Algebra & Number Theory最新文献

For a smooth affine group scheme $G$ over the ring of $p$-adic integers ${\mathbb{Z}}_p$ and a cocharacter $\mu$ of $G$, we study $G$-$\mu$-displays over the prismatic site of Bhatt and Scholze. In particular, we obtain several descent results for them. If $G�={\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$, then our $G$-$\mu$-displays can be thought of as Breuil–Kisin modules with some additional conditions. The relation between our $G$-$\mu$-displays and prismatic $F$-gauges introduced by Drinfeld and Bhatt–Lurie is also discussed.

In fact, our main results are formulated and proved for smooth affine group schemes over the ring of integers ${\mathcal{𝒪}}_E$ of any finite extension $E$ of ${\mathbb{Q}}_p$ by using ${\mathcal{𝒪}}_E$-prisms, which are ${\mathcal{𝒪}}_E$-analogues of prisms.

We prove that the Torelli, Prym and spin-Torelli morphisms, as well as covering maps between moduli stacks of smooth projective curves, cannot be deformed. The proofs use properties of the Fujita decomposition of the Hodge bundle of families of curves.

We establish the arithmetic Siegel–Weil formula on the modular curve ${\mathcal{𝒳}}_0(N)$ for arbitrary level $N$, i.e., we relate the arithmetic degrees of special cycles on ${\mathcal{𝒳}}_0(N)$ to the derivatives of Fourier coefficients of a genus-2 Eisenstein series. We prove this formula by a precise identity between the local arithmetic intersection numbers on the Rapoport–Zink space associated to ${\mathcal{𝒳}}_0(N)$ and the derivatives of local representation densities of quadratic forms. When $N$ is odd and square-free, this gives a different proof of the main results in work of Sankaran, Shi and Yang. This local identity is proved by relating it to an identity in one dimension higher, but at hyperspecial level.

We prove a power saving upper bound for the sum of Fourier coefficients ${\rho }_f(\cdot )$ of a fixed cubic metaplectic cusp form $f$ over primes. Our result is the cubic analogue of a celebrated 1990 theorem of Duke and Iwaniec, and the cuspidal analogue of a theorem due to the author and Radziwiłł for the bias in cubic Gauss sums.

The proof has two main inputs, both of independent interest. Firstly, we prove a new large sieve estimate for a bilinear form whose kernel function is ${\rho }_f(\cdot )$. The proof of the bilinear estimate uses a number field version of circle method due to Browning and Vishe, Voronoi summation, and Gauss–Ramanujan sums. Secondly, we use Voronoi summation and the cubic large sieve of Heath-Brown to prove an estimate for a linear form involving ${\rho }_f(\cdot )$. Our linear estimate overcomes a bottleneck occurring at level of distribution $\frac{2}{3}$.

The core of an ideal is defined as the intersection of all of its reductions. We provide an explicit description for the core of a monomial ideal $I$ satisfying certain residual conditions, showing that $\mathrm{core}<!--FUNCTION\; APPLICATION-->(I)$ coincides with the largest monomial ideal contained in a general reduction of $I$. We prove that the class of lex-segment ideals satisfies these residual conditions and study the core of lex-segment ideals generated in one degree. For monomial ideals that do not necessarily satisfy the residual conditions and that are generated in one degree, we conjecture an explicit formula for the core, and make progress towards this conjecture.

We establish a rigid-analytic analog of the Pila–Wilkie counting theorem, giving subpolynomial upper bounds for the number of rational points in the transcendental part of a ${\mathbb{Q}}_p$-analytic set and the number of rational functions in a ${\mathbb{𝔽}}_q((t))$-analytic set. For $\mathbb{Z}((t))$-analytic sets, we prove such bounds uniformly for the specialization to every nonarchimedean local field.

We generalise a result of Montgomery and Vaughan regarding exponential sums with multiplicative coefficients to the setting of Weyl sums. As applications, we establish a joint equidistribution result for roots of polynomial congruences and polynomial values which generalises a result of Hooley. We also obtain some new results for mixed character sums.

We study the harmonically weighted one-level density of low-lying zeros of $L\text{ -}$functions in the family of holomorphic newforms of fixed even weight $k$ and prime level $N$ tending to infinity. For this family, Iwaniec, Luo and Sarnak proved that the Katz–Sarnak prediction for the one-level density holds unconditionally when the support of the Fourier transform of the implied test function is contained in $(-3∕2,3∕2)$. This result was improved by Ricotta–Royer, who increased the admissible support for $k�\ge 4$ in a way that is asymptotically as good as the best known GRH result. We extend the admissible support for all $k�\ge 2$ to $(-{\Theta }_k,{\Theta }_k)$, where ${\Theta }_2�=1.866\dots <!--FUNCTION\; APPLICATION-->$ and ${\Theta }_k$ tends monotonically to $2$ asymptotically five times faster than what was previously known. The main novelty in our analysis is the use of zero-density estimates for Dirichlet $L$-functions.

The gonality of a smooth geometrically connected curve over a field $k$ is the smallest degree of a nonconstant $k$-morphism from the curve to the projective line. In general, the gonality of a curve of genus $g�\ge 2$ is at most $2g�-2$. Over finite fields, a result of F. K. Schmidt from the 1930s can be used to prove that the gonality is at most $g�+1$. Via a mixture of geometry and computation, we improve this bound: for a curve of genus $g�\ge 5$ over a finite field, the gonality is at most $g$. For genus $g�=3$ and $g�=4$, the same result holds with exactly $217$ exceptions: there are two curves of genus $4$ and gonality $5$, and $215$ curves of genus $3$ and gonality $4$. The genus-$4$ examples were found in other papers, and we reproduce their equations here; in supplementary material, we provide equations for the genus-$3$ examples.