Journal of Number Theory最新文献

In this paper, we establish a crucial requirement for a number of the form n, having two prime factors p and q such that $(p,q)\equiv (1,3)\left(\mathrm{mod}8\right)$, to qualify as a congruent number. Specifically, we present congruence relations modulo 16 for the 2-part of the class number of the imaginary quadratic field $Q\left(\sqrt{-2pq}\right)$ when n is congruent.

For an isotropic subgroup H of a discriminant form D there exists a lift from modular forms for the Weil representation of the discriminant form $H^{\perp }/H$ to modular forms for the Weil representation of D. We determine a set of discriminant forms such that all modular forms for any discriminant form are induced from the discriminant forms in this set. Furthermore for any discriminant form in this set there exist modular forms that are not induced from smaller discriminant forms.

Let k be a real quadratic number field, and $k_{\infty }$ its cyclotomic $Z_2$-extension. We study the cyclicity of the Galois group $X_{\infty }^{\prime }$ over $k_{\infty }$ of the maximal abelian unramified 2-extension, in which all 2-adic primes of $k_{\infty }$ split completely. As consequence, we determine the complete list of real quadratic number fields for which $X_{\infty }^{\prime }$ is cyclic.

When $X_{\infty }^{\prime }$ is cyclic non-trivial, we give a new infinite family of real quadratic number fields, for which Greenberg's conjecture is valid.

We classify and enumerate all rational numbers with approximation constant at least $\frac{1}{3}$ using hyperbolic geometry. Rational numbers correspond to geodesics in the modular torus with both ends in the cusp, and the approximation constant measures how far they stay out of the cusp neighborhood in between. Compared to the original approach, the geometric point of view eliminates the need to discuss the intricate symbolic dynamics of continued fraction representations, and it clarifies the distinction between the two types of worst approximable rationals: (1) There is a plane forest of Markov fractions whose denominators are Markov numbers. They correspond to simple geodesics in the modular torus with both ends in the cusp. (2) For each Markov fraction, there are two infinite sequences of companions, which correspond to non-simple geodesics with both ends in the cusp that do not intersect a pair of disjoint simple geodesics, one with both ends in the cusp and one closed.

We extend the dictionary between Fontaine rings and p-adic functionnal analysis, and we give a refinement of the p-adic local Langlands correspondence for principal series representations of ${\text{GL}}_2\left(Q_p\right)$.

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 $s\equiv 3\left(\mathrm{mod}4\right)$.