This article studies uniform stabilization and social optimality for linear quadratic (LQ) mean field control problems with multiplicative noise, where agents are coupled via dynamics individual costs. The state weights in cost functionals not limited to be positive semidefinite. leads an indefinit...

We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suitable algorithm for computing the class number of an arbitrary function field. The ideas underlying the clas...

We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in n variables. The main tool is combinatorial polarization, and the approach is applicable even when n! is not invertible in the underlying field.

Let φ be a Drinfeld module of rank 2 over the field of rational functions F = Fq(T ), with EndF̄ (φ) = Fq[T ]. Let K be a fixed imaginary quadratic field over F and d a positive integer. For each prime p of good reduction for φ, let πp(φ) be a root of the characteristic polynomial of the Frobenius endomorphism of φ over the finite field Fq[T ]/p. Let Πφ(K; d) be the number of primes p of degree ...

Let F be a number field with odd class number, and suppose that k/F is a quadratic extension. We will deal with the problem of constructing parts of the Hilbert 2-class field towers of such fields k.

In this paper, we study a class of risk-sensitive mean-field stochastic differential games. Under regularity assumptions, we use results from standard risk-sensitive differential game theory to show that the mean-field value of the exponentiated cost functional coincides with the value function of a Hamilton-Jacobi-Bellman-Fleming (HJBF) equation with an additional quadratic term. We provide an...

Hasse-Minkowski’s theorem asserts that a quadratic form over a number field k admits a nontrivial zero if it does over completions at all places of k. One could look for analogues of Hasse principle for function fields. Let k be a field of characteristic not 2 and Ω a set of discrete valuations of k. Let k̂v denote the completion of k at v. We say that k satisfies Hasse principle with respect to...

We determine all real quadratic number fields with 2-class field tower of length at most 1.

In this paper we characterise involutions that become metabolic over a quadratic field extension attained by adjoining a square root.

Motivated by the conjectures of “Mazur-Tate-Teitelbaum type” formulated in [BD1] and by the main result of [BD3], we describe a conjectural construction of a global point PK ∈ E(K), where E is a (modular) elliptic curve over Q of prime conductor p, and K is a real quadratic field satisfying suitable conditions. The point PK is constructed by applying the Tate p-adic uniformization of E to an ex...