Related papers: On some theta constants and class fields

Let L/F be a dihedral extension of degree 2p, where p is an odd prime. Let K/F and k/F be subextensions of L/F with degrees p and 2, respectively. Then we will study relations between the p-ranks of the class groups Cl(K) and Cl(k).

We extend the result of Blumberg and Mandell on K-theoretic Tate-Poitou duality at odd primes which serves as a spectral refinement of the classical arithmetic Tate-Poitou duality. The duality is formulated for the $K(1)$-localized…

We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and…

The Miller-Morita-Mumford classes associate to an oriented surface bundle $E\to B$ a class $\kappa_i(E) \in H^{2i}(B;\Z)$. In this note we define for each prime $p$ and each integer $i\geq 1$ a secondary characteristic class $\lambda_i(E)…

We consider the semiclassical asymptotics of the sum of negative eigenvalues of the three-dimensional Pauli operator with an external potential and a self-generated magnetic field $B$. We also add the field energy $\beta \int B^2$ and we…

If kappa is strongly compact, lambda > kappa is regular, then (2^{< lambda})^+ --> (lambda+eta)^2_theta holds for eta,theta<kappa.

We introduce `canonical' classes in the Selmer groups of certain Galois representations with a conjugate-symplectic symmetry. They are images of special cycles in unitary Shimura varieties, and defined uniquely up to a scalar. The…

We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $\phi(n)$ and the sum-of-proper-divisors function $s(n):=\sigma(n)-n$. We prove that the values $\phi(n)$, for $n\le x$, that are coprime to $p$…

Let $k$ be a real abelian number field and $p$ an odd prime not dividing $[k:\mathbb{Q}]$. For a natural number $d$, let $E_d$ denote the group of units of $k$ congruent to $1$ modulo $d$, $C_d$ the subgroup of $d$-circular units of $E_d$,…

We deduce a formula enumerating the isomorphism classes of extensions of a $\kp$-adic field $K$ with given ramification $e$ and inertia $f$. The formula follows from a simple group-theoretic lemma, plus the Krasner formula and an elementary…

Dirichlet's $L$-functions are natural extensions of the Riemann zeta function. In this paper we first give a brief survey of Ap\'ery-like series for some special values of the zeta function and certain $L$-functions. Then, we establish two…

Let $p$ be a prime. For $p=2$, the fields of values of the complex irreducible characters of finite groups whose degrees are not divisible by $p$ have been classified; for odd primes $p$, a conjectural classification has been proposed. In…

Let $p$ be a prime and let $S_2(\Gamma(p))$ be the space of weight $2$ cusp forms for the principal congruence subgroup $\Gamma(p)$. Then $\mathrm{SL}_2(\mathbb{F}_p)$ acts on $S_2(\Gamma(p))$ in a natural way. Around 1928, Hecke proved…

We determine the conditions under which singular values of multiple $\eta$-quotients of square-free level, not necessarily prime to~6, yield class invariants, that is, algebraic numbers in ring class fields of imaginary-quadratic number…

The possibility of getting a Radon-Nikodym type theorem and a Lebesgue-like decomposition for a non necessarily positive sesquilinear $\Omega$ form defined on a vector space $\mathcal D$, with respect to a given positive form $\Theta$…

The Euler--Kronecker constant of a number field $K$ is the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_K(s)$ at $s=1$. We study the distribution of the Euler--Kronecker constant…

Let $K/F$ be a CM extension satisfying the ordinary assumption for an odd prime $p$ and let $\psi$ be a finite order anticyclotomic Hecke character of $K$. When $K$ has a place above $p$ of degree one, we apply Urban's method and the…

Let $p^k m^2$ be an odd perfect number with special prime $p$. In this article, we provide an alternative proof for the biconditional that $\sigma(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We then give an application…

Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|theta) relative to p. We call each member B_theta of this…

We say that a function f defined on R or Qp has a well defined weak Mellin transform (or weak zeta integral) if there exists some function $M\_f(s)$ so that we have $Mell(\phi \star f,s) = Mell(\phi,s)M\_f(s)$ for all test functions $\phi$…