Here are some suggested topics for the paper. You can also choose your own
topic, but make sure to get it approved first.
(1) Rational points on conics. In class we stated Legendre’s theorem, which
characterizes when a conic has a rational point. Give a proof of this theorem. If you know something about p-adic numbers, then you can give
a proof of the following equivalent characterization, known as the Hasse
principle: a homogeneous quadratic equation has a nonzero solution over
Q iff it has a nonzero solution over R and a nonzero solution over Qp for
all p. References: [8, Section 5.5] for an elemetary proof, [3, Chapters 1-5]
for the Hasse principle.
(2) The congruent number problem. A rational number n is called congruent
if it is the area of a right triangle with rational side lengths. One can
show that n is congruent if and only if the equation y 2 = x3 − n2 x has a
nontrivial rational solution. Describe some of the progress that has been
made in determining which numbers are congruent. Reference: [7]
(3) Elliptic curve cryptography. Describe some uses of elliptic curves in cryptography. Reference: [2, Part III], [6, Chapter 6], [14, Chapters 5-6]
(4) Diophantine approximation and sums of two cubes. The theory of diophantine approximation measures how well an irrational number can be
approximated by rational numbers (as a function of the size of the denominator of the rational number). Use diophantine approximation to show
that the equation ax3 + bx3 = c has only finitely many solutions for fixed
a, b, c. Reference: [13, Chapter V]
(5) Complex tori. As mentioned in class, any elliptic curve over C can be
considered as the quotient of a complex plane by a lattice. Make this correspondence explicit by defining Weierstrass ellitpic functions and proving
some of their properties. References: [1, Chapter 7] and [12, Chapter VI].
(6) Complex multiplication. An endomorphism of an elliptic curve E is a map
E → E that preserves the identity element. One can show that any endomorphism of elliptic curves preserves the group law. For any integer n,
the map P 7→ nP is an endomorphism of E. Most elliptic curves do not
have any more endomorphisms. However, there are some special curves
that do have more endomorphisms. For example, the curve y 2 = x3 − x has
the endomorphism (x, y) 7→ (−x, iy). Applying this endomorphism twice is
the same as multiplying by −1, so we can think of this endomorphism as
“multiplication by i”. Define complex multiplication and describe some of
its applications. References: [13, Chapter VI].
(7) Abstract algebraic curves. In class, we have been considering curves as
as subspaces of the projective plane. However, it is also possible to define
curves without reference to any larger ambient space. One advantage of this
more general construction is that it makes the elliptic curve group law seem
less mysterious. Explain elliptic curves from this point of view. Reference:
[11, Chapters I-III]
(8) Modular forms. This topic also involves considering elliptic curves over C
as quotients of the complex plane by a lattice. A modular form of weight k
is a (sufficiently
nice) function on the upper half complex plane satisfying
k
f az+b
cz+d = (cz +d) f (z) for all integers a, b, c, d such that ad−bc = 1. One
can also think of a modular form as a function on the set of lattices in the
1
2
complex plane. Give some basic properties of modular forms, and perhaps
describe their relation to elliptic curves. References: [4], [7, Chapter III],
[10, Chapter VII].
P∞ n
(9) Carlitz modules. You are familiar with the exponetial function ez = n=0 zn! ,
which maps R → R (or C → C). This function has many nice properties. For example, ex+y = ex ey , and e2πiz is a root of unity if z is rational. Carlitz discovered an analogue of the exponential function that
is defined over Fq ((T )) (the field of Laurent series with coefficients in
the finite field Fq ) rather than R. The Carlitz exponential eC satisfies
eC (T x) = T eC (x) + eC (x)q . Furthermore, there exists ζ (an analogue of
2πi) so that if x is a rational function of T , then eC (ζx) lies in an abelian
extension of Fq ((T )). Carlitz modules are the simplest examples of Drinfeld
modules, which are a sort of function field analogue of elliptic curves. Define the Carlitz exponential and explore some of its basic properties. This
topic is probably on the difficult side. References: [5, Chapters 1-3] and [9,
Chapter 12].
References
[1] L. Ahlfors. Complex Analysis.
[2] A. K. Bhandari, D. Nagraj, B. Ramakrishnan, and T. N. Venkataramana. Elliptic Curves,
Modular Forms and Cryptography.
[3] J. W. S. Cassels. Lectures on Elliptic Curves.
[4] F. Diamond and J. Shurman. A First Course in Modular Forms.
[5] D. Goss. Basic Structures of Function Field Arithmetic.
[6] N. Koblitz. A Course in Number Theory and Cryptography.
[7] N. Koblitz. An Introduction to Elliptic Curves and Modular Forms.
[8] I. Niven, H. S. Zuckerman, and H. L. Montgomery. An Introduction to the Theory of Numbers.
[9] M. Rosen. Number Theory in Function Fields.
[10] J. P. Serre. A Course in Arithmetic.
[11] I. R. Shafarevich. Basic Algebraic Geometry.
[12] J. H. Silverman. The Arithmetic of Elliptic Curves.
[13] J. H. Silverman and J. T. Tate. Rational Points on Elliptic Curves.
[14] L. Washington. Elliptic Curves: Number Theory and Cryptography.