Algebra & Number Theory最新文献

We study instances of Beilinson–Tate conjectures for automorphic representations of ${\mathrm{PGSp}<!--FUNCTION\; APPLICATION-->}_6$ whose spin $L$-function has a pole at $s�=1$. We construct algebraic cycles of codimension 3 in the Siegel–Shimura variety of dimension 6 and we relate its regulator to the residue at $s�=1$ of the $L$-function of certain cuspidal forms of ${\mathrm{PGSp}<!--FUNCTION\; APPLICATION-->}_6$. Using the exceptional theta correspondence between the split group of type $G_2$ and ${\mathrm{PGSp}<!--FUNCTION\; APPLICATION-->}_6$ and assuming the nonvanishing of a certain archimedean integral, this allows us to confirm a conjecture of Gross and Savin on rank-$7$ motives of type $G_2$.

We find lower bounds on higher moments of the error term in the Chebotarev density theorem. Inspired by the work of Bellaïche, we consider general class functions and prove bounds which depend on norms associated to these functions. Our bounds also involve the ramification and Galois theoretical information of the underlying extension $L∕K$. Under a natural condition on class functions (which appeared in earlier work), we obtain that those moments are at least Gaussian. The key tools in our approach are the application of positivity in the explicit formula followed by combinatorics on zeros of Artin $L$-functions (which generalize previous work), as well as precise bounds on Artin conductors.

We formulate an analogue of the Breuil–Mézard conjecture for the group of units of a central division algebra over a $p$-adic local field, and we prove that it follows from the conjecture for ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$. To do so we construct a transfer of inertial types and Serre weights between the maximal compact subgroups of these two groups, in terms of Deligne–Lusztig theory, and we prove its compatibility with mod $p$ reduction, via the inertial Jacquet–Langlands correspondence and certain explicit character formulas. We also prove analogous statements for $\ell$-adic coefficients.

We prove the Pappas–Rapoport conjecture on the existence of canonical integral models of Shimura varieties with parahoric level structure in the case where the Shimura variety is defined by a torus. As an important ingredient, we show, using the Bhatt–Scholze theory of prismatic $F$-crystals, that there is a fully faithful functor from $\mathcal{𝒢}$-valued crystalline representations of $\mathrm{Gal}<!--FUNCTION\; APPLICATION-->(\overline{K}∕K)$ to $\mathcal{𝒢}$-shtukas over $\mathrm{Spd}<!--FUNCTION\; APPLICATION-->({\mathcal{𝒪}}_K)$, where $\mathcal{𝒢}$ is a parahoric group scheme over ${\mathbb{Z}}_p$ and ${\mathcal{𝒪}}_K$ is the ring of integers in a $p$-adic field $K$.

We study the index of log Calabi–Yau pairs $(X,B)$ of coregularity 0. We show that $2\lambda (K_X�+�B)�\sim 0$, where $\lambda$ is the Weil index of $(X,B)$. This is in contrast to the case of klt Calabi–Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends to the context of generalized log Calabi–Yau pairs, semi-log canonical pairs, and isolated log canonical singularities of coregularity 0. As a consequence, we show that the index of a variety appearing in the Gross–Siebert program or in the Kontsevich–Soibelman program is at most $2$. Finally, we discuss applications to Calabi–Yau varieties endowed with a finite group action, including holomorphic symplectic varieties endowed with a purely nonsymplectic automorphism.

An arc space of an affine cone over a projective toric variety is known to be nonreduced in general. It was demonstrated recently that the reduced scheme structure of arc spaces is very meaningful from algebro-geometric, representation-theoretic and combinatorial points of view. In this paper we develop a general machinery for the description of the reduced arc spaces of affine cones over toric varieties. We apply our techniques to a number of classical cases and explore some connections with representation theory of current algebras.

Let $k$ be a positive integer. We show that, as $n$ goes to infinity, almost every entry of the character table of $S_n$ is divisible by $k$. This proves a conjecture of Miller.

We establish a geometrization of the Breuil–Mézard conjecture for potentially Barsotti–Tate representations, as well as of the weight part of Serre’s conjecture, for moduli stacks of two-dimensional mod $p$ representations of the absolute Galois group of a $p$-adic local field. These results are first proved for the stacks of our earlier papers, and then transferred to the stacks of Emerton and Gee by means of a comparison of versal rings.

We prove vanishing results for the coherent cohomology of the good reduction modulo $p$ of the Siegel modular variety with coefficients in some automorphic bundles. We show that for an automorphic bundle with highest weight $\lambda$ near the walls of the antidominant Weyl chamber, there is an integer $e�\ge 0$ such that the cohomology is concentrated in degrees $[0,e]$. The accessible weights with our method are not necessarily regular and not necessarily $p$-small. Since our method is technical, we also provide an algorithm written in SageMath that computes explicitly the vanishing results.

Let $X$ be a K3 surface over a number field. We prove that $X$ has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar, Shankar and Tang to the case where $X$ has bad reduction. We prove a similar result for generically ordinary nonisotrivial families of K3 surfaces over curves over ${\overline{\mathbb{𝔽}}}_p$ which extends previous work of Maulik, Shankar and Tang. As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties.