## 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]}We provide a complete classification of well-rounded ideal lattices arising from real quadratic fields.

^{[18]}Applying the genus theory of Gauss on the real quadratic fields, we derive identities involving quadratic forms and genus characters.

^{[19]}We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i) $16$ -rank of the class group $\text{Cl}(-4p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-4p})$ ; (ii) $8$ -rank of the ordinary class group $\text{Cl}(8p)$ of the real quadratic field $\mathbb{Q}(\sqrt{8p})$ ; (iii) the solvability of the negative Pell equation $x^{2}-2py^{2}=-1$ over the integers; (iv) $2$ -part of the Tate–Safarevic group $\unicode[STIX]{x0428}(E_{p})$ of the congruent number elliptic curve $E_{p}:y^{2}=x^{3}-p^{2}x$.

^{[20]}Milnor and Thurston’s famous paper proved monotonicity of the topological entropy for the real quadratic family.

^{[21]}We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields.

^{[22]}We consider here the case of a real quadratic field.

^{[23]}We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics.

^{[24]}In this paper, we classify all the real quadratic v-functions.

^{[25]}Using half-integral weight modular forms we give a criterion for the existence of real quadratic $p$-rational fields.

^{[26]}We give a new definition of a $p$-adic $L$-function for a mixed signature character of a real quadratic field and for a nontrivial ray class character of an imaginary quadratic field.

^{[27]}Let $$\rho $$ρ be an even two-dimensional representation of the Galois group $${{\mathrm{Gal}}}(\overline{\mathbb Q}/\mathbb Q)$$Gal(Q¯/Q) which is induced from a character $$\chi $$χ of odd order of the absolute Galois group of a real quadratic field K.

^{[28]}In this paper, we give a nontrivial lower bound for the fundamental unit of norm $$-1$$ of a real quadratic field of class number 1.

^{[29]}We classify and then attempt to count the real quadratic fields (ordered by the size of the totally positive fundamental unit, as in Sarnak [14], [15]) from which quaternionic Artin representations of minimal conductor can be induced.

^{[30]}We then employ our techniques in the situation where $K$ is a totally real, abelian, ramified cubic extension of a real quadratic field.

^{[31]}(Invent Math 91(3):391–407, 1988) and Cohen (Invent Math 91(3):409–422, 1988), relating arithmetic in \({{\mathbb {Q}}}(\sqrt{6})\) to modularity of Ramanujan’s function \(\sigma (q)\), in the context of the general family of Richaud–Degert real quadratic fields \({{\mathbb {Q}}}(\sqrt{2p})\).

^{[32]}As a byproduct, we show that there exist infinitely many real quadratic fields with period ` of minimal type for each even `≥ 6.

^{[33]}For the real analogue of orthogonal groups, we take into account the signature of a real quadratic form to determine bounds in every case.

^{[34]}Our formula has a simple formulation for real quadratic number fields.

^{[35]}The theory of $\mathbb{R}$ viewed as an ordered $K$-vector space and expanded by a predicate for $\mathbb{Z}$ is decidable if and only if $K$ is a real quadratic field.

^{[36]}We demonstrate a connection to the results of Gross and Zagier on the factorization of differences of singular moduli by finding a real quadratic analogue of one of their results.

^{[37]}determining fundamental units and Yokoi’s d-invariants nd and md in the relation to continued fraction expansion of wd where ` (d) is a period length of wd for the such type of real quadratic number fields Q( √ d).

^{[38]}We get a bound on the $\ell$-rank of the class group of $H$(under some conditions) in terms of the $\ell$-rank of the class group of real quadratic subfield contained in $H$.

^{[39]}We study kernel functions of L -functions and products of L -functions of Hilbert cusp forms over real quadratic fields.

^{[40]}In particular, we apply all this to real quadratic number fields whose discriminants are a sum of two squares.

^{[41]}We show that the dyonic BSM orbits with $U$-duality invariant $\Delta<0$ are in exact correspondence with the solution sets of the Brahmagupta-Pell equation, which implies that they are isomorphic to the group of units in the order $\mathbb{Z}[\sqrt{|\Delta|}]$ in the real quadratic field $\mathbb{Q}(\sqrt{|\Delta|})$.

^{[42]}In this paper we study a continuity of the "values" of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics.

^{[43]}This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.

^{[44]}In particular, we discuss several systems where the normaliser is an infinite extension of the centraliser, including the visible lattice points or the $k$-free integers in some real quadratic number fields.

^{[45]}For $\mu<0$, we find a hyperacoustic branch and a nonhypoacoustic branch, with a limit $2\Delta$ and a purely real quadratic start at $q=0$ for $\Delta/|\mu|<0,222$.

^{[46]}Namely, we prove that given K{Q a real quadratic extension or an imaginary dihedral extension of degree 6, if the generalized abc-conjecture holds in K, then there exist at least c logX prime numbers p d X for which K is p-rational, here c is some nonzero constant depending on K.

^{[47]}We show that Hecke L-function of a real quadratic field can be expressed as a diagonal part of some normalized Shintani L-function of several variables.

^{[48]}We prove that there is a Hecke-equivariant linear map from the space of elliptic cusp forms of integer weight k, level $$N, ((N,D)=1)$$N,((N,D)=1) to Hilbert cusp forms of weight k, level N associated to a real quadratic field of discriminant D ($$D\equiv 1\pmod {4}$$D≡1(mod4)) with class number one.

^{[49]}A key role in our analysis is played by a theorem characterizing those spaces of type S for which the function exp(iQ(x)) is a pointwise multiplier for any real quadratic form Q.

^{[50]}

## 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]}We provide a complete classification of well-rounded ideal lattices arising from real quadratic fields.

^{[9]}Applying the genus theory of Gauss on the real quadratic fields, we derive identities involving quadratic forms and genus characters.

^{[10]}We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i) $16$ -rank of the class group $\text{Cl}(-4p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-4p})$ ; (ii) $8$ -rank of the ordinary class group $\text{Cl}(8p)$ of the real quadratic field $\mathbb{Q}(\sqrt{8p})$ ; (iii) the solvability of the negative Pell equation $x^{2}-2py^{2}=-1$ over the integers; (iv) $2$ -part of the Tate–Safarevic group $\unicode[STIX]{x0428}(E_{p})$ of the congruent number elliptic curve $E_{p}:y^{2}=x^{3}-p^{2}x$.

^{[11]}We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields.

^{[12]}We consider here the case of a real quadratic field.

^{[13]}We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics.

^{[14]}We give a new definition of a $p$-adic $L$-function for a mixed signature character of a real quadratic field and for a nontrivial ray class character of an imaginary quadratic field.

^{[15]}Let $$\rho $$ρ be an even two-dimensional representation of the Galois group $${{\mathrm{Gal}}}(\overline{\mathbb Q}/\mathbb Q)$$Gal(Q¯/Q) which is induced from a character $$\chi $$χ of odd order of the absolute Galois group of a real quadratic field K.

^{[16]}In this paper, we give a nontrivial lower bound for the fundamental unit of norm $$-1$$ of a real quadratic field of class number 1.

^{[17]}We classify and then attempt to count the real quadratic fields (ordered by the size of the totally positive fundamental unit, as in Sarnak [14], [15]) from which quaternionic Artin representations of minimal conductor can be induced.

^{[18]}We then employ our techniques in the situation where $K$ is a totally real, abelian, ramified cubic extension of a real quadratic field.

^{[19]}(Invent Math 91(3):391–407, 1988) and Cohen (Invent Math 91(3):409–422, 1988), relating arithmetic in \({{\mathbb {Q}}}(\sqrt{6})\) to modularity of Ramanujan’s function \(\sigma (q)\), in the context of the general family of Richaud–Degert real quadratic fields \({{\mathbb {Q}}}(\sqrt{2p})\).

^{[20]}As a byproduct, we show that there exist infinitely many real quadratic fields with period ` of minimal type for each even `≥ 6.

^{[21]}The theory of $\mathbb{R}$ viewed as an ordered $K$-vector space and expanded by a predicate for $\mathbb{Z}$ is decidable if and only if $K$ is a real quadratic field.

^{[22]}We study kernel functions of L -functions and products of L -functions of Hilbert cusp forms over real quadratic fields.

^{[23]}We show that the dyonic BSM orbits with $U$-duality invariant $\Delta<0$ are in exact correspondence with the solution sets of the Brahmagupta-Pell equation, which implies that they are isomorphic to the group of units in the order $\mathbb{Z}[\sqrt{|\Delta|}]$ in the real quadratic field $\mathbb{Q}(\sqrt{|\Delta|})$.

^{[24]}This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.

^{[25]}We show that Hecke L-function of a real quadratic field can be expressed as a diagonal part of some normalized Shintani L-function of several variables.

^{[26]}We prove that there is a Hecke-equivariant linear map from the space of elliptic cusp forms of integer weight k, level $$N, ((N,D)=1)$$N,((N,D)=1) to Hilbert cusp forms of weight k, level N associated to a real quadratic field of discriminant D ($$D\equiv 1\pmod {4}$$D≡1(mod4)) with class number one.

^{[27]}Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields.

^{[28]}

## real quadratic number

Our formula has a simple formulation for real quadratic number fields.^{[1]}determining fundamental units and Yokoi’s d-invariants nd and md in the relation to continued fraction expansion of wd where ` (d) is a period length of wd for the such type of real quadratic number fields Q( √ d).

^{[2]}In particular, we apply all this to real quadratic number fields whose discriminants are a sum of two squares.

^{[3]}In this paper we study a continuity of the "values" of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics.

^{[4]}In particular, we discuss several systems where the normaliser is an infinite extension of the centraliser, including the visible lattice points or the $k$-free integers in some real quadratic number fields.

^{[5]}In this paper, we study the torsion structures of elliptic curves over the real quadratic number fields $$\mathbb{Q}\left( {\sqrt 2 } \right)$$ℚ(2) and $$\mathbb{Q}\left( {\sqrt 5 } \right)$$ℚ(5), which have the smallest discriminants among all real quadratic fields $$\mathbb{Q}\left( {\sqrt d } \right)$$ℚ(d) with d ≢ 1 mod 4 and d ≡ 1 mod 4 respectively.

^{[6]}

## 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]}Namely, we prove that given K{Q a real quadratic extension or an imaginary dihedral extension of degree 6, if the generalized abc-conjecture holds in K, then there exist at least c logX prime numbers p d X for which K is p-rational, here c is some nonzero constant depending on K.

^{[3]}

## 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 family

Milnor and Thurston’s famous paper proved monotonicity of the topological entropy for the real quadratic family.^{[1]}As in Tsujii's approach \cite{Tsu0,Tsu1}, for real maps we obtain {\em positive} transversality (where $>0$ holds instead of just $\ne 0$), and thus monotonicity of entropy for these families, and also (as an easy application) for the real quadratic family.

^{[2]}

## real quadratic form

For the real analogue of orthogonal groups, we take into account the signature of a real quadratic form to determine bounds in every case.^{[1]}A key role in our analysis is played by a theorem characterizing those spaces of type S for which the function exp(iQ(x)) is a pointwise multiplier for any real quadratic form Q.

^{[2]}