11,906 research outputs found

    Quadratic Fields Admitting Elliptic Curves with Rational jj-Invariant and Good Reduction Everywhere

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    Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by xx over which there exist elliptic curves with good reduction everywhere and rational jj-invariant is ≫xlogβ‘βˆ’1/2(x)\gg x\log^{-1/2}(x). In this paper, we assume the abcabc-conjecture to show the sharp asymptotic ∼cxlogβ‘βˆ’1/2(x)\sim cx\log^{-1/2}(x) for this number, obtaining formulae for cc 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 E/QE/\mathbb{Q} with short Weierstrass equation y2=f(x)y^2 = f(x) for reducible f∈Z[x]f \in \mathbb{Z}[x], we show that the number of integers dd, ∣dβˆ£β‰€D|d| \leq D, for which the quadratic twist dy2=f(x)dy^2 = f(x) has an integral non-22-torsion point is at most D2/3+o(1)D^{2/3+o(1)}, assuming the abcabc-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 xx is O(x1βˆ’Ο΅)O(x^{1-\epsilon}) for some Ο΅>0\epsilon > 0. We also exhibit a matching lower bound. If instead of the abcabc-conjecture we assume a particular tail bound, we can prove all the aforementioned results and that the coefficient cc 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

    On Fermat's equation over some quadratic imaginary number fields

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    Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat's Last Theorem over Q(i)\mathbb Q(i). Under the same assumption, we also prove that, for all prime exponents pβ‰₯5p \geq 5, Fermat's equation ap+bp+cp=0a^p+b^p+c^p=0 does not have non-trivial solutions over Q(βˆ’2)\mathbb Q(\sqrt{-2}) and Q(βˆ’7)\mathbb Q(\sqrt{-7}).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

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    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 Q(2)\mathbb{Q}(\sqrt{2}).Comment: 36 pages (revised version of 2002 preprint
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