a CM point on a modular curve (with T. Genao, P. Pollack and F. Saia) J. Lond. Math. Soc. (2) 105 (2022), no. 2, 825–883. (pdf)
[52] Chevalley-Warning at the boundary (with T. Genao and F. Saia) Expo. Math. 39 (2021), no. 4, 604–623. (pdf)
[51] Acyclotomy of torsion in the CM case (with M. Chou and M. Milosevic) Ramanujan J. 55 (2021), no. 3, 1015–1037. (pdf)
[50] Torsion points and isogenies on CM elliptic curves (with A. Bourdon) J. Lond. Math. Soc. (2) 102 (2020), 580–622. (pdf)
[49] Torsion points and Galois representations on CM elliptic curves (with A. Bourdon) Pacific J. Math. 305 (2020), 43–88. (pdf)
[48] Reciprocity by resultant in k[t] (with P. Pollack) Enseign. Math. 65 (2019), 101–116. (pdf)
[47] The Instructor's Guide to Real Induction Math. Mag. 92 (2019), 136-150. (pdf)
[46] Rabinowitsch times six Rocky Mountain J. Math. 49 (2019), 433–485. (pdf)
[45] Warning's Second Theorem with relaxed outputs (Previous title: "Fattening up Warning's Second Theorem.") J. Algebraic Combin. 48 (2018), 325–349. (pdf)
[44] ABC and the Hasse principle for quadratic twists of hyperelliptic curves (with L.D. Watson) C. R. Math. Acad. Sci. Paris 356 (2018), 911–915. (pdf)
[43] Typically bounding torsion (with M. Milosevic and P. Pollack) (pdf) J. Number Theory 192 (2018), 150–167.
[42] Varga's theorem in number fields (with L.D. Watson) INTEGERS 18 (2018), A74, 11 pp. (pdf)
[41] A generalization of the theorems of Chevalley-Warning and Ax-Katz via polynomial substitutions. (with I.N. Baoulina and A. Bishnoi) Proc. Amer. Math. Soc. 147 (2019), 4107–4122. (arxiv preprint)
[40] A Note on Golomb topologies (with N. Lebowitz-Lockard and P. Pollack) Quaest. Math. 42 (2019), 73–86. (pdf)
[39] There are genus one curves of every index over every infinite, finitely generated field (with A. Lacy). J. Reine Angew. Math. 749 (2019), 65–86. (pdf)
[38] Hasse Principle Violations for Atkin-Lehner Twists of Shimura Curves (with J. Stankewicz) Proc. Amer. Math. Soc. 146 (2018), 2839–2851. (pdf)
[37] Pursuing polynomial bounds on torsion (with P. Pollack) Israel J. Math. 227 (2018), 889–909. (pdf)
[36] A note on rings of finite rank Comm. Algebra 46 (2018), 4223–4232. (pdf)
[35] The truth about torsion in the CM case, II (with P. Pollack) Q. J. Math. 68 (2017), 1313–1333. (pdf)
[34] The number of atoms in a primefree atomic domain (with S. Gosavi and P. Pollack). Comm. Algebra 45 (2017), 5431–5442. (pdf)
[33] On zeros of a polynomial in a finite grid (with A. Bishnoi, A. Potukuchi and J.R. Schmitt). Combin. Probab. Comput. 27 (2018), no. 3, 310–333. (pdf).
[32] Algebraic curves uniformized by congruence subgroups of triangle groups (with J. Voight). Trans. Amer. Math. Soc. 371 (2019), 33–-82. (pdf)
[31] The cardinal Krull dimension of a ring of holomorphic functions. Expo. Math. 35 (2017), 350–356. (pdf)
[30] Anatomy of torsion in the CM case (with A. Bourdon and P. Pollack) Math. Z. 285 (2017), no. 795–820. (pdf)
[29] The Euclidean criterion for irreducibles. Amer. Math. Monthly 124 (2017), 198–216. (pdf)
[28] Torsion points on CM elliptic curves over real number fields (with A. Bourdon and J. Stankewicz). Trans. Amer. Math. Soc. 369 (2017), 8457–8496. (pdf)
[27] The truth about torsion in the CM case (with P. Pollack). (pdf) C. R. Math. Acad. Sci. Paris 353 (2015), 683–688.
[26] Warning's second theorem with restricted variables (with A. Forrow and J. Schmitt). Combinatorica 37 (2017), 397–417. (pdf)
[25] A Note on Euclidean order types. Order 32 (2015), 157–178. (pdf)
[24] Extending the Zolotarev-Frobenius approach to quadratic reciprocity (with A. Brunyate). Ramanujan J. 37 (2015), 25–50. (pdf)
[23] The Combinatorial Nullstellensätze revisited. Electronic Journal of Combinatorics. Volume 21, Issue 4 (2014). Paper #P4.15 (pdf)
[22] Computation on elliptic curves with complex multiplication (with P. Corn, A. Rice and J. Stankewicz). LMS Journal of Computation and Mathematics 17 (2014), 509-535. (link to the arxiv)
[21] Euclidean quadratic forms and ADC-Forms II: integral forms (with W. Jagy). Acta Arithmetica 164 (2014), 265-308. (pdf)
[20] Absolute convergence in ordered fields. (with N.J. Diepeveen). American Math. Monthly 121 (2014), 909-916. (pdf)
[19] Graph derangements. Open Journal of Discrete Mathematics 3 (2013), 183-191. (link to the arxiv)
[18] GoNI: Primes represented by binary quadratic forms) (with J. Hicks, H. Parshall and K. Thompson). INTEGERS 13 (2013), A37, 18pp. (pdf)
[17] Torsion points on elliptic curves with complex multiplication (with an appendix by Alex Rice) (with B. Cook and J. Stankewicz). International Journal of Number Theory 9 (2013), 447-479. (pdf)
[16] GoNII: Universal quaternary quadratic forms (with J. Hicks, K. Thompson and N. Walters). INTEGERS 12 (2012), A50, 16 pp. (pdf)
[15D] Euclidean quadratic forms are ADC forms: a short proof (by F. Dacar (only!)). (pdf)
[15] Euclidean quadratic forms and ADC forms I. (pdf), Acta Arithmetica 154 (2012), 137-159.
[14] Covering numbers in linear algebra, American Math. Monthly 119 (2012), 65-67. (pdf)
[13] Period-index problems in WC-groups IV: a local transition theorem, J. Théor. Nombres Bordeaux 18 (2010), 583-606.(pdf)
[12] Period, index and potential Sha (with S. Sharif), Algebra Number Theory 4 (2010), 151–174. (pdf)
[11] Elliptic Dedekind domains revisited. Enseignement Math. 55 (2009), 213-225. (pdf)
[10] On the Hasse principle for Shimura curves, Israel J. Math. 171 (2009), no. 1, 349-365. (pdf)
[9] An ''Anti-Hasse Principle'' for prime twists. Int. J. of Number Theory 4 (2008), 627-637. (pdf)
[8] Local bounds for torsion points on abelian varieties (with X. Xarles). Canad. J. Math. 60 (2008), no. 3, 532-555. (pdf)
[7] Abelian points on algebraic varieties. Math. Res. Lett. 14 (2007), no. 5, 731-743. (pdf)
[6] On the indices of curves over local fields. Manuscripta Math. 124 (2007), no. 4, 411-426. (pdf)
[5] Galois groups via Atkin-Lehner twists. Proc. Amer. Math. Soc. 135 (2007), no. 3, 617-624 (pdf)
[4] There are genus one curves of every index over every number field. J. Reine Angew. Math. 594 (2006), 201-206. (pdf)
[3] On elementary equivalence, isogeny and isomorphism. J. Théor. Nombres Bordeaux 18 (2006), no. 1, 29-58. (pdf)
[2] On the Number of Representations of an Integer by a Linear Form (with G. Alon). Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.2 (free online publication)
[1] Period-index problems in WC-groups I: elliptic curves. J. Number Theory 114 (2005), 193-208. (pdf)

The versions of the papers listed here are isomorphic, but not necessarily identical, to the final published versions. In particular they incorporate all responses to referees' comments, so in many cases contain substantial expository and mathematical improvements over the corresponding arXived versions.

Expository papers:

See also my page of expositions. The papers listed here are more polished and contain novelties of exposition (or so I hope).

[E3] Wilson's Theorem: an Algebraic Approach. (pdf)
[E2] Factorization in Integral Domains. (pdf)
[E1] A site-theoretic characterization of points in a topological space (with Andrew Archibald and David Savitt). (pdf)

Thesis:

[0] Rational points on Atkin-Lehner quotients of Shimura curves. Ph.D. thesis, 2003. (pdf)

Caveat emptor: You are advised not to read either Chapter 1 or Chapter 2. There is a substantial error: Proposition 90 is false, which invalidates Main Theorem 1. (Indeed, later work of Rotger and Murabayashi showed that this theorem is also false as stated. The truth is significantly more complicated than I had expected at the time!) To the best of my knowledge, the other results are essentially correct. Also, Main Theorem 3 was derived independently by Ogg in the Mauvaise reduction paper cited in the references, although the proof I gave is different and, perhaps, of some interest. In short, it is a bit of a mess, and I would advise you not to use this thesis as a reference in the formal sense: if you wish to use a result in the thesis, cite an earlier or (see below) later paper, if possible. If you cannot find another reference, ask me for one!

Published thesis work: Although I have not written any papers which solely contain material from my thesis, several papers are strongly influenced by my thesis work and contain results which are improvments / simplifications / mild corrections of certain results of my thesis. Specifically, [6] began as a writeup of Main Theorem 5 of [0], but when I looked back I found that I was able to exploit new techniques to prove a much more attractive result. (On the other hand, the results of [6] do not literally include Main Theorem 5.) Main Theorem 6 and its application to X^D₁(N) appears in [7]. Main Theorem 2 is applied in [10]; as discussed there, in the meantime Rotger, Skorobogatov and Yafaev proved a more general result (by methods which include and generalize those that I used) and graciously acknowledged my priority on the special case, so in writing up [10] I felt that there would be no value added to include my proof.

Still unpublished thesis work: Main Theorem 4 -- a QM analogue of Mazur's theorem on rational isogenies of prime degree, which I thought of as the "main" Main Theorem of [0] -- has never been published in any form. So far as I know the proof is correct (up to some minor changes necessary to circumvent the erroneous result in Chapter 1) and seems to be of some interest. However, I have always viewed this theorem as a rough first step to a more satisfactory and complete analysis of rational points on X^D₀(N). Since many experts in the field have read my thesis and are hence aware of Main Theorem 4, there doesn't seem to be much advantage to publishing an "unripe" theorem. Finally, the more precise bounds on torsion on QM abelian surfaces have never been published. In this regard, it seems to me that the results would be much more interesting in conjunction with some nontrivial examples, and that this is a reasonable future project for someone (but maybe not for me).

Notes from Seminar Talks

Probabilistic Ideas and Methods in Analytic Number Theory. (pdf)
From a 11/5/08 talk given in the UGA number theory seminar.
Almost Sure Limit Sets of Random Series. (pdf)
From a 9/30/08 talk given in the UGA VIGRE seminar.
Ramanujan Graphs Part II. (pdf)
From a 3/21/07 talk given in the UGA number theory seminar.
Recent thoughts on abelian points. (pdf)
From a 2/7/2007 talk given in the UGA number theory seminar. The talk describes the work from my above "abelian points" paper. Details of proofs -- and especially, Galois cohomological calculations -- were often suppressed in favor of providing additional (and a posteriori) motivation for these results in terms of Kodaira dimension and field arithmetic. I have recklessly propounded a conjecture about metabelian points on algebraic varieties and explained a connection with the inverse Galois problem.
Mumford-Tate groups and abelian varieties. (pdf)
From a 11/9/2006 talk given in Elham Izadi's VIGRE seminar on the Hodge Conjecture.
Selections from the arithmetic geometry of modular and Shimura curves. (pdf)
From two talks given at the UGA number theory seminar in August and September 2006. Fielding a request from my new colleagues, I took a crack at ``going from zero to Shimura curves in two talks.''
Rational points on Atkin-Lehner twists of modular curves. (pdf)
From a talk given at the University of Pennsylvania in July of 2006. If you think you might be interested in twisted modular curves and applications to Galois theory, perhaps you will enjoy reading this document more than the two research papers (Anti-Hasse Principle... and Galois groups...) of mine which cover the same material.
On period-index problems. (pdf)
These are the "long form" lecture notes of a two hour talk I gave on April 24, 2006 at MSRI. This is the most ambitious talk I have ever given: I described all known period-index problems, their interrelationships, and I pointed out how the period-index problem in higher-dimensional WC-groups is related to Brauer groups, Shafarevich-Tate groups, and the so-called "elementary obstruction." The notes need to be polished (posting them here makes it much more likely that this polishing will take place). It was while preparing this talk that I began to appreciate how fruitfully the thesis work of Shahed Sharif and my own work could be combined, and portions of this paper give a preview of our forthcoming joint work. (click here for the short version)
An introductory lecture on WC-groups. (pdf)
These are notes inspired by a (too) short talk I gave on WC-groups at MSRI on March 17, 2006. After the talk the task became to record everything I wanted to say about WC-groups, so of course the document got longer and longer as I continued to work on it. At present I seem to have approximately half (the easier half) of a fairly substantial survey of the structure of WC-groups in general. Perhaps later I will develop this into a formal paper or a monograph.
Acquisition of rational points on algebraic curves. (pdf)
These are the slides for my UGA job talk. It turned out that I had at least 50% too much material (so that the talk was in fact not that good; I am thankful to UGA for hiring me anyway!), some of which was not really what I should have been discussing in a job talk but was nevertheless sort of interesting (and does not appear in any of my other writings), so here it is.
Introduction to the real spectrum. (pdf)
Notes from the second talk I gave at Colorado College in early December 2004 (the first was on the discrete geometry of Chicken McNuggets). The purpose of these notes is to convince a general mathematical audience that the collection of all orderings on a field is an interesting geometric object. Moreover, it develops some of the background material that one needs to appreciate "real arithmetic geometry," a topic of some very interesting recent papers of Colliot-Thelene, Parimala, and Scheiderer.
Notes on the DeRham cohomology of varieties. (pdf)
Notes from a series of talks given in Eyal Goren's 2003 cohomology theories seminar at McGill University (click here for the seminar webpage). These notes give a discussion of classical Hodge theory and DeRham cohomology first in the analytic category and then algebraically, as a preparation for crystalline cohomology. Not exactly light reading. Chapter 3 still seems to be a reasonably good short introduction to the Hodge Theorem, although in its free use of sheaves and cohomology, it is not maximally elementary.
On beyond line bundles: the Brauer group of almost anything. (pdf)
Notes from the last talk I gave in Harvard's Trivial Notions seminar (the grad student seminar: organized, lectured and attended by grad students -- and the odd wunderkind undergrad -- no faculty allowed). Here I push the idea that, although "everybody" (at least, everybody at Harvard, where algebraic geometry is tested on the first-year qualifying exams) is familiar with line bundles and the Picard group in a wide variety of geometric contexts, the idea that one can study the projective bundles modulo projectivizations of vector bundles (on almost anything) is much less familiar. All material is taken from Grothendieck's three papers on the subject.
Fundamental groups in characteristic p. (pdf)
Notes from a guest lecture in Romyar Sharifi's Galois actions on fundamental groups seminar, at Harvard in 2002.