Journal of Number Theory最新文献

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.

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.

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.

We give an axiomatic characterization of multiple Dirichlet series over the function field $F_q\left(T\right)$, generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature.

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b, the set$\{p⩽x:p\text{ prime, }\phantom{m}a\mathrm{mod}p\in 〈b〉\mathrm{mod}p\}$ has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs $(a,b)$$\#\{p⩽x:p\text{ prime, }\phantom{m}a\mathrm{mod}p\in 〈b〉\mathrm{mod}p\}\gg \frac{x}{{\log }^{2}x}.$ In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers $S=\{m_1,m_2,m_3\}$ such that