## What is/are Real Quadratic?

Real Quadratic - We obtain divisibility conditions on the multiplicative orders of elements of the form $\zeta + \zeta^{-1}$ in a finite field by exploiting a link to the arithmetic of real quadratic fields.^{[1]}One of the goals of this book is to enumerate all possible geometrical configurations of singularities, finite and infinite, of real quadratic differential systems.

^{[2]}Let [Formula: see text] be a positive square-free integer and [Formula: see text] be the fundamental unit of the real quadratic field [Formula: see text].

^{[3]}In the same spirit, we prove that for a given integer $t \geq 1$ with $t \equiv 0 \pmod {4}$, a positive proportion of fundamental discriminants $D > 0$ exist for which the class numbers of both the real quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D + t})$ are indivisible by $3$.

^{[4]}Using this, we compute slopes for weights in the center and near the boundary of weight space for certain real quadratic fields.

^{[5]}Illustration is given by some real quadratic fields.

^{[6]}Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra.

^{[7]}This is the first of a series of two papers in which we present a solution to Manin’s Real Multiplication program (Manin in: Laudal and Piene (eds) The Legacy of Niels Henrik Abel, Springer, Berlin, 2004) —an approach to Hilbert’s 12th problem for real quadratic extensions of $$\mathbb Q$$ Q —in positive characteristic, using quantum analogs of the modular invariant and the exponential function.

^{[8]}We also study the case of odd weights $k\geq 1$ newforms where the nebentypus is given by a real quadratic Dirichlet character.

^{[9]}For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL(2,Z)$PSL(2,\mathbb{Z})$-orbits of real quadratic fields.

^{[10]}We develop further the theory of q-deformations of real numbers introduced in [10] and [9] and focus in particular on the class of real quadratic irrationals.

^{[11]}In this paper we develop an effective procedure for expressing Stark units in real quadratic extensions of totally real fields as values of the Barnes multiple Gamma function at algebraic points.

^{[12]}We compare the homology of a congruence subgroup Gamma of GL_2(Z) with coefficients in the Steinberg modules over Q and over E, where E is a real quadratic field.

^{[13]}The monotonicity of entropy is investigated for real quadratic rational maps on the real circle R∪{∞} based on the natural partition of the corresponding moduli space M2(R) into its monotonic, covering, unimodal and bimodal regions.

^{[14]}Our goal is to make a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite and infinite singularities.

^{[15]}This paper presents a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite singularity and an infinite singularity.

^{[16]}[1] Pierre Charollois, Yingkun Li, Harmonic Maass forms associated to real quadratic fields, JEMS 22, 1115–1148 (2020).

^{[17]}

## real quadratic field

We obtain divisibility conditions on the multiplicative orders of elements of the form $\zeta + \zeta^{-1}$ in a finite field by exploiting a link to the arithmetic of real quadratic fields.^{[1]}Let [Formula: see text] be a positive square-free integer and [Formula: see text] be the fundamental unit of the real quadratic field [Formula: see text].

^{[2]}In the same spirit, we prove that for a given integer $t \geq 1$ with $t \equiv 0 \pmod {4}$, a positive proportion of fundamental discriminants $D > 0$ exist for which the class numbers of both the real quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D + t})$ are indivisible by $3$.

^{[3]}Using this, we compute slopes for weights in the center and near the boundary of weight space for certain real quadratic fields.

^{[4]}Illustration is given by some real quadratic fields.

^{[5]}For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL(2,Z)$PSL(2,\mathbb{Z})$-orbits of real quadratic fields.

^{[6]}We compare the homology of a congruence subgroup Gamma of GL_2(Z) with coefficients in the Steinberg modules over Q and over E, where E is a real quadratic field.

^{[7]}[1] Pierre Charollois, Yingkun Li, Harmonic Maass forms associated to real quadratic fields, JEMS 22, 1115–1148 (2020).

^{[8]}

## real quadratic polynomial

Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra.^{[1]}Our goal is to make a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite and infinite singularities.

^{[2]}This paper presents a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite singularity and an infinite singularity.

^{[3]}

## real quadratic extension

This is the first of a series of two papers in which we present a solution to Manin’s Real Multiplication program (Manin in: Laudal and Piene (eds) The Legacy of Niels Henrik Abel, Springer, Berlin, 2004) —an approach to Hilbert’s 12th problem for real quadratic extensions of $$\mathbb Q$$ Q —in positive characteristic, using quantum analogs of the modular invariant and the exponential function.^{[1]}In this paper we develop an effective procedure for expressing Stark units in real quadratic extensions of totally real fields as values of the Barnes multiple Gamma function at algebraic points.

^{[2]}