Algebra & Number Theory最新文献

This paper is an extension of Kim et al. (2020a), and we prove equidistribution theorems for families of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur’s invariant trace formula in terms of Shintani zeta functions in a uniform way. Several applications, including the vertical Sato–Tate theorem and low-lying zeros for standard $L$-functions of holomorphic Siegel cusp forms, are discussed. We also show that the “nongenuine forms”, which come from nontrivial endoscopic contributions by Langlands functoriality classified by Arthur, are negligible.

The equation $x^m�=0$ defines a fat point on a line. The algebra of regular functions on the arc space of this scheme is the quotient of $k[x,x^{\prime },x^{(2)},\dots <!--FUNCTION\; APPLICATION-->]$ by all differential consequences of $x^m�=0$. This infinite-dimensional algebra admits a natural filtration by finite-dimensional algebras corresponding to the truncations of arcs. We show that the generating series for their dimensions equals $m∕(1�-�mt)$. We also determine the lexicographic initial ideal of the defining ideal of the arc space. These results are motivated by the nonreduced version of the geometric motivic Poincaré series, multiplicities in differential algebra, and connections between arc spaces and the Rogers–Ramanujan identities. We also prove a recent conjecture put forth by Afsharijoo in the latter context.

We give a streamlined and effective proof of Ozaki’s theorem that any finite $p$-group $\Gamma$ is the Galois group of the $p$-Hilbert class field tower of some number field $F<!--FUNCTION\; APPLICATION-->$. Our work is inspired by Ozaki’s and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field ${k<!--FUNCTION\; APPLICATION-->}_0$ with class number prime to $p$. We construct $F<!--FUNCTION\; APPLICATION-->∕{k<!--FUNCTION\; APPLICATION-->}_0$ by a sequence of $\mathbb{Z}∕p$-extensions ramified only at finite tame primes and also give explicit bounds on $[F<!--FUNCTION\; APPLICATION-->�:{k<!--FUNCTION\; APPLICATION-->}_0]$ and the number of ramified primes of $F<!--FUNCTION\; APPLICATION-->∕{k<!--FUNCTION\; APPLICATION-->}_0$ in terms of $\#\Gamma$.

We calculate certain “wide moments” of central values of Rankin–Selberg $L$-functions $L\left(\pi �\otimes \mathrm{\Omega },\frac{1}{2}\right)$ where $\pi$ is a cuspidal automorphic representation of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_2$ over $\mathbb{Q}$ and $\mathrm{\Omega }$ is a Hecke character (of conductor $1$) of an imaginary quadratic field. This moment calculation is applied to obtain “weak simultaneous” nonvanishing results, which are nonvanishing results for different Rankin–Selberg $L$-functions where the product of the twists is trivial.

The proof relies on relating the wide moments of $L$-functions to the usual moments of automorphic forms evaluated at Heegner points using Waldspurger’s formula. To achieve this, a classical version of Waldspurger’s formula for general weight automorphic forms is derived, which might be of independent interest. A key input is equidistribution of Heegner points (with explicit error terms), together with nonvanishing results for certain period integrals. In particular, we develop a soft technique for obtaining the nonvanishing of triple convolution $L$-functions.

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.