Congruences involving $\binom{2k}k^2\binom{4k}{2k}m^{-k}$

Source: arXiv

ABSTRACT Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the
paper, by using the work of Ishii and Deuring's theorem for elliptic curves
with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning
$\sum_{k=0}^{p-1}\binom{2k}k^2\binom{4k}{2k}m^{-k}\mod {p^2}$.

  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Let p>3p>3 be a prime and a,b∈Za,b∈Z. In the paper we mainly determine the number Vp(x4+ax2+bx)Vp(x4+ax2+bx) of incongruent residues of x4+ax2+bx(x∈Z) modulo p and reveal the connections with elliptic curves over the field FpFp of p elements.
    Journal of Number Theory 08/2006; 119(2):210-241. DOI:10.1016/j.jnt.2005.10.012 · 0.52 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: New explicit formulas are given for the supersingular polynomial ssp(t) and the Hasse invariant Hˆp(E) of an elliptic curve E in characteristic p. These formulas are used to derive identities for the Hasse invariants of elliptic curves En in Tate normal form with distinguished points of order n. This yields a proof that Hˆ(E4) and Hˆ(E5) are projective invariants (mod p) for the octahedral group and the icosahedral group, respectively; and that the set of fourth roots λ1/4 of supersingular parameters of the Legendre normal form Y2=X(X−1)(X−λ) in characteristic p has octahedral symmetry. For general n⩾4, the field of definition of a supersingular En is determined, along with the field of definition of the points of order n on En.
    Journal of Number Theory 10/2006; 120(2):234-271. DOI:10.1016/j.jnt.2005.12.008 · 0.52 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D<-4 of an imaginary quadratic field K . If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal P of K dividing p and there exist positive integers u and v such that 4p=u^2-Dv^2. It is well known that square of the trace a_P of Frobenius endomorphism of the reduction of E modulo P is equal to u^2. We determine whether a_P=u or a_P=-u in the case the class number of R is 2 or 3 and D is divided by 3,4 or 5. This article is a revised version of the authors' preprint DMIS-RR-02-5,2002.
    Bulletin of the Australian Mathematical Society 02/2004; DOI:10.1017/S0004972700035875 · 0.48 Impact Factor

Preview (2 Sources)

Available from