This page collects some problems which are, so far as I know, open. I have spent some time thinking about these problems, and some of them are closely related to aspects of my past
research. However, I am not currently working on any of them per se. If you would like to work on any of these problems with me or under my supervision (or if you would like to tell me the answers!), please let me know.
Transcendental Galois Theory
Suppose we define an arbitrary field extension K/F to be Galois if, for all
subextensions L of K/F, we have KAut(K/L) = L. In words: for any element x of
K \setminus L, there exists an automorphism s of K such that s(l) = l for all l in L, but s(x)
\neq x. (Note that in case K/F is algebraic, this is indeed a characteristic property of Galois extensions.) What are the transcendental Galois extensions?
In Transcendental Galois Theory I show that if F has characteristic 0 and K is algebraically closed, then K/F is Galois in the above sense. I also conjectured: if K/F is Galois, then either K/F is algebraic, normal and separable, or F has characteristic 0 and K is algebraically closed. Is this true??
Hitting Problems For Varieties
Consider the class of all varieties defined over a field K (K = Q is always an
interesting choice). If X and Y are varieties over K, let us say that X hits Y if
there is a K-morphism from X to Y. Note that, since constant morphisms are not prohibited,
the "hitting relation" is only interesting for varieties without rational points: indeed, a
variety Y has a K-rational point iff it is hit by every K-variety X. If F_1 and F_2 are
two families (i.e., sets) of varieties, we say F_1 hits F_2 if every variety in
F_2 is hit by some variety in F_1.
Question: Is every Severi-Brauer variety W hit by some genus one curve?
Comments: (1) Let n be the order of the Brauer class [W] of W. Then W is hit by every genus one curve whose Jacobian has n^2 K-rational points of order n. Un/fortunately this is only
possible when K contains the nth roots of unity, so only for n = 2 when K = Q.
(2) When considering source varieties V of dimension greater than one, it might be more
interesting to weaken the notion of hitting to the existence of a rational map V ---> W.
With this change, the Brauer classes of Severi-Brauer varieties hit by V are precisely
the subgroup of Br(K) generated by the obstructions to Galois-invariant line bundles on the
algebraic closure of V descending to K-rational line bundles on V.
Ramanujan Graphs, Zeta Functions, and Shimura Varieties
A hyperbolic tiling of a compact Riemann surface determines a pair of "dual" finite graphs G_0 (in which the vertices correspond to the 0-cells of the tiling) and G_2 (in which the vertices correspond to the 2-cells). What is the spectral relationship between such a "dual pair"? Explore these graphs in the case of the
tilings associated to congruence subgroups of hyperbolic triangle groups, especially in the non-arithmetic case. The simplest case -- X(p) -- is already intriguing, in that is known that the G_0 graphs are Ramanujan of varying vertex degree (Gunnells) and the G_2 graphs are non-Ramanujan but expanding cubic graphs (Selberg). Strangely, the proof of the latter statement lies much deeper than the first -- and uses the arithmeticity of Gamma(p) in an apparently crucial way. Is there some way to deduce the expanding property for the G_2's
from the Ramanujan property for the G_0's? How are these constructions related to their "function field analogues"?
Rational Points on Atkin-Lehner twists of Modular Curves
Let X be an algebraic curve over a field K, and let w: X -> X be a K-rational involution. For
any d in K*/K*2 one can define a twisted curve X(w,d). The case of
X = X_0(N) a modular curve and w = w_N the main Atkin-Lehner involution is especially interesting, as X(w,d) parameterizes quadratic Q-curves and has applications to the Inverse
Galois Problem: see Galois groups via Atkin-Lehner twists and An "Anti-Hasse Principle for prime twists". These papers contain many open problems:
(1) Conjecture: Let m and M be coprime positive integers, and let F(X) be the number of primes
p <= X which are congruent to m modulo M and such that the class number of Q(\sqrt{-3p}) is
not divisible by 3. Then F(X) >> X/log X.
As mentioned in "Galois groups...", a proof of this conjecture would give an unconditional proof of a new set of primes p, of positive density, such that PSL_2(F_p) occurs as a Galois group over Q.
A closely related problem is:
(2) Show by a Selmer group calculation that, in the family of twists X(11,p) of X_0(11) by
w_N and prime numbers p congruent to 1 mod 4, the rank is always equal to 1, or at least that
it is at most 2.
(3) (Ellenberg's Problem A): For any Atkin-Lehner twisted modular curve X(N,d),
determine the set of primes p such that the curve fails to have points over Q_p.
(4) (Closely related) For any prime p, give an explicit description of the minimal regular Z_p-model of X(N,d).
(5) More generally, let R be a Dedekind domain with fraction field K, C a nice curve over K (say of genus at least one), and w a K-rational involution on C Let d be in K*/K*2. Suppose we are "given" the minimal regular R-model of C. How explicitly can we describe the minimal regular R-model
of the twisted curve C(w,d)? [One can ask the same question for twists by an arbitrary finite order automorphism of C.]
Metabelian Points on Algebraic Varieties
In Abelian points on algebraic varieties I give many examples of geometrically integral algebraic varieties defined over
Q which have no rational points over the maximal abelian extension Qab of Q. All of these examples have metabelian points, i.e., points rational over the maximal abelian extension Qmetab = (Qab)ab of Qab. Several years ago now, B. Mazur
mentioned to me the idea that every genus one curve over Q should have metabelian points, and recent work of Ciperiani-Wiles gives some results in this direction. On the other hand, there is no compelling reason to believe that every geometrically integral variety X/Q should have
metabelian points (even the comparatively much weaker conjecture that every such variety has solvable points is viewed with a fair amount of skepticism). Can someone construct an example of a variety X/Q without metabelian points? Note that if it turns out that every geometrically integral variety over Qmetab has rational points, then every finite group occurs as a Galois group over Qmetab, which would be a sensational result.