Real Quadratic Fields Research Articles

Measured Foliations and Hilbert 12th Problem

Yu. I. Manin conjectured that the maximal abelian extensions of the real quadratic number fields are generated by the pseudo-lattices with real multiplication. We prove this conjecture using theory of measured foliations on the modular curves.

On abelian 2-ramification torsion modules of quadratic fields

For a number field F and a prime number p, the ℤp-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F, denoted by \({{\cal T}_p}(F)\), is an important subject in abelian p-ramification theory. In this paper, we study the group \({{\cal T}_2}(F) = {{\cal T}_2}(m)\) of the quadratic field \(F = \mathbb{Q}(\sqrt m )\). Firstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that \({\rm{r}}{{\rm{k}}_4}({{\cal T}_2}( - m)) = {\rm{r}}{{\rm{k}}_2}({{\cal T}_2}( - m)) - {\rm{rank}}(R)\), where R is a certain explicitly described Rédei matrix over \({\mathbb{F}_2}\). Furthermore, using this formula, we obtain the 4-rank density formula of \({{\cal T}_2}\)-groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain results about the 2-power divisibility of orders of \({{\cal T}_2}( \pm l)\) and \({{\cal T}_2}( \pm 2l)\), both of which are cyclic 2-groups. In particular, we find that \(\# {{\cal T}_2}(l) \equiv 2\# {{\cal T}_2}(2l) \equiv {h_2}( - 2l)\) (mod 16) if l ≡ 7 (mod 8), where h2(−2l) is the 2-class number of \(\mathbb{Q}(\sqrt { - 2l} )\). We then obtain density results for \({{\cal T}_2}( \pm l)\) and \({{\cal T}_2}( \pm 2l)\) when the orders are small. Finally, based on our density results and numerical data, we propose distribution conjectures about \({{\cal T}_p}(F)\) when F varies over real or imaginary quadratic fields for any prime p, and about \({{\cal T}_2}( \pm l)\) and \({{\cal T}_2}( \pm 2l)\) when l varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the \({{\cal T}_2}(l)\) case is closely connected to Shanks-Sime-Washington’s speculation on the distributions of the zeros of 2-adic L-functions and to the distributions of the fundamental units.

Reductions of abelian surfaces over global function fields

Let $A$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. We prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves.

The universal ordinary deformation ring associated to a real quadratic field

The universal ordinary deformation ring associated to a real quadratic field

The Barnes-Hurwitz zeta cocycle on [formula omitted

We introduce a 1-cocycle Z of the group G=PGL2(Q) with values in a module D of distributions (in the sense of Stevens and Hu-Solomon). This cocycle is essentially constructed from the Barnes' double zeta function and it has the advantage of defining a family of maps that depend meromorphically on the usual parameter s∈C. In particular, this permits the extension of the cocycle property to any Taylor coefficient of such zeta function at s=0. Furthermore, we show that the class of Z in the first cohomology group H1(G,D) is nonzero, and we use basic facts about the arithmetic of real quadratic fields to prove the vanishing of H0(G,D), the group of G-invariant elements in D.

Restriction of Eisenstein series and Stark–Heegner points

In a recent work of Darmon, Pozzi and Vonk, the authors consider a particular p-adic family of Hilbert–Eisenstein series E k (1,ϕ) associated with an odd character ϕ of the narrow ideal class group of a real quadratic field F and compute the first derivative of a certain one-variable twisted triple product p-adic L-series attached to E k (1,ϕ) and an elliptic newform f of weight 2 on Γ 0 (p). In this paper, we generalize their construction to include the cyclotomic variable and thus obtain a two-variable twisted triple product p-adic L-series. Moreover, when f is associated with an elliptic curve E over ℚ, we prove that the first derivative of this p-adic L-series along the weight direction is a product of the p-adic logarithm of a Stark–Heegner point of E over F introduced by Darmon and the cyclotomic p-adic L-function for E.

Determination of elliptic curves with everywhere good reduction over real quadratic fields, II

Determination of elliptic curves with everywhere good reduction over real quadratic fields, II

Fermat’s Last Theorem and modular curves over real quadratic fields

In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree $d$ in this range there are no non-trivial solutions to this equation for $n \geq 4$.

Greenberg’s conjecture for real quadratic fields and the cyclotomic ℤ₂-extensions

Let A n \mathcal {A}_n be the 2 2 -part of the ideal class group of the n n -th layer of the cyclotomic Z 2 \mathbb {Z}_2 -extension of a real quadratic number field F F . The cardinality of A n \mathcal {A}_n is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the A n \mathcal {A}_n ’s stabilizes for the real fields F = Q ( f ) F=\mathbb {Q}(\sqrt {f}) for any integer 0 > f > 10000 0>f>10000 . Equivalently Greenberg’s conjecture holds for those fields.

On the Distribution of αp Modulo One in Quadratic Number Fields

Abstract We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.

The Diophantine Equation $X^3=u+v$ over Real Quadratic Fields, II

Let $k$ be a real quadratic field. The Diophantine equation $X^3=u+27v$ in $X\in\mathcal{O}_k$ (the ring of integers of $k$), $u,v\in\mathcal{O}_k^\times$ (the group of units of $k$) is solved under some assumptions on $k$.

On metabelian 2-class field towers over -extensions of real quadratic fields

AbstractWe give a family of real quadratic fields such that the 2-class field towers over their cyclotomic $\mathbb Z_2$ -extensions have metabelian Galois groups of abelian invariants $[2,2,2]$ . We also consider the boundedness of the Galois groups in relation to Greenberg’s conjecture, and calculate their class-2 quotients with an explicit example.

Abelian bm$p$-ramification groups and new Cohen-Lenstra heuristics

In this paper, we first review the $\\T_p$-groups in the abelian $p$-ramification theory of general number fields.We also review a general setting of the Cohen-Lenstra heuristic. Then we propose various new Cohen-Lenstra heuristics for distributions of$\\T_p$-groups of quadratic fields, which in particular explains the speculation of Shanks et al. (1999) on the distributions of zeros of $2$-adic $L$-functions and also reveals the distribution of fundamental units in certain real quadratic fields. Theoretical and numerical evidence of our conjectures is also presented.

A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group

This work deals with the well-known group-theoretic graphs called coset graphs for the modular group G and its applications. The group action of G on real quadratic fields forms infinite coset graphs. These graphs are made up of closed paths. When M acts on the finite field Zp, the coset graph appears through the contraction of the vertices of these infinite graphs. Thus, finite coset graphs are composed of homomorphic copies of closed paths in infinite coset graphs. In this work, we have presented a comprehensive overview of the formation of homomorphic copies.

Dihedral group and classification of G-circuits of length 10

Abstract In this paper, we classify G-circuits of length 10 with the help of the location of the reduced numbers lying on G-circuit. The reduced numbers play an important role in the study of modular group action on P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -subset of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ . For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of real quadratic fields. In particular, we classify P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ = ⋃ k ∈ N Q * k 2 m ${=}{\bigcup }_{k\in N}{Q}^{\ast }\left(\sqrt{{k}^{2}m}\right)$ containing G-circuits of length 10 and determine that the number of equivalence classes of G-circuits of length 10 is 41 in number. We also use dihedral group to explore cyclically equivalence classes of circuits and use cyclic group to explore similar G-circuits of length 10 corresponding to 10 of these circuits. By using cyclically equivalent classes of circuits and similar circuits, we obtain the exact number of G-orbits and the structure of G-circuits corresponding to cyclically equivalent classes. This study also helps us in classifying the reduced numbers lying in the P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits.

On Yokoi’s Invariants and the Ankeny–Artin–Chowla conjecture

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]. The Ankeny–Artin–Chowla (AAC) conjecture asserts that [Formula: see text] (mod [Formula: see text]) for primes [Formula: see text] (mod 4), which still remains unsolved. In this paper, sufficient conditions for [Formula: see text] have been given in terms of Yokoi’s invariants [Formula: see text] and [Formula: see text], and it has been shown that the AAC conjecture is true in some special cases.

Simultaneous indivisibility of class numbers of pairs of real quadratic fields

For a square-free integer t, Byeon (Proc. Am. Math. Soc. 132:3137–3140, 2004) proved the existence of infinitely many pairs of quadratic fields $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{tD})$$ with $$D > 0$$ such that the class numbers of all of them are indivisible by 3. In the same spirit, we prove that for a given integer $$t \ge 1$$ 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. This also addresses the complement of a weak form of a conjecture of Iizuka (J. Numb. Theory, 184:122–127, 2018). As an application of our main result, we obtain that for any integer $$t \ge 1$$ with $$t \equiv 0 \pmod {12}$$ , there are infinitely many pairs of real quadratic fields $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{D + t})$$ such that the Iwasawa $$\lambda $$ -invariants associated with the basic $$\mathbb {Z}_{3}$$ -extensions of both $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{D + t})$$ are 0. For $$p = 3$$ , this supports Greenberg’s conjecture which asserts that $$\lambda _{p}(K) = 0$$ for any prime number p and any totally real number field K.

On the key-exchange protocol using real quadratic fields

To prevent an exhaustive key-search attack of the key-exchange protocol using real quadratic fields, we need to ensure that the number l of reduced principal ideals in K is sufficiently large. In this paper we present an example of a family which are not valid for this protocol.

A quaternionic construction of p-adic singular moduli

Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural p-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of \({{\,\mathrm{SL}\,}}_2(\mathbb {Z}[1/p])\) which can be evaluated at real quadratic irrationalities, and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article, we present a construction of cohomology classes inspired by that of Darmon–Vonk, in which \({{\,\mathrm{SL}\,}}_2(\mathbb {Z}[1/p])\) is replaced by an order in an indefinite quaternion algebra over a totally real number field F. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions K of F, and we conjecture that the corresponding values lie in algebraic extensions of K. We also report on extensive numerical evidence for this algebraicity conjecture.

Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group $G$, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to $G$, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined class number formula for all finite groups. As an application, we prove the existence of superspecial abelian surfaces whose endomorphism algebras coincide with $\mathbb{Q}(\sqrt{p})$ in all positive characteristic $p\not\equiv 1\pmod{24}$.