The second moment of the number of integral points on elliptic curves is bounded

In this paper, we show that the second moment of the number of integral points on elliptic curves over $\mathbb{Q}$ is bounded. In particular, we prove that, for any $0 < s < \log_2 5 = 2.3219 \ldots$, the $s$-th moment of the number of integral points is bounded for many families of elliptic curves --- e.g., for the family of all integral short Weierstrass curves ordered by naive height, for the family of only minimal such Weierstrass curves, for the family of semistable curves, or for subfamilies thereof defined by finitely many congruence conditions. For certain other families of elliptic curves, such as those with a marked point or a marked $2$-torsion point, the same methods show that for $0 < s < \log_2 3 = 1.5850\ldots$, the $s$-th moment of the number of integral points is bounded. The main new ingredient in our proof is an upper bound on the number of integral points on an affine integral Weierstrass model of an elliptic curve depending only on the rank of the curve and the number of square divisors of the discriminant. We obtain the bound by studying a bijection first observed by Mordell between integral points on these curves and certain types of binary quartic forms. The theorems on moments then follow from H\"older's inequality, analytic techniques, and results on bounds on the average sizes of Selmer groups in the families.Comment: 14 pages, comments welcome

Analysis of Parallel Montgomery Multiplication in CUDA

For a given level of security, elliptic curve cryptography (ECC) offers improved efficiency over classic public key implementations. Point multiplication is the most common operation in ECC and, consequently, any significant improvement in perfor- mance will likely require accelerating point multiplication. In ECC, the Montgomery algorithm is widely used for point multiplication. The primary purpose of this project is to implement and analyze a parallel implementation of the Montgomery algorithm as it is used in ECC. Specifically, the performance of CPU-based Montgomery multiplication and a GPU-based implementation in CUDA are compared

Nuclear profile dependence of elliptic flow from a parton cascade

The transverse profile dependence of elliptic flow is studied in a parton cascade model. We compare results from the binary scaling profile to results from the wounded nucleon scaling profile. The impact parameter dependence of elliptic flow is shown to depend sensitively on the transverse profile of initial particles, however, if elliptic flow is plotted as a function of the relative multiplicity, the nuclear profile dependence disappears. The insensitivity was found previously in a hydrodynamical calculation. Our calculations indicate that the insensitivity is also valid with additional viscous corrections. In addition, the minimum bias differential elliptic flow is demonstrated to be insensitive to the nuclear profile of the system