11,906 research outputs found
Quadratic Fields Admitting Elliptic Curves with Rational -Invariant and Good Reduction Everywhere
Clemm and Trebat-Leder (2014) proved that the number of quadratic number
fields with absolute discriminant bounded by over which there exist
elliptic curves with good reduction everywhere and rational -invariant is
. In this paper, we assume the -conjecture to show
the sharp asymptotic for this number, obtaining
formulae for in both the real and imaginary cases. Our method has three
ingredients:
(1) We make progress towards a conjecture of Granville: Given a fixed
elliptic curve with short Weierstrass equation for
reducible , we show that the number of integers , , for which the quadratic twist has an integral
non--torsion point is at most , assuming the -conjecture.
(2) We apply the Selberg--Delange method to obtain a Tauberian theorem which
allows us to count integers satisfying certain congruences while also being
divisible only by certain primes.
(3) We show that for a polynomially sparse subset of the natural numbers, the
number of pairs of elements with least common multiple at most is
for some . We also exhibit a matching lower
bound.
If instead of the -conjecture we assume a particular tail bound, we can
prove all the aforementioned results and that the coefficient above is
greater in the real quadratic case than in the imaginary quadratic case, in
agreement with an experimentally observed bias.Comment: 35 pages, 1 figur
Elliptic Curves over Real Quadratic Fields are Modular
We prove that all elliptic curves defined over real quadratic fields are
modular.Comment: 38 pages. Magma scripts available as ancillary files with this arXiv
versio
On Fermat's equation over some quadratic imaginary number fields
Assuming a deep but standard conjecture in the Langlands programme, we prove
Fermat's Last Theorem over . Under the same assumption, we also
prove that, for all prime exponents , Fermat's equation
does not have non-trivial solutions over
and .Comment: The present is a revised version, including suggestions from
referees, that was accepted for publication in Research in Number Theory; 16
page
On the modularity of supersingular elliptic curves over certain totally real number fields
We study generalisations to totally real fields of methods originating with
Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic
curves with ordinary reduction, we focus here on the case of supersingular
reduction. Combining these, we then obtain some partial results on the
modularity problem for semistable elliptic curves, and end by giving some
applications of our results, for example proving the modularity of all
semistable elliptic curves over .Comment: 36 pages (revised version of 2002 preprint
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