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Material Related to Current Projects

  • August 22 2023: Jennifer Balakrishnan and I posted the paper Ogg’s Torsion conjecture: Fifty years later as an arXiv note (it has an an appendix by Netan Dogra). Here’s an up-dated version: : https://sites.harvard.edu/barry-mazur/wp-admin/upload.php?item=1272
  • August 22 2023: Star-like configurations in data related to the computation of L-values is the title of the lecture that I gave at the Murmurations in Arithmetic conference held at ICERM in Jul 6 – 8, 2023 (where I talked about a joint project with Karl Rubin regarding the statistical behavior of L-function values).
  • August 19 2023: An Experiment in Class Field Theory consists of the slides for a lecture I gave at the Iwasawa 2023 conference (in memory of John Coates) University of Cambridge, July 17 – 21 2023. It describes a joint project with Tony Feng, Michael Harris, and Arpon Raksit Derived Class Field Theory (as in our arXiv note https://arxiv.org/pdf/2304.14161.pdf).
  • August 9 2023: Sasha Shlapentokh, Karl Rubin, and I have just posted two (linked) papers on arXiv: Existential definability and diophantine stability and Defining ℤ using unit groups.
  • January 7, 2014: Karl Rubin and I have posted two (linked) papers on arXiv: “Controlling Selmer groups in the higher core rank case” and “Refined class number formulas for G_m”. In the second of these papers we formulate a generalization of a `refined class number formula’ of Darmon and we prove a large part of this conjecture when the order of vanishing of the corresponding complex L-function is 1.
  • October 12, 2013: Here are notes for the Erdos Memorial Lecture, given at Temple University on {Arithmetic statistics: elliptic curves and other things} ([PDF]). These notes also relate to the lecture I gave the week before at the Quebec/Maine Number Theory Conference.
  • October 5, 2013: “Some comments on elliptic curves over general number fields and Brill-Noether modular varieties” are rough notes for a lecture I gave at the Quebec/Maine Number Theory Conference.
  • March 2012: “Selmer Companion Curves” is a paper that Karl Rubin and I wrote that discusses elliptic curves whose quadratic twist families have exactly the same Selmer data, let alone statistics.
  • February 27, 2012: Here [PDF] are rough notes to my Basic Notions Seminar talk “Elliptic curves and their statistics” on February 28, 2012
  • December 13 2011: Here is a link to a paper “The spin of prime ideals” with John Friedlander, Henryk Iwaniec, and Karl Rubin, where we study spin densities of prime ideals with applications to Selmer ranks. It has appeared in the journal Inventiones Mathematicae, 193, Issue 3 (2013) 697-749
  • June 6, 2011: Here are rough notes to the talk “Disparity in the statistics of elliptic curves” I gave on joint work with Zev Klagsbrun and Karl Rubin in the April 2011 workshop of the program “Arithmetic Statistics” at MSRI.
  • June 17 2010: “How can we construct abelian Galois extensions of basic number fields?” is an article based on a lecture I gave at the 60th birthday conference for Ken Ribet in Berkeley (2008). This has been published (Bulletin of the American Mathematical Society, volume 48 (2011) 155-210).Here is a list of corrections and comments for it.
  • September 25, 2009: In an article entitled “Refined class number formulas and Kolyvagin systems” Karl Rubin and I use our prior work on Kolyvagin systems to prove (most of) a refined class number formula conjectured by Henri Darmon.
  • July 27, 2009: A paper “Deforming Galois Representations” is available.
  • April 25, 2009: In the article “Ranks of twists of elliptic curves and Hilbert’s Tenth Problem” (Inventiones mathematicae volume 181, 541�575 (2010)) Karl Rubin and I study the variation of the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.
  • Aug. 7 2007: The joint article with Frank Calegari on topics related to eigenvarieties for GL2 over quadratic imaginary fields entitled “Nearly Ordinary Galois Deformations over Arbitrary Number Fields” is available here on arXiv.org.
  • In an article written with Karl Rubin and Alice Silverberg entitled “Twisting commutative algebraic groups” we make explicit the construction of twisting commutative algebraic groups by characters of the Galois group of the base field, for applications in number theory (specifically, for use in articles that Karl Rubin and I are writing) and cryptography (specifically, for use in articles that Karl Rubin and Alice Silverberg are writing). It is available on ArXiv and has been published in the Journal of Algebra 314 (2007) 419-43.
  • Here is an expository article in PDF form entitled “Average ranks of elliptic curves” that I wrote with Baur Bektimirov, William Stein, and Mark Watkins. Its aim is to discuss the data that has recently been accumulated (by Stein and Watkins) to test current conjectures about average ranks. It has appeared in the Bulletin of the American Mathematical Society 44, (2007).
  • A revised version (July 29, 2006) of the article “When is one thing equal to some other thing” is available.
  • In “Computation of p-Adic Heights and Log Convergence”, William Stein, John Tate and I provide a fast algorithm for the computation of p-adic heights of rational points on elliptic curves (using work of Kedlaya and others). We also discuss related convergence questions concerning the p-adic modular form given by the Eisenstein series of weight two (whose computation is essential for p-adic heights).
  • On the arithmetic of elliptic curves. I have written one short book and eight articles with Karl Rubin. The book and the first of these eight articles are about systems of cohomology classes, such as those that come from Euler systems, via the theory of Kolyvagin. The next four articles are about the construction (for triples (p,K,E) satisfying some hypotheses, where p is a prime number, K is a number field, and E is an elliptic curve over K) of what we call an organization of the arithmetic of (p,K,E). This organization consists of a single skew-Hermitian matrix with entries in the Iwasawa algebra associated to L/K, the maximal Zp power extension of K, that provides a complete description of the Selmer modules, and the relevant p-adic height pairings, and Cassels-Tate pairings for all layers of L/K. In the initial of these four articles the existence of this skew-Hermitian matrix was conjectured, and its properties explored, while in the last of this series of four articles such an organizing skew-Hermitian matrix is actually constructed, under mild hypotheses. All the articles in this series, except for the last two, have already appeared. The remaining two articles, neither of which has yet been published, have to do with the problem of obtaining (unconditional) lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields, and over more general nonabelian extensions, respectively. The book and these eight articles, all joint with Karl Rubin, are described in more detail in the items below.
  • Kolyvagin Systems [PDF] [DVI]. This is a treatise published in the AMS memoir series (Mem. Amer. Math. Soc. 168 (2004), no. 799, viii+96 pp.). It gives the details of our way of thinking about the “coherent systems of cohomology classes” that come, via Kolyvagin’s construction, from Euler Systems. We show that these systems obtained by Kolyvagin satisfy even stronger “coherence relations” than were previously satisfied. By “Kolyvagin Systems” we then mean “systems of cohomology classes satisfying these strong coherence relations”, whether or not they arise from a classical Euler System. “Kolyvagin Systems” attached to p-adic Galois representations are extremely rigid, and manageable; they behave somewhat as if they were (refined) leading terms in an L-function, and they control quite precisely the size and shape of the corresponding dual Selmer module. They also are quite amenable to p-adic deformation, and using a result of Ben Howard, one sees that Kolyvagin Systems (at least attached to residual representations) exist quite generally.
  • Introduction to Kolyvagin systems [PDF], [DVI]. This is a short expository piece (pp. 207-221 in Stark’s conjectures: recent work and new directions, \, Contemp. Math., 358, Amer. Math. Soc., Providence, RI, 2004) intended to give the general ideas behind our treatise “Kolyvagin Systems” and to work these ideas out in some detail in a concrete classical instance. For somewhat older material on this topic, see the Arizona Winter School 2001 website which contains the notes for a project in Euler Systems directed by Tom Weston and myself; included there is a general expository article “Introduction to Euler Systems” [DVI], material regarding the Heegner Euler System and Kato’s Euler System, and lecture notes [DVI] for a course on Euler Systems that I recently gave.
  • Elliptic curves and class field theory, appeared in the Proceedings of the International Congress of Mathematicians, ICM 2002, Beijing, Ta Tsien Li, ed., vol II. Beijing: Higher Education Press (2002) 185-195. The published version or updated version ([PDF] [DVI]) with corrected references. This is a survey of open problems regarding, for the most part, the (p-adic) anti-cyclotomic arithmetic of elliptic curves, in view of the recent breakthroughs due to Cornut and Vatsal, building on the work of many other people, including Kolyvagin. We introduce here a single conjectural structure (which we refer to as an “organization” of the p-adic anti-cyclotomic arithmetic of an elliptic curve E over a quadratic field K) which, if it exists, incorporates all the known standard conjectures, some in somewhat strengthened forms. The text was delivered as a plenary address at the ICM in Beijing by Rubin.
  • “Pairings occurring in the arithmetic of elliptic curves” is available. This has appeared in Modular Curves and Abelian Varieties, J. Cremona et al., eds., Progress in Math. 224, Basel: Birkh�user (2004) 151- 163. Proceedings of the conference on arithmetic algebraic geometry, held in Barcelona, July 2002. This is a fuller account of our theory of “organizations” of the p-adic anti-cyclotomic arithmetic of an elliptic curve E over a quadratic field K. It is the text of a lecture given by me at the Barcelona conference.
  • Studying the growth of Mordell-Weil (available as [PDF] or [DVI]) has appeared in Documenta Math. extra volume (2003) 585-607, a volume in honor of K. Kato. Here, motivated by the recent work of Cornut and Vatsal, we investigate in a more general context cases, where the coherence of negative signs in the appropriate functional equations (together with the conjectures of Birch and Swinnerton-Dyer) point to the possibility that there be nontrivial universal norms of (p-adic completions of) Mordell-Weil groups relative to specific Z_p-extensions. This phenomenon is somewhat rarer than one might first imagine, and seems to be pointing quite specifically to contexts that deserve further close attention.
  • “Organizing the arithmetic of elliptic curves”. Here we construct the skew-Hermitian modules conjectured to exist in the previous articles. (Adv. Math. 198 (2005), no. 2, 504–546) Here is the PDF file of a semi-final version.
  • “Finding Large Selmer Groups” ( J. Differential Geom. 70 (2005), no. 1, 1–22) is available in PDF form (updated April 9, 2005). Here we apply the theory we have built up in the previous articles to prove that the rank of Selmer grows in a large quantity of Zp-extensions where we would “expect” growth because of functional-equation-sign reasons. The article Finding large Selmer rank via an arithmetic theory of local constants (Ann. of Math. (2) 166 (2007), no. 2, 579–612) is available at on ArXiv (with Karl Rubin). Here we offer a self-contained proof, by a new method, of a significant generalization of previous results that guarantee large Selmer rank when the corresponding (conjectured) functional equation would predict odd rank. Specifically we obtain (unconditional) lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let L denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(L/K) by -1). We prove (under mild hypotheses) that if the Zp-rank of the pro-p Selmer group Sp(E/K) is odd, then the Zp rank of Sp(E/F) is greater than or equal to [F:K] for every finite extension F of K in L.
  • Some notes for my basic notions seminar (now called Faculty Colloquium) on Functional equations, signs, and parity on Monday February 6 are here.
  • Growth of Selmer rank in nonabelian extensions of number fields: Here we extend our theory to offer growth results of ranks of Selmer groups of elliptic curves over Galois number field extensions of degree twice a power of an odd prime. Our article is available on ArXiv.
  • On rational connectivity in algebraic geometry and arithmetic. This is a project with Tom Graber, Joe Harris and Jason Starr:
  • T. Graber, J. Harris, B. Mazur and J. Starr: “Rational connectivity and sections of families over curves” ([PDF], [ps]). Here we prove a theorem that we call the “converse theorem.” It gives sufficient conditions for families of varieties over (high-dimensional bases) to possess what we call “pseudo–sections.” These are subfamilies dominating the base whose fibers are (generically) rationally connected varieties. We view this result as a “converse” to the theorem of Graber-Harris-Starr that guarantees that proper families of rationally connected varieties over smooth curves have sections.
  • T. Graber, J. Harris, B. Mazur, J. Starr: “Arithmetic questions related to rationally connected varieties” is a continuation of our joint work on the “converse theorem,” ([DVI] [PDF]) in the theory of rationally connected varieties. It has appeared in the proceedings of the conference in honor of Abel, held in Oslo. The legacy of Niels Henrik Abel, 531-542, Springer, Berlin, 2004.
  • T. Graber, J. Harris, B. Mazur, J. Starr: “Jumps in Mordell-Weil rank and arithmetic surjectivity” is a short discussion of some of the open problems in the previously cited article. It is a partial account of a lecture I gave at the AIM conference on “Rational points on varieties” held in Palo Alto, in December 2002. It has appeared in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 141-147, Progr. Math., 226, Birkhäuser Boston, Boston, MA, 2004.