Algebra & Number Theory最新文献

We investigate the problem of computing Tamagawa numbers of CM tori. This problem arises naturally from the problem of counting polarized abelian varieties with commutative endomorphism algebras over finite fields, and polarized CM abelian varieties and components of unitary Shimura varieties in the works of Achter, Altug, Garcia and Gordon and of Guo, Sheu and Yu, respectively. We make a systematic study on Galois cohomology groups in a more general setting and compute the Tamagawa numbers of CM tori associated to various Galois CM fields. Furthermore, we show that every (positive or negative) power of $2$ is the Tamagawa number of a CM tori, proving the analogous conjecture of Ono for CM tori.

We prove various explicit formulas concerning $p$-rank of $p$-coverings of pointed semistable curves over discrete valuation rings. In particular, we obtain a full generalization of Raynaud’s formula for $p$-rank of fibers over nonmarked smooth closed points in the case of arbitrary closed points. As an application, for abelian $p$-coverings, we give an affirmative answer to an open problem concerning boundedness of $p$-rank asked by Saïdi more than twenty years ago.

We prove a relative decidability result for perfectoid fields. This applies to show that the fields ${\mathbb{Q}}_p(p^{1∕p^{\infty }�})$ and ${\mathbb{Q}}_p({\zeta }_{p^{\infty }})$ are (existentially) decidable relative to the perfect hull of ${\mathbb{𝔽}}_p((t))$ and ${\mathbb{Q}}_p^{ab}$ is (existentially) decidable relative to the perfect hull of ${\overline{\mathbb{𝔽}}}_p((t))$. We also prove some unconditional decidability results in mixed characteristic via reduction to characteristic $p$.

We prove that the $k$-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval $(x,x�+�H]$ matches the corresponding Gaussian moment, as long as $H�\ll �x∕{(\log <!--FUNCTION\; APPLICATION-->x)}^{2k^2+2+o(1)�}$ and $H$ tends to infinity with $x$. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals $(x,x�+�H]$ with $H�\ll �X∕{(\log <!--FUNCTION\; APPLICATION-->X)}^{W(X)}$ tending to infinity with $X$, where $x$ is uniformly chosen from $\{1,2,\dots <!--FUNCTION\; APPLICATION-->,X\}$, and $W(X)$ tends to infinity with $X$ arbitrarily slowly. This makes some initial progress on a recent question of Harper.

Let $M$ be a representation of an acyclic quiver $Q$ over an infinite field $k$. We establish a deterministic algorithm for computing the Harder–Narasimhan filtration of $M$. The algorithm is polynomial in the dimensions of $M$, the weights that induce the Harder–Narasimhan filtration of $M$, and the number of paths in $Q$. As a direct application, we also show that when $k$ is algebraically closed and when $M$ is unstable, the same algorithm produces Kempf’s maximally destabilizing one parameter subgroups for $M$.

We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of “tame” differential fields. We state several characterizations and exhibit plenty of examples and applications. Our results strongly indicate that differentially large fields will play a key role in differential field arithmetic. For instance, we characterize differential largeness in terms of being existentially closed in their power series field (furnished with natural derivations), we give explicit constructions of differentially large fields in terms of iterated powers series, we prove that the class of differentially large fields is elementary, and we show that differential largeness is preserved under algebraic extensions, therefore showing that their algebraic closure is differentially closed.

Nous établissons le principe de Hasse et l’approximation faible pour certains espaces homogènes de ${\mathrm{SL}<!--FUNCTION\; APPLICATION-->}_m$ à stabilisateur géométrique nilpotent de classe 2, construits par Borovoi et Kunyavskii. Ces espaces homogènes vérifient donc une conjecture de Colliot-Thélène concernant l’obstruction de Brauer–Manin pour les variétés géométriquement rationnellement connexes.

We establish the Hasse principle and the weak approximation property for certain homogeneous spaces of ${\mathrm{SL}<!--FUNCTION\; APPLICATION-->}_m$ whose geometric stabilizer is of nilpotency class 2, which were constructed by Borovoi and Kunyavskii. These homogeneous spaces verify thus a conjecture of Colliot-Thélène on the Brauer–Manin obstruction for geometrically rationally connected varieties.

We study projective irreducible symplectic orbifolds of dimension four that are deformations of partial resolutions of quotients of hyperkähler manifolds of $K{3}^{[2]}$-type by symplectic involutions; we call them orbifolds of Nikulin type. We first classify those projective orbifolds that are really quotients, by describing all families of projective fourfolds of $K{3}^{[2]}$-type with a symplectic involution and the relation with their quotients, and then study their deformations. We compute the Riemann–Roch formula for Weil divisors on orbifolds of Nikulin type and using this we describe the first known locally complete family of singular irreducible symplectic varieties as double covers of special complete intersections $(3,4)$ in ${\mathbb{P}}^6$.

This paper is devoted to a weighted version of the one-level density of the nontrivial zeros of $L$-functions, tilted by a power of the $L$-function evaluated at the central point. Assuming the Riemann hypothesis and the ratio conjecture, for some specific families of $L$-functions, we prove that the same structure suggested by the density conjecture also holds in this weighted investigation, if the exponent of the weight is small enough. Moreover, we speculate about the general case, conjecturing explicit formulae for the weighted kernels.