arXiv · 1005.3448
We show that for any even positive integer d there exist polynomials x and y with integer coefficients such that deg(x) = 2d, deg(y) = 3d and deg(x^3 - y^2) = d + 5.
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Andrej Dujella. 2010-07-09. On Hall's conjecture. https://doi.org/10.4064/aa147-4-5
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We study the pair correlation between zeros of a shifted auxiliary $ L $-function attached to a non-CM newform, the scale of which is a fixed constant. We prove an unconditional asymptotic result for the pair correlation and introduce a simplicity hypothesis for the zeros of this function, which if true means that multiple zeros of the original $ L $-function cannot be separated by the same fixed distance. Our results provide macroscopic information in contrast to the pair correlation of the original $ L $-function which is of microscopic nature.
We obtain an asymptotic formula with a power-saving error term for counting the integer points $(a,b,c,d)$ in an expanding box that satisfy the determinant equation $x_1x_2-x_3x_4 =r$ for $r \neq 0 $ with two of entries to be prime. Finally, these estimates are applied to evaluate mixed fourth moments of Dirichlet character sums over integers and primes, yielding non-trivial bounds. The method involves the Poisson summation formula and the estimation for the average of the sums of the Kloosterman fractions over primes.
Given a maximal order $\mathcal{D}$ of a central division algebra $D$ over a global function field $F$, we prove an explicit sufficient condition for moduli stacks of $\mathcal{D}^\times$-shtukas to be proper over a finite field (modulo a suitable central action) in terms of the \emph{local invariants} of $D$ and \emph{bounds}. Our proof is a refinement of E.~Lau's result (Duke Math. J. \textbf{140} (2007)), which showed the properness of the \emph{leg morphism} (or \emph{characteristic morphism}) away from the ramification locus of $D$. %, by carefully measuring the contribution of ``ramified legs''. We also establish non-emptiness of Newton and Kottwitz--Rapoport strata for moduli stacks of $\mathcal{B}^\times$-shtukas, where $\mathcal{B}$ is a maximal order of a central simple algebra over $F$.