We exploit the properties of Legendre polynomials defined by the contour integral $\bold P_n(z)=(2\pi {\rm i})^{-1} \oint (1-2tz+t^2)^{-1/2}t^{-n-1} {\rm d} t,$ where the contour encloses the origin and is traversed in the counterclockwise direction, to obtain congruences of certain sums of central binomial coefficients. More explicitly, by comparing various expressions of the values of Legendre polynomials, it can be proved that for any positive integer $r$, a prime $p \geqslant 5$ and $n=rp^2-1$, we have $\sum _{k=0}^{\lfloor n/2\rfloor }{2k \choose k}\equiv 0, 1\text { or }-1 \pmod {p^2}$, depending on the value of $r \pmod 6$.
Theorem 1 of J.-J. Lee, Congruences for certain binomial sums. Czech. Math. J. 63 (2013), 65–71, is incorrect as it stands. We correct this here. The final result is changed, but the essential idea of above mentioned paper remains valid.
A conjecture due to Honda predicts that given any abelian variety over a number field $K$, all of its quadratic twists (or twists of a fixed order in general) have bounded Mordell-Weil rank. About 15 years ago, Rubin and Silverberg obtained an analytic criterion for Honda's conjecture for a family of quadratic twists of an elliptic curve defined over the field of rational numbers. In this paper, we consider this problem over number fields. We will prove that the existence of a uniform upper bound for the ranks of elliptic curves in this family is equivalent to the convergence of a certain infinite series. This result extends the work of Rubin and Silverberg over $\mathbb Q$.