Algebra & Number Theory最新文献

We prove that the Galois pseudorepresentation valued in the mod $p^n$ cuspidal Hecke algebra for $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(2)$ over a totally real number field $F$, of parallel weight $1$ and level prime to $p$, is unramified at any place above $p$. The same is true for the noncuspidal Hecke algebra at places above $p$ whose ramification index is not divisible by $p-1$. A novel geometric ingredient, which is also of independent interest, is the construction and study, in the case when $p$ ramifies in $F$, of generalised $\Theta$-operators using Reduzzi and Xiao’s generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights.

Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu–Farkas strong maximal rank conjecture, in genus $22$ and $23$. This constitutes a major step forward in Farkas’ program to prove that the moduli spaces of curves of genus $22$ and $23$ are of general type. Our techniques involve a combination of the Eisenbud–Harris theory of limit linear series, and the notion of linked linear series developed by Osserman.

We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms in $2^n$ variables over function fields of transcendence degree $n�\ge 2$ over an algebraically closed field of characteristic $\ne 2$. Our construction involves the generalized Kummer varieties considered by Borcea and by Cynk and Hulek as well as new results on the nontriviality of unramified cohomology of products of elliptic curves over discretely valued fields.

We provide a simple procedure for resolving, in characteristic 0, singularities of a variety $X$ embedded in a smooth variety $Y$ by repeatedly blowing up the worst singularities, in the sense of stack-theoretic weighted blowings up. No history, no exceptional divisors, and no logarithmic structures are necessary to carry this out; the steps are explicit geometric operations requiring no choices; and the resulting algorithm is efficient.

A similar result was discovered independently by McQuillan (2020).

We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove nontrivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters we show that there are infinitely many primes of the form $a^2�+b^8$.

We combine two of Igusa’s conjectures with recent semicontinuity results by Mustaţă and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in terms of degrees and dimensions only. We provide evidence consisting partly of adaptations of already known results about Igusa’s conjecture on exponential sums, but also some new evidence like for all polynomials in up to $4$ variables. We show that, in turn, these bounds imply consequences for Igusa’s (strong) monodromy conjecture. The bounds are related to estimates for major arcs appearing in the circle method for local-global principles.

Given a finite group $G$ and a prime $p$, let ${\mathcal{𝒜}}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. The group $G$ satisfies the Quillen dimension property at $p$ if ${\mathcal{𝒜}}_p(G)$ has nonzero homology in the maximal possible degree, which is the $p$-rank of $G$ minus $1$. For example, D. Quillen showed that solvable groups with trivial $p$-core satisfy this property, and later, M. Aschbacher and S. D. Smith provided a list of all $p$-extensions of simple groups that may fail this property if $p$ is odd. In particular, a group $G$ with this property satisfies Quillen’s conjecture: $G$ has trivial $p$-core and the poset ${\mathcal{𝒜}}_p(G)$ is not contractible.

In this article, we focus on the prime $p�=2$ and prove that the $2$-extensions of finite simple groups of exceptional Lie type in odd characteristic satisfy the Quillen dimension property, wit

We formulate and prove the weight part of Serre’s conjecture for three-dimensional mod $p$ Galois representations under a genericity condition when the field is unramified at $p$. This removes the assumption made previously that the representation be tamely ramified at $p$. We also prove a version of Breuil’s lattice conjecture and a mod $p$ multiplicity one result for the cohomology of $U(3)$-arithmetic manifolds. The key input is a study of the geometry of the Emerton–Gee stacks using the local models we introduced previously (2023).

A flag positroid of ranks $r�:=�(r_1�<�\cdots �<r_k)$ on $[n]$ is a flag matroid that can be realized by a real $r_k�\times �n$ matrix $A$ such that the $r_i�\times r_i$ minors of $A$ involving rows $1,2,\dots <!--FUNCTION\; APPLICATION-->,r_i$ are nonnegative for all $1�\le �i�\le �k$. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when $r�:=�(a,a�+1,\dots <!--FUNCTION\; APPLICATION-->,b)$ is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety ${\mathrm{TrFl}<!--FUNCTION\; APPLICATION-->}_{r,n}^{\ge 0}$ equals the nonnegative flag Dressian ${\mathrm{FlDr}<!--FUNCTION\; APPLICATION-->}_{r,n}^{\ge 0}$, and that the points $\mu �=�({\mu }_a,\dots <!--FUNCTION\; APPLICATION-->,{\mu }_{}$

We establish functoriality of higher Coleman theory for certain unitary Shimura varieties and use this to construct a $p$-adic analytic function interpolating unitary Friedberg–Jacquet periods.