Generalized explicit descent and its application to curves of genus 3

We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically-defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over Q of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus-3 examples defined by polynomials with small coefficients.Comment: 58 pages; added a few references, and updated a few other

Note on the nonvanishing of L(1)

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Prediction of Adolescent Sexual Reoffending: A Meta-Analysis of the J-SOAP-II, ERASOR, J-SORRAT-II, and Static-99

Several risk assessment tools, including the Juvenile Sex Offender Assessment Protocol-II (Prentky & Righthand, 2003), the Estimate of Risk of Adolescent Sexual Offense Recidivism (Worling & Curwen, 2001), the Juvenile Sexual Offense Recidivism Risk Assessment Tool-II (Epperson, ralston, Fowers, DeWitt, & Gore, 2006), and the Static-99 (Hanson & Thornton, 1999), have been used to assess reoffense risk among adolescents who have committed sexual offenses. Given that research on these tools has yielded somewhat mixed results, we empirically synthesized 33 published and unpublished studies involving 6,196 male adolescents who had committed a sexual offense. We conducted two separate meta-analyses, first with correlations and then with AUCs. Total scores on each of the tools significantly predicted sexual reoffending, with aggregated correlations ranging from .12 - .20 and aggregated area under the receiver operating characteristic curve (AUC) scores ranging from .64 - .67. In many cases, however, heterogeneity across studies was moderate to high. There were no significant differences between tools, and although the Static-99 was developed for adults, it achieved similar results as the adolescent tools. To help interpret these findings, results are compared to other meta-analyses of risk tools used in the area of violence risk assessment and in other fields

Incomplete Quadratic Exponential Sums in Several Variables

We consider incomplete exponential sums in several variables of the form S(f,n,m) = \frac{1}{2^n} \sum_{x_1 \in \{-1,1\}} ... \sum_{x_n \in \{-1,1\}} x_1 ... x_n e^{2\pi i f(x)/p}, where m>1 is odd and f is a polynomial of degree d with coefficients in Z/mZ. We investigate the conjecture, originating in a problem in computational complexity, that for each fixed d and m the maximum norm of S(f,n,m) converges exponentially fast to 0 as n grows to infinity. The conjecture is known to hold in the case when m=3 and d=2, but existing methods for studying incomplete exponential sums appear to be insufficient to resolve the question for an arbitrary odd modulus m, even when d=2. In the present paper we develop three separate techniques for studying the problem in the case of quadratic f, each of which establishes a different special case of the conjecture. We show that a bound of the required sort holds for almost all quadratic polynomials, a stronger form of the conjecture holds for all quadratic polynomials with no more than 10 variables, and for arbitrarily many variables the conjecture is true for a class of quadratic polynomials having a special form.Comment: 31 pages (minor corrections from original draft, references to new results in the subject, publication information

"The numerical accuracy of truncated Ewald sums for periodic systems with long-range Coulomb interactions"

Ewald summation is widely used to calculate electrostatic interactions in computer simulations of condensed-matter systems. We present an analysis of the errors arising from truncating the infinite real- and Fourier-space lattice sums in the Ewald formulation. We derive an optimal choice for the Fourier-space cutoff given a screening parameter $\eta$. We find that the number of vectors in Fourier space required to achieve a given accuracy scales with $\eta^3$. The proposed method can be used to determine computationally efficient parameters for Ewald sums, to assess the quality of Ewald-sum implementations, and to compare different implementations.Comment: 6 pages, 3 figures (Encapsulated PostScript), LaTe

Constructions of diagonal quartic and sextic surfaces with infinitely many rational points

In this note we construct several infinite families of diagonal quartic surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where $a,b,c,d\in\Z\setminus\{0\}$ with infinitely many rational points and satisfying the condition $abcd\neq \square$. In particular, we present an infinite family of diagonal quartic surfaces defined over \Q with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type $ax^6+by^6+cz^6+dw^i=0$, $i=2$, $3$, or $6$, with infinitely many rational points.Comment: revised version will appear in International Journal of Number Theor

On the Quantum Invariant for the Brieskorn Homology Spheres

We study an exact asymptotic behavior of the Witten-Reshetikhin-Turaev invariant for the Brieskorn homology spheres $\Sigma(p_1,p_2,p_3)$ by use of properties of the modular form following a method proposed by Lawrence and Zagier. Key observation is that the invariant coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We show that the Casson invariant is related to the number of the Eichler integrals which do not vanish in a limit $\tau\to N \in \mathbb{Z}$. Correspondingly there is a one-to-one correspondence between the non-vanishing Eichler integrals and the irreducible representation of the fundamental group, and the Chern-Simons invariant is given from the Eichler integral in this limit. It is also shown that the Ohtsuki invariant follows from a nearly modular property of the Eichler integral, and we give an explicit form in terms of the L-function.Comment: 26 pages, 2 figure

Constructing Elliptic Curves over Q(T)\mathbb{Q}(T) with Moderate Rank

We give several new constructions for moderate rank elliptic curves over $\mathbb{Q}(T)$. In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over $\mathbb{Q}$ using polynomials of degree two in $T$. While our method generates linearly independent points, we are able to show the rank is exactly 6 \emph{without} having to verify the points are independent. The method generalizes; however, the higher rank surfaces are not rational, and we need to check that the constructed points are linearly independent.Comment: 11 page