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P. B. Vinod Kumar
coefficients mod p. If the equation of E has rational but non-integral coefficients, we
would need to assume none of their denominators are divisible by p, so we might as
well assume all coefficients to be integral to begin with (since if the denominators
are prime to p they have inverses mod p). Further, the definition of an elliptic curve
requires that there are no repeated roots of the polynomial in x, and this may fail to
be true when reducing mod p for some primes. Such primes are said to have “bad
reduction”. There will be only a finite number of these for any particular curve (they
will divide the discriminant), but they have to be dealt with specially.
For any prime p where E has good reduction, we can consider the elliptic curve
E(F P ) over F P . Since F P is finite, there are only a finite number of points on E(F P )),
so it is a finite group. The order of this group, (E(F P )), turns out to be a very important
number.
There is a general approach in number theory of trying to deal with “global”
problems, such investigating the structure of E(Q), by looking at a closely related
“local” problem mod p for all primes p. This is why we are interested in E(F P ). In
particular, if E(F P ) is “large” for most p, we would expect E(Q) to be large too.
We will see that the numbers (E(F P )) are studied by relating them to coefficients
of the Dirichlet series of L(E, s), the L-function of E.
The most important fact about the minimal discriminant is that the primes which
divide it are precisely the ones at which the curve has bad reduction. In other words,
except for those primes, the reduced curve is an elliptic curve over F P .
There is still another invariant of an elliptic curve E, called its conductor, and
often denoted simply by N . The exact definition is rather technical, but basically the
conductor is, like the minimal discriminant, a product of primes at which the curve
has bad reduction. Recall that E has bad reduction when it has a singularity modulo
p. The type of singularity determines the power of p that occurs in the conductor.
If the singularity is a “node”, corresponding to a double root of the polynomial, the
curve is said to have “multiplicative reduction” and p occurs to the first power in the
conductor. If the singularity is a “cusp”, corresponding to a triple root, E is said to
have “additive reduction”, and p occurs in the conductor with a power of 2 or more.
If the conductor of E is N , then it will turn out that N is the “level” of certain functions called modular forms (not yet defined) with which, according to the
Taniyama-Shimura conjecture, E is intimately connected.
If N is square-free, then all cases of bad reduction are of the multiplicative type.
An elliptic curve of this sort is called semistable. It is for elliptic curves of this sort
that Wiles proved the Taniyama-Shimura conjecture.
Theorem 2.2 Any finite field with characteristic p has p
n elements for some positive
integer n.
Proof: Let L be the finite field and K the prime subfield of L. The vector space of L
over K is of some finite dimension, say n, and there exists a basis P 1 , P 2 , . . . , P n of L
over K . Since every element of L can be expressed uniquely as a linear combination
of the p i over K i.e., every a in L can be written as a =
β i P i , with β i in K , and
since K has p elements, L must have p n elements.
P. B. Vinod Kumar
coefficients mod p. If the equation of E has rational but non-integral coefficients, we
would need to assume none of their denominators are divisible by p, so we might as
well assume all coefficients to be integral to begin with (since if the denominators
are prime to p they have inverses mod p). Further, the definition of an elliptic curve
requires that there are no repeated roots of the polynomial in x, and this may fail to
be true when reducing mod p for some primes. Such primes are said to have “bad
reduction”. There will be only a finite number of these for any particular curve (they
will divide the discriminant), but they have to be dealt with specially.
For any prime p where E has good reduction, we can consider the elliptic curve
E(F P ) over F P . Since F P is finite, there are only a finite number of points on E(F P )),
so it is a finite group. The order of this group, (E(F P )), turns out to be a very important
number.
There is a general approach in number theory of trying to deal with “global”
problems, such investigating the structure of E(Q), by looking at a closely related
“local” problem mod p for all primes p. This is why we are interested in E(F P ). In
particular, if E(F P ) is “large” for most p, we would expect E(Q) to be large too.
We will see that the numbers (E(F P )) are studied by relating them to coefficients
of the Dirichlet series of L(E, s), the L-function of E.
The most important fact about the minimal discriminant is that the primes which
divide it are precisely the ones at which the curve has bad reduction. In other words,
except for those primes, the reduced curve is an elliptic curve over F P .
There is still another invariant of an elliptic curve E, called its conductor, and
often denoted simply by N . The exact definition is rather technical, but basically the
conductor is, like the minimal discriminant, a product of primes at which the curve
has bad reduction. Recall that E has bad reduction when it has a singularity modulo
p. The type of singularity determines the power of p that occurs in the conductor.
If the singularity is a “node”, corresponding to a double root of the polynomial, the
curve is said to have “multiplicative reduction” and p occurs to the first power in the
conductor. If the singularity is a “cusp”, corresponding to a triple root, E is said to
have “additive reduction”, and p occurs in the conductor with a power of 2 or more.
If the conductor of E is N , then it will turn out that N is the “level” of certain functions called modular forms (not yet defined) with which, according to the
Taniyama-Shimura conjecture, E is intimately connected.
If N is square-free, then all cases of bad reduction are of the multiplicative type.
An elliptic curve of this sort is called semistable. It is for elliptic curves of this sort
that Wiles proved the Taniyama-Shimura conjecture.
Theorem 2.2 Any finite field with characteristic p has p
n elements for some positive
integer n.
Proof: Let L be the finite field and K the prime subfield of L. The vector space of L
over K is of some finite dimension, say n, and there exists a basis P 1 , P 2 , . . . , P n of L
over K . Since every element of L can be expressed uniquely as a linear combination
of the p i over K i.e., every a in L can be written as a =
β i P i , with β i in K , and
since K has p elements, L must have p n elements.