Material Related to Current Projects

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" ([PDF]) is available.

In "Computation of p-Adic Heights and Log Convergence" ([PDF]), 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 in [DVI], or [PDF] format. This has appeared in Modular Curves and Abelian Varieties, J. Cremona et al., eds., Progress in Math. 224, Basel: Birkhäuser (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.

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" [PDF]) 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.