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Problem List
# Problem List

## Updated: November 11, 1996

Here are some problems of interest to me.
I welcome discussion. Send me email: [email protected]
## Fields with a rational point for a given curve.

- Let X be an algebraic variety over a field K.
Find all finite field extensions L of K in which X has a rational point.
If X is the curve x^2+y^2=-1 and K=Q the field of rational numbers then
the problem is to characterize all
number fields L in which x^2+y^2=-1 has a solution.

The problem can be rephrased in terms of finding so-called
minimal splitting fields.

- Let X be an algebraic variety over a field K.
Suppose L is an extension field of K such that the set of
L-rational points of X, denoted X(L), is nonempty.
Assume furthermore that L is minimal with respect to this property.
Thus if F (not equal to L) is an intermediate field
contained between K and L then X(F) is empty.
Then L is called a minimal splitting field for X.
Find all minimal splitting fields for a given variety X/K.

- Fix a number field K and a positive integer m. Does
there exist a genus one curve over K whose minimal splitting
field is of degree m over K? This is question was partially
resolved in [Lang-Tate, 1958].

- Does period=index for H^1 of an abelian variety over a number
field? ANSWER: NO. Cassels found a counterexample in the 60's.

- Let E be a genus one curve over a field K. Let L_1 and L_2
be extensions of K in which E has a rational point.
Using the Riemann-Roch theorem one sees that E has a point in
an extension of degree d=gcd([L_1:K],[L_2:K]). Now suppose L
is a Galois extension of K which contains both L_1 and L_2.
Does there exist a subfield of L of degree d over which E has
a rational point? [I doubt it but I haven't quite put together
a counterexample yet. It could be true though.]

## Algebra problem.

- (Communicated by H.W. Lenstra, proposed by June)
Let R be a finite ring (i.e. R has finitely many elements) and
fix finite R-modules M and N. Let M^n denote the direct sum of M with itself
n times. Suppose that there is some positive integer n along with an
R-module injection M^n ---> N^n. Question: Does it follow that there exists
an R-module injection from M into N?
The answer is well known in the special case when M^n is isomorphic to N^n.
In this case the Krull-Schmidt theorem implies that the indecomposable
factors of M and N are the same. Since R is finite this is enough to
show that M is isomorphic to N. See Lang's Algebra.