Abstract. We prove that the modular symbols appropriately normalized and ordered have a Gaussian distribution for all cofinite subgroups of SL 2 (R). We use spectral deformations to study the poles and the residues of Eisenstein series twisted by power of modular symbols.
SUMMARYFrost hardiness in Calluna vulgaris (L.) Hull, which had received ammonium nitrate applications in the field for 30 months, was assessed using scores of visible injury and measurements of the rate of total electrolyte leakage from excised shoots following controlled freezing treatments in the laboratory. There was good overall correlation between the two methods (Spearman correlation coefficient 0-77), but leakage measurements were more sensitive than injury scores to the efFects of nitrogen. Visible injury was not significantly altered by nitrogen supply (Kruskal-Wallis non-parametric test). Ion leakage was analyzed in different ways, using either calculations of the first-order rate coefficients or expressions of relative conductivity. These analyses produced similar results with respect to the effect of frost and nitrogen. Shoots of nitrogen-fertilized (40, 80 and 120 kg ha"^ yr'^) C. vulgaris sampled in October 1991 showed significantly {P < 0-05) less leakage after overnight frosts of -15 and -20 °C than did the water-treated controls. In October the temperature which killed 50% of the shoots (LT,,,,), derived from the leakage data, was raised by at least 4 °C by the highest nitrogen treatments compared with the control plants. Frost treatments to pot-grown C. vulgaris in November produced similar visible injury to attached and excised shoots from the same plants, both being significantly less damaged by a -18 °C frost after a 7-month exposure to an NaNOg mist solution (1-0 mM, pH 4-5) than were water-misted controls. Ammonium-mist treatments also marginally reduced frost injury, but the effects were not statistically significant. These results demonstrate that frost hardiness of C. vulgaris in the field can be assessed rapidly and accurately in the laboratory by analysis of electrolyte leakage from excised shoots.
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.
We study a variant of a problem considered by Dinaburg and Sinaȋ on the statistics of the minimal solution to a linear Diophantine equation. We show that the signed ratio between the Euclidean norms of the minimal solution and the coefficient vector is uniformly distributed modulo one. We reduce the problem to an equidistribution theorem of Anton Good concerning the orbits of a point in the upper half-plane under the action of a Fuchsian group.
The hyperbolic lattice point problem asks to estimate the size of the orbit Γz inside a hyperbolic disk of radius cosh −1 (X/2) for Γ a discrete subgroup of PSL2(R). Selberg proved the estimate O(X 2/3 ) for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for PSL2(Z). The result is that the error term can be improved to O(X 7/12+ε ). The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maaß cusp forms and estimates on spectral exponential sums. We also prove omega results for this averaging, consistent with the conjectural best error bound O(X 1/2+ε ). In the appendix the relevant exponential sum over the spectral parameters is investigated.
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