Elliptic curves our fascinating! For example,
Fermat's last theroem was solved by Andrew Wiles
in 1995 using elliptic curves. The problem
remained unsolved for over 300 years.
The Fermat's last theorem states that equation:
x^n + y^n= z^n has NO solution for n > 2.
Secondly, Clay institute  of mathematics
has suggested seven different mathematics
problems for solving each problem
Clay institute of mathematics offers a million
dollars reward. These problems include
P versus NP, Riemann hypothesis, Yang-Mills and Mass Gap problem,  Poincare
conjecture, which already solved by a Russian 
mathematician. One of these 7 problems is
is "Birch and Swinnerton-Dyer Conjecture".  This
problem deals with finding real value
solutions of higher degree polynomials,
such as elliptic curves.  So you can say that
the elliptic curve can help you to get a Abel prize
which is considered to be equivalent to Nobel prize
in mathematics. As Andrew Wiles was rewarded 
with Abel price and similarly
elliptic curve can reward you 1 million dollars
if you may solve one of the
problems offered by Clay institute of mathematics.
However, the most exciting part is elliptic curve based cryptography.
Because of many known
attacks against RSA, we have to use a
very large key, in order to obtain
reasonable security.
Similarly discrete logarithmic problem
over finite fields also uses
a very large key size because of
"general number field sieve problem"
However, this attack does not work on
 elliptic curves over finite fields.
Therefore, in elliptic curves we can use
very small key size. For instance,
key size of 512 bits is enough to
provide security equivalent to 256 bits of
advanced encryption standard (AES). So ECs
provide us enhanced security using small key sizes.
Therefore in your resource constraint
devices (e.g. Smart cards) having small energy
we cannot use RSA and DL over finite fields, instead
we may us elliptic curves.
Elliptic curves are also
faster because of their small key size as
compared to other asymmetric algorithms.
These are just three examples. In fact by
learning about elliptic curves and about curves of
higher degrees and multiple variables,
will open new gateway of research and
mathematical understanding.
In the next lecture, I will introduce
 elliptic curves over finite fields.
So stay tuned!
