In my PhD thesis
Cyclicity of elliptic curves modulo p,
written under the supervision of
Prof. M. Ram Murty
and completed in the Spring of 2002,
I investigated how often the group of points of the
reduction modulo a prime p of an elliptic curve defined over the rationals
is cyclic.
This problem may be viewed as a weaker version of the
elliptic curve analogue of Artin's
primitive root conjecture formulated by Lang and Trotter in 1976.
THESIS ABSTRACT
The main focus of the thesis is on the cyclicity of the group of points
of the reduction
modulo primes p of an elliptic curve defined over the rationals.
More precisely, given an elliptic
curve E defined over Q, we investigate the asymptotic behaviour of the
number f(x, Q) of primes
p < x for which the group of points of the reduction of E modulo p is cyclic.
An asymptotic formula of the form f(x, Q) ~ c_E li(x),
where c_E is some constant depending
on E and li(x) is the logarithmic integral, was first obtained by
Jean-Pierre Serre in
1976, under the assumption of a Generalized Riemann Hypothesis (denoted GRH).
Important work on the
asymptotic behaviour of f(x, Q) was further done by Ram Murty in 1979
and 1987, and by
Rajiv Gupta and Ram Murty in 1990.
In this thesis we consider the problems of obtaining an unconditional
asymptotic formula for f(x, Q), if possible, and of providing
explicit error terms
in such a formula. The different properties of elliptic curves with or
without complex multiplication
(denoted CM) lead us to different analyses and results in the two situations.
In the case of a non-CM elliptic curve we obtain an effective asymptotic
formula for f(x, Q)
under the assumption of a quasi-GRH. In the case of a CM elliptic curve
we obtain an unconditional
effective asymptotic formula for f(x, Q). Using the ideas involved in
the proofs of these
results, we make significant improvements in the size of the error terms in
the asymptotic
formulae for f(x, Q), under GRH. Consequently, we obtain interesting
upper estimates in terms of the conductor of E for the
smallest prime p for which the group of points of E modulo p is cyclic.
By observing that the cyclicity of the group of points of E modulo p is
ensured if the group
has square-free order, we are naturally led to considering the problem
of finding
(effective) asymptotic formulae for the number h(x, Q) of primes p < x for
which the order
of the group E modulo p is square-free. It turns out that this problem is
more challenging than
the original one on cyclicity, and that new ideas are needed to resolve it.
In the case of a non-CM
elliptic curve we obtain a conditional asymptotic formula for h(x, Q).
In the case
of a CM elliptic curve we obtain an unconditional effective asymptotic
formula for h(x, Q), and, under GRH, we also obtain a significantly
small error term
in this formula.
Our different treatments of the above problems in the CM and non-CM situations
are determined by
some major differences between CM and non-CM elliptic curves.
In this thesis we briefly
discuss two such differences. The first one concerns a famous result
of Jean-Pierre Serre from 1972
on the surjectivity of the Galois representations phi_l associated to a
non-CM elliptic
curve defined over the rationals.
We obtain conditional and unconditional upper estimates
for the primes l for which phi_l is not surjective.
The second difference concerns a
conjecture of Serge Lang and Hale Trotter from 1976 about
the fields
generated by the Frobenius endomorphisms of the reduction modulo primes p
of a non-CM
elliptic curve defined over the rationals.
Under GRH, we obtain a nontrivial estimate towards this conjecture.
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