Related papers: The number of $S_4$ fields with given discriminant

The goal is to obtain an asymptotic formula for the number of quadratic extensions with bounded discriminant of a some quadratic number field with odd class number. This extends an already known result for Q.

Let $p$ be an odd prime number. We construct explicit uniformizers for the totally ramified extension $\mathbb{Q}_p(\zeta_{p^2},\sqrt[p]{p})$ of the field of $p$-adic numbers $\mathbb{Q}_p$, where $\zeta_{p^2}$ is a primitive $p^2$-th root…

We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree $p$ over a field $F$ with $\operatorname{char}(F)=p$ is a finite integer $d$ greater than 1 then the symbol length of $p$-algebras of…

Second order ordinary differential equations of the form $y'' = P(x,y) + 4 Q(x,y) y' + 6 R(x,y) y'^2 + 4 S(x,y) y'^3 + L(x,y) y'^4$ are considered and their point-expansions are constructed. Geometrical structures connected with these…

The Dushnik--Miller dimension of a poset $P$ is the least $d$ for which $P$ can be embedded into a product of $d$ chains. Lewis and Souza showed that the dimension of the divisibility order on the interval of integers $[N/\kappa, N]$ is…

For each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of $\mathbb{Q}$ of degree $d$, for any fixed…

We construct a maximal discrete extension of the paramodular group with a full level-2 structure. The corresponding Siegel variety parametrizes (birationally) the space of Kummer surfaces associated to (1,p)-polarized abelian surfaces with…

Assume that the field $K$ is $p$-rational. We study the freeness of the $\Lambda(G_{\infty,S})$-module $\mathcal{X}=\mathcal{H}^{ab}=\mathrm{\mathrm{G}al}(K_{S\cup S_p}/K_{\infty,S})^{ab}$. For numerical evidence to our result we consider…

Let $p$ be an odd prime and $\mathbb{F}_p$ be the prime field of order $p$. Consider a $2$-dimensional orthogonal group $G$ over $\mathbb{F}_p$ acting on the standard representation $V$ and the dual space $V^*$. We compute the invariant…

We will invest quite some computer power to find double octic threefolds that are connected to weight four modular forms.

Let $X\subseteq \mathbb{P}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in…

In this paper, we give an upper bound on the number of extensions of a triple to a quadruple for the Diophantine $m$-tuples with the property $D(4)$. We also confirm the conjecture of the uniqueness of such an extension in some special…

Let E be a real quadratic field with discriminant d and let p be an odd prime not dividing d. For \rho=1 or -1, we determine $\prod_{0<c<d, (d/c)=\rho} binomial coeff.{p-1}{\lfloor pc/d\rfloor}$ modulo p^2 in terms of Lucas numbers, the…

We show that for a large class of potential functions and big coupling constant $\lambda$ the Schr\"odinger cocycle over the expanding map $x\mapsto bx ~( \text{mod} 1)$ on $\mathbb{T}$ has a Lyapunov exponent $>(\log\lambda)/4$ for all…

In this paper we investigate the growth with respect to $p$ of dimensions of irreducible representations of a semisimple Lie algebra $\mathfrak{g}$ over $\overline{\mathbb{F}}_p$. More precisely, it is known that for $p\gg 0$, the…

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_{q}(x)$ whose class groups have elements of a fixed odd order. More precisely, for $q$, a power of an odd prime, and…

Let $p$ be a prime number. If a number field $k$ has at least one complex place, there are infinitely many $\mathbb{Z}_p$-extensions over $k$, and some authors studied the behavior of Iwasawa invariants of these $\mathbb{Z}_p$-extensions.…

In this paper, we present an algorithm to compute a basis of the space of algebraic modular forms on the maximal order of the definite quaternion algebra of discriminant $2$, and provide a database of such bases. One of our motivations is…

We give several new moduli interpretations of the fibers of certain Shimura varieties over several prime numbers. As a consequence (of our theorem 9.1) one obtains that for every prescribed odd prime characteristic $p$ every bounded…

We establish an equidistribution result for push-forwards of certain locally finite algebraic measures in the adelic extension of the space of lattices in the plane. As an application of our analysis we obtain new results regarding the…