Material Related to Current Projects
- 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:
are rough notes to my Basic Notions Seminar talk "Elliptic curves and their statistics" on February 28, 2012
- June 6, 2011:
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.
- September 25, 2009:
In an article entitled "Refined class number formulas and Kolyvagin systems"
([Arxiv]) 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 [PDF] is
- April 25, 2009:
In the article Ranks of twists of elliptic curves and Hilbert's Tenth Problem
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.
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"
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
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]
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
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
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
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
- Studying the growth of Mordell-Weil (available as
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
(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
- 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
- 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],
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,"
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,
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),
226, Birkhäuser Boston, Boston, MA, 2004.