Journal of Number Theory最新文献

Let L be a lattice. We exhibit algorithms for calculating Tits buildings and orbits of vectors in L for certain subgroups of the orthogonal group $O\left(L\right)$. We discuss how these algorithms can be applied to determine the configuration of boundary components in the Baily-Borel compactification of orthogonal modular varieties and to improve the performance of computer arithmetic of orthogonal modular forms.

We prove a conjecture of B. Gross and D. Prasad about determination of generic L-packets in terms of the analytic properties of the adjoint L-function for p-adic general even spin groups of semi-simple ranks 2 and 3. We also explicitly write the adjoint L-function for each L-packet in terms of the local Langlands L-functions for the general linear groups.

Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete “stick breaking”) subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior.

In this article we investigate the action of (ramified and unramified) Hecke operators on automorphic forms for the function field of the projective line defined over $F_q$ and for the group ${\mathrm{GL}}_2$. We first compute the dimension of the Hecke eigenspaces for every generator of the unramified Hecke algebra. Thus, we consider the ramification in a point of degree one and explicitly describe the action of certain ramified Hecke operators on automorphic forms. Moreover, we also compute the dimensions of its eigenspaces for those ramified Hecke operators. We finish the article considering more general ramifications, namely those one attached to a closed point of higher degree.

A conjecture recently stated by Flach and Morin relates the action of the monodromy on the Galois invariant part of the p-adic Beilinson–Hyodo–Kato cohomology of the generic fiber of a scheme defined over a DVR of mixed characteristic to (the cohomology of) its special fiber. We prove the conjecture in the case that the special fiber of the given arithmetic scheme is also a fiber of a geometric family over a curve in positive characteristic.

Over function fields of p-adic curves, we construct stably rational varieties in the form of homogeneous spaces of ${\mathrm{SL}}_n$ with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places fails for such varieties. The construction combines the Lichtenbaum duality and the degree 3 cohomological invariants of the stabilizers. We then establish a reciprocity obstruction which accounts for this failure of strong approximation. We show that this reciprocity obstruction to strong approximation is the only one for counterexamples we constructed, and also for classifying varieties of tori. We also show that this reciprocity obstruction to strong approximation is compatible with known results for tori. At the end, we explain how a similar point of view shows that the reciprocity obstruction to weak approximation is the only one for classifying varieties of tori over p-adic function fields.

In this article, we prove that for a convergent sequence of residually absolutely irreducible representations of the absolute Galois group of a number field F with coefficients in a domain, which admits a finite monomorphism from a power series ring over a p-adic integer ring, the set of places of F where some of the representations ramifies has density zero. Using this, we extend a result of Das–Rajan to such convergent sequences. We also establish a strong multiplicity one theorem for big Galois representations.

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of λ and μ, where $\mu \ge C\log \lambda$, such that intervals $[\lambda ,\lambda +\mu ]$ do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in $R^2$ that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in $R^2$. Specifically, we establish the existence of annuli $\{x\in R^2:\lambda \le |x{|}^2\le \lambda +\kappa \}$ with arbitrarily large λ and $\kappa \ge C{\lambda }^s$ for $0<s<\frac{1}{4}$, satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold $s=\frac{1}{4}$. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in $R^3$.