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Abstract We describe an algorithm for computing, for all primes$$p \le X$$ $p\le X$, the trace of Frobenius atpof a hypergeometric motive over$$\mathbb {Q}$$ $Q$in time quasilinear inX. This involves computing the trace modulo$$p^e$$ $p^e$for suitablee; as in our previous work treating the case$$e=1$$ $e=1$, we combine the Beukers–Cohen–Mellit trace formula with average polynomial time techniques of Harvey and Harvey–Sutherland. The key new ingredient for$$e>1$$ $e>1$is an expanded version of Harvey’s “generic prime” construction, making it possible to incorporate certainp-adic transcendental functions into the computation; one of these is thep-adic Gamma function, whose average polynomial time computation is an intermediate step which may be of independent interest. We also provide an implementation in Sage and discuss the remaining computational issues around tabulating hypergeometricL-series. 

Abstract Given a prime powerqand$$n \gg 1$$ $n\gg 1$, we prove that every integer in a large subinterval of the Hasse–Weil interval$$[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$$ $[{(\sqrt{q}-1)}^{2n},{(\sqrt{q}+1)}^{2n}]$is$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some ordinary geometrically simple principally polarized abelian varietyAof dimensionnover$${\mathbb {F}}_q$$ $F_q$. As a consequence, we generalize a result of Howe and Kedlaya for$${\mathbb {F}}_2$$ $F_2$to show that for each prime powerq, every sufficiently large positive integer is realizable, i.e.,$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some abelian varietyAover$${\mathbb {F}}_q$$ $F_q$. Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse–Weil interval. A separate argument determines, for fixedn, the largest subinterval of the Hasse–Weil interval consisting of realizable integers, asymptotically as$$q \rightarrow \infty $$ $q\to \infty$; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if$$q \le 5$$ $q\le 5$, then every positive integer is realizable, and for arbitraryq, every positive integer$$\ge q^{3 \sqrt{q} \log q}$$ $\ge q^{3\sqrt{q}logq}$is realizable. 

Abstract We present efficient algorithms for counting points on a smooth plane quartic curve X modulo a prime p . We address both the case where X is defined over  $${\mathbb {F}}_p$$ F p and the case where X is defined over $${\mathbb {Q}}$$ Q and p is a prime of good reduction. We consider two approaches for computing $$\#X({\mathbb {F}}_p)$$ # X ( F p ) , one which runs in $$O(p\log p\log \log p)$$ O ( p log p log log p ) time using $$O(\log p)$$ O ( log p ) space and one which runs in $$O(p^{1/2}\log ^2p)$$ O ( p 1 / 2 log 2 p ) time using $$O(p^{1/2}\log p)$$ O ( p 1 / 2 log p ) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $$X/{\mathbb {Q}}$$ X / Q that compute $$\#X({\mathbb {F}}_p)$$ # X ( F p ) for good primes $$p\leqslant N$$ p ⩽ N in $$O(N\log ^3 N)$$ O ( N log 3 N ) time using O ( N ) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $${\mathbb {P}}^1$$ P 1 , which in combination with previous results addresses all curves of genus $$g\leqslant 3$$ g ⩽ 3 . Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest. 

Abstract We report on our results concerning the distribution of the geometric Picard ranks of K 3 surfaces under reduction modulo various primes. In the situation that $${\mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{K}}}}$$ rk Pic S K ¯ is even, we introduce a quadratic character, called the jump character, such that $${\mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{{\mathbb {F}}}}_{\!{{\mathfrak {p}}}}} > \mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{K}}}}$$ rk Pic S F ¯ p > rk Pic S K ¯ for all good primes at which the character evaluates to $$(-1)$$ ( - 1 ) . 

Abstract Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x , y , z and w ; let $$\mathcal{X}$$ be the generic element of the family of surfaces in ℙ given by \begin{equation*}X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2.\end{equation*} The surface $$\mathcal{X}$$ is a K3 surface over the function field ℚ( t ). In this paper, we explicitly compute the geometric Picard lattice of $$\mathcal{X}$$ , together with its Galois module structure, as well as derive more results on the arithmetic of $$\mathcal{X}$$ and other elements of the family X . 

Abstract The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ C . We develop and present a new technique to compute the expected value of tr(Θ C n ) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [ Rud10 ] and Chinis [ Chi16 ]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [ BDF + 16 ] and [ Zha ]. We extend [ BDF + 16 ] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L -functions L (1/2 + it , χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers.