Journal of Number Theory最新文献

We derive a local formula for the parity of the $2^{\infty }$-Selmer rank of Jacobians of curves of genus 2 or 3 which admit an unramified double cover. We give an explicit example to show how this local formula gives rank parity predictions against which the 2-parity conjecture may be tested. Our results yield applications to the parity conjecture for semistable curves of genus 3.

In this paper, we study properties of factorization in orders of a PID via the computation of algebraic invariants that measure the failure of unique factorization. The focus is on the numerical semigroup rings over a finite field and the orders of imaginary quadratic fields with class number 1. We also give a complete description of the class group structure of those rings.

For a real-valued sequence ${\left(x_n\right)}_{n=1}^{\infty }$, denote by $S_N\left(\ell \right)$ the number of its first N fractional parts lying in a random interval of size $\ell :=L/N$, where $L=o\left(N\right)$ as $N\to \infty$. We study the variance of $S_N\left(\ell \right)$ (the number variance) for sequences of the form $x_n=\alpha a_n$, where ${\left(a_n\right)}_{n=1}^{\infty }$ is a sequence of distinct integers. We show that if the additive energy of the sequence ${\left(a_n\right)}_{n=1}^{\infty }$ is bounded from above by $N^{3-\epsilon }/L^2$ for some $\epsilon >0$, then for almost all α, the number variance is asymptotic to L (Poissonian number variance). This holds in particular for the sequence $x_n=\alpha n^d,d\ge 2$ whenever $L=N^{\beta }$ with $0\le \beta <1/2$.

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function ${\sigma }_{\alpha }\left(n\right):={\sum }_{d|n}{d}^{\alpha }$. We obtain an exact identity relating the Dirichlet series $\zeta \left(s\right)\zeta (s-\alpha )$ and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip.

In this paper we prove that polynomials $F(x_1,\cdots ,x_n)\in Z[x_1,\cdots ,x_n]$ of degree $d\ge 3$, satisfying certain hypotheses, take on the expected density of $(d-1)$-free values. This extends the authors' earlier result in [14] where a different method implied the similar statement for polynomials of degree $d\ge 5$.

Working over a base number field K, we study the attractive question of Zariski non-density for $(D,S)$-integral points in $O_f\left(x\right)$ the forward f-orbit of a rational point $x\in X\left(K\right)$. Here, $f:X\to X$ is a regular surjective self-map for X a geometrically irreducible projective variety over K. Given a non-zero and effective f-quasi-polarizable Cartier divisor D on X and defined over K, our main result gives a sufficient condition, that is formulated in terms of the f-dynamics of D, for non-Zariski density of certain dynamically defined subsets of $O_f\left(x\right)$. For the case of $(D,S)$-integral points, this result gives a sufficient condition for non-Zariski density of integral points in $O_f\left(x\right)$. Our approach expands on that of Yasufuku, [13], building on earlier work of Silverman [11]. Our main result gives an unconditional form of the main results of [13]; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in [10] and expanded upon in [3] and [6].

In this paper, we study the asymptotic distribution of coefficients of general L-functions over arithmetic progressions without the Ramanujan conjecture. As an application, we consider the high mean of Fourier coefficients of holomorphic forms or Maass forms for $\Gamma =\mathrm{SL}(2,Z)$ over arithmetic progressions, and improve the results of Jiang and Lü [10]. Our new results remove the restriction to prime module and improve the interval length of module q.

We give asymptotics for shifted convolutions of the form$\sum _{n<X}\frac{{\sigma }_{2u}(n,\chi ){\sigma }_{2v}(n+k,\psi )}{n^{u+v}}$ for nonzero complex numbers $u,v$ and nontrivial Dirichlet characters $\chi ,\psi$. We use the technique of automorphic regularization to find the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable. The error term we obtain is in some cases smaller than what the method we use typically yields.

For all ${\alpha }_1,{\alpha }_2\in (1,2)$ with $1/{\alpha }_1+1/{\alpha }_2>5/3$, we show that the number of pairs $(n_1,n_2)$ of positive integers with $N=\lfloor n_1^{{\alpha }_1}\rfloor +\lfloor n_2^{{\alpha }_2}\rfloor$ is equal to $\Gamma (1+1/{\alpha }_1)\Gamma (1+1/{\alpha }_2)\Gamma {(1/{\alpha }_1+1/{\alpha }_2)}^{-1}{N}^{1/{\alpha }_1+1/{\alpha }_2-1}+o\left(N^{1/{\alpha }_1+1/{\alpha }_2-1}\right)$ as $N\to \infty$, where Γ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when ${\alpha }_1,{\alpha }_2\in (1,2)$ lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the p-th Fermat curve where 2 is a quadratic residue modulo p, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.