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Register a new userAveraging over Heegner points in the hyperbolic circle problem
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For $Gamma={hbox{PSL}_2( {mathbb Z})}$ the hyperbolic circle problem aims to estimate the number of elements of the orbit $Gamma z$ inside the hyperbolic disc centered at $z$ with radius $cosh^{-1}(X/2)$. We show that, by averaging over Heegner points $z$ of discriminant $D$, Selbergs error term estimate can be improved, if $D$ is large enough. The proof uses bounds on spectral exponential sums, and results towards the sup-norm conjecture of eigenfunctions, and the Lindelof conjecture for twists of the $L$-functions attached to Maa{ss} cusp forms.
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Let $e(s)$ be the error term of the hyperbolic circle problem, and denote by $e_alpha(s)$ the fractional integral to order $alpha$ of $e(s)$. We prove that for any small $alpha>0$ the asymptotic variance of $e_alpha(s)$ is finite, and given by an explicit expression. Moreover, we prove that $e_alpha(s)$ has a limiting distribution.
Given a newform f, we extend Howards results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekovav{r}s extended Selmer groups attached to suitable big Galois representations and we formulate two-variable Iwasawa main conjectures both in the definite case and in the indefinite case. Moreover, in the definite case we propose refined conjectures `a la Greenberg on the vanishing at the critical points of (twists of) the L-functions of the modular forms in the Hida family of f living on the same branch as f.
The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_k(n)^2$, where $P_k(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of radius $sqrt{n}$ and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including $sum P_k(n)^2 e^{-n/X}$ and the Laplace transform $int_0^infty P_k(t)^2 e^{-t/X}dt$, in dimensions $k geq 3$. We also obtain main terms and power-saving error terms for the sharp sums $sum_{n leq X} P_k(n)^2$, along with similar results for the sharp integral $int_0^X P_3(t)^2 dt$. This includes producing the first power-saving error term in mean square for the dimension-three Gauss circle problem.
The Gauss circle problem concerns the difference $P_2(n)$ between the area of a circle of radius $sqrt{n}$ and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients $P_2(n)^2$, and prove that this series has meromorphic continuation to $mathbb{C}$. Using this series, we prove that the Laplace transform of $P_2(n)^2$ satisfies $int_0^infty P_2(t)^2 e^{-t/X} , dt = C X^{3/2} -X + O(X^{1/2+epsilon})$, which gives a power-savings improvement to a previous result of Ivic [Ivic1996]. Similarly, we study the meromorphic continuation of the Dirichlet series associated to the correlations $r_2(n+h)r_2(n)$, where $h$ is fixed and $r_2(n)$ denotes the number of representations of $n$ as a sum of two squares. We use this Dirichlet series to prove asymptotics for $sum_{n geq 1} r_2(n+h)r_2(n) e^{-n/X}$, and to provide an additional evaluation of the leading coefficient in the asymptotic for $sum_{n leq X} r_2(n+h)r_2(n)$.
If the $ell$-adic cohomology of a projective smooth variety, defined over a $frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$).
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