Journal of Number Theory最新文献

In this paper, we show that ∞-adic multiple zeta values associated to the function field of an algebraic curve of higher genus over a finite field are not zero, under certain assumption on the gap sequence associated to the rational point ∞ on the given curve. Using arguments and results of Sheats and Thakur for the case of the projective line, we calculate the absolute values of power sums in the series defining multiple zeta values, and show that the calculation implies the non-vanishing result.

We prove finiteness and base change properties for analytic cohomology of families of L-analytic $({\phi }_L,{\Gamma }_L)$-modules parametrised by affinoid algebras in the sense of Tate. For technical reasons we work over a field K containing a period of the Lubin-Tate group, which allows us to describe analytic cohomology in terms of an explicit generalised Herr complex.

We investigate the rationality of Weil sums of binomials of the form $W_u^{K,s}={\sum }_{x\in K}\psi (x^s-ux)$, where K is a finite field whose canonical additive character is ψ, and where u is an element of $K^{\times }$ and s is a positive integer relatively prime to $\left|K^{\times }\right|$, so that $x\mapsto x^s$ is a permutation of K. The Weil spectrum for K and s, which is the family of values $W_u^{K,s}$ as u runs through $K^{\times }$, is of interest in arithmetic geometry and in several information-theoretic applications. The Weil spectrum always contains at least three distinct values if s is nondegenerate (i.e., if s is not a power of p modulo $\left|K^{\times }\right|$, where p is the characteristic of K). It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers. We show that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where $\left|K\right|=5$ and $s\equiv 3\left(\mathrm{mod}4\right)$.

For reductive groups G over a number field we discuss automorphic liftings of cohomological cuspidal irreducible automorphic representations π of $G\left(A\right)$ to irreducible cohomological automorphic representations of $H\left(A\right)$ for the quasi-split inner form H of G, and other inner forms as well. We show the existence of nontrivial weak global cohomological liftings in many cases, in particular for the case where G is anisotropic at the archimedean places. A priori, for these weak liftings we do not give a description of the precise nature of the corresponding local liftings at the ramified places, nor do we characterize the image of the lifting. For inner forms of the group $H=\mathrm{GSp}\left(4\right)$ however we address these finer questions. Especially, we prove the recent conjectures of Ibukiyama and Kitayama on paramodular newforms of square-free level.

We study minimal and toroidal compactifications of p-integral models of Hilbert modular varieties. We review the theory in the setting of Iwahori level at primes over p, and extend it to certain finer level structures. We also prove extensions to compactifications of recent results on Iwahori-level Kodaira–Spencer isomorphisms and cohomological vanishing for degeneracy maps. Finally we apply the theory to study q-expansions of Hilbert modular forms, especially the effect of Hecke operators at primes over p over general base rings.

In this paper, we give characterization of quadratic ε-canonical number system (ε−CNS) polynomials for all values $\epsilon \in [0,1)$. Our characterization provides a unified view of the well-known characterizations of the classical quadratic CNS polynomials ($\epsilon =0$) and quadratic SCNS polynomials ($\epsilon =1/2$). This result is a consequence of our new characterization results of ε-shift radix systems (ε−SRS) in the two-dimensional case and their relation to quadratic ε−CNS polynomials.

Let $F$ be the function field of a curve over an algebraically closed field with $\mathrm{char}\left(F\right)\ne 2,3$, and let $E/F$ be a non-isotrivial elliptic curve. Then for all finite extensions $K/F$ and all non-torsion points $P\in E\left(K\right)$, the $F$-normalized canonical height of P is bounded below by${\hat{h}}_E\left(P\right)\ge \frac{1}{10500\cdot h_F{\left(j_E\right)}^2\cdot {[K:F]}^2}.$

We describe the foundations of a Hecke theory for the orthogonal group $S{O}^{+}(2,n+2)$. In particular we consider the Hermitian modular group of degree 2 as a special example of $S{O}^{+}(2,4)$. As an application we show that the attached Maaß space is invariant under Hecke operators. This implies that the Eisenstein series belongs to the Maaß space. If the underlying lattice is even and unimodular, our approach allows us to reprove the explicit formula of its Fourier coefficients.