Algebra & Number Theory最新文献

We construct infinitesimal invariants of thickened one dimensional cycles in three dimensional space, which are the simplest cycles that are not in the Milnor range. This generalizes Park’s work on the regulators of additive cycles. The construction also allows us to prove the infinitesimal version of the strong reciprocity conjecture for thickenings of all orders. Classical analogs of our invariants are based on the dilogarithm function and our invariant could be seen as their infinitesimal version. Despite this analogy, the infinitesimal version cannot be obtained from their classical counterparts through a limiting process.

The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups — the usual étale fundamental group ${\pi }_1^{\text{ ét}}$ defined in SGA1 and the more general ${\pi }_1^{\mathrm{SGA3}<!--FUNCTION\; APPLICATION-->}$. It controls local systems in the pro-étale topology and leads to an interesting class of “geometric coverings” of schemes, generalizing finite étale coverings.

We prove exactness of the fundamental sequence for the pro-étale fundamental group of a geometrically connected scheme $X$ of finite type over a field $k$, i.e., that the sequence

is exact as abstract groups and the map ${\pi }_1^{\text{ proét}}(X_{\overline{k}})�\to {\pi }_1^{\text{ proét}}(X)$ is a topological embedding.

On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group.

We construct an analogue of the classical descent theory of Colliot-Thélène and Sansuc in which algebraic tori are replaced with finite supersolvable groups. As an application, we show that rational points are dense in the Brauer–Manin set for smooth compactifications of certain quotients of homogeneous spaces by finite supersolvable groups. For suitably chosen homogeneous spaces, this implies the existence of supersolvable Galois extensions of number fields with prescribed norms, generalising work of Frei, Loughran and Newton.

Let $K$ be the function field of a curve $C$ over a $p$-adic field $k$. We prove that, for each $n,d�\ge 1$ and for each hypersurface $Z$ in ${\mathbb{P}}_K^n$ of degree $d$ with $d^2�\le �n$, the second Milnor $K$-theory group of $K$ is spanned by the images of the norms coming from finite extensions $L$ of $K$ over which $Z$ has a rational point. When the curve $C$ has a point in the maximal unramified extension of $k$, we generalize this result to hypersurfaces $Z$ in ${\mathbb{P}}_K^n$ of degree $d$ with $d�\le �n$.

We prove a Künneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic $p�>0$ for various coefficients, including finite discrete rings, algebraic field extensions $E�\supset {\mathbb{Q}}_{\ell }$, $\ell \ne p$, and their rings of integers ${\mathcal{𝒪}}_E$. We also consider a variant for ind-constructible sheaves which applies to the cohomology of moduli stacks of shtukas over global function fields.

We give a self-contained treatment of finite group quotients of admissible (formal) schemes and adic spaces that are locally topologically finite type over a locally strongly noetherian adic space.

Let $F$ be a $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(3)$ Hecke–Maass cusp form of prime level $P_1$ and let $f$ be a $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(2)$ Hecke–Maass cuspform of prime level $P_2$. We will prove a subconvex bound for the $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(3)�\times \mathrm{GL}<!--FUNCTION\; APPLICATION-->(2)$ Rankin–Selberg $L$-function $L(s,F�\times �f)$ in the level aspect for certain ranges of the parameters $P_1$ and $P_2$.

We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new rational functions, and thus also the local zeta functions under consideration, enjoy a self-reciprocity property, expressed in terms of a functional equation upon inversion of variables. We establish a conjecture of Grunewald, Segal, and Smith on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups.

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.