Nonexistence of certain Galois representations for quadratic fields

The problem of estimating the dimension of the space of (holomorphic) cusp forms of weight one and a given level was first considered by Serre [Ser]. The problem is markedly different from the case when the weight k is two or more, in which case the dimension is O(kq), q being the level [Shim]. In the case of weight one, it is believed (see [Duke], [MV]) that the dimension should be O(q½+ε) where the implied constant is absolute. Recall that by the Deligne-Serre theorem (Theorem 4), normalized newforms of weight one correspond to isomorphism classes of complex, two-dimensional, irreducible, continuous Galois representations. Here, the absolute Galois group is given the profinite topology, which means that continuity of such a representation is equivalent to its image in GL(2, ℂ) being finite. So the image of the projectivization of one such representation is a finite subgroup in PGL(2, ℂ) and therefore, by a classical result of Klein [Klein], must be either a dihedral group or one of A4, S4, and A5. Depending on the images of the associated Galois representations, the weight one forms are referred to as of the dihedral, tetrahedral, octahedral, or icosahedral type. It is not difficult to prove that the dimension of the subspace of forms of dihedral type is O(q½+ε) (see [Hec], [Duke]). And it is expected that the dimension of the space spanned by those new forms (of a fixed nebentypus) that correspond to a non-dihedral Galois representation should only be O(qε) (see [Duke], [MV]).2000 Mathematics Subject ClassificationPrimary 11F7211F8011F30Secondary 11N35Key words and phrasesmodular forms of weight onelarge sieve inequalityArtin representations

The non-existence is proved of 2-dimensional mod 3 irreducible representations of $\operatorname {Gal}(\overline {\mathbb Q}/\mathbb Q)$ of Artin conductor dividing 4.

In this paper, we generalise to the family of Fermat quartics $X^4 + Y^4 = 2^m, m \in \mathbb {Z}$ , a result of Aigner [‘Über die Möglichkeit von $x^4 + y^4 = z^4$ in quadratischen Körpern’, Jahresber. Deutsch. Math.-Ver. 43 (1934), 226–228], which proves that there is only one quadratic field, namely $\mathbb {Q}(\sqrt {-7})$ , that contains solutions to the Fermat quartic $X^4 + Y^4 = 1$ . The $m \equiv 0 \pmod 4$ case is due to Aigner. The $m \equiv 2 \pmod 4$ case follows from a result of Emory [‘The Diophantine equation $X^4 + Y^4 = D^2Z^4$ in quadratic fields’, Integers 12 (2012), Article no. A65, 8 pages]. This paper focuses on the two cases $m \equiv 1, 3 \pmod 4$ , classifying for $m \equiv 1 \pmod 4$ the infinitely many quadratic number fields that contain solutions, and proving for $m \equiv 3 \pmod 4$ that $\mathbb {Q}(\sqrt {2})$ and $\mathbb {Q}(\sqrt {-2})$ are the only quadratic number fields that contain solutions.

Modularity of hypertetrahedral representations

Fix K a p-adic field and denote by G_K its absolute Galois group. Let K_infty be the extension of K obtained by adding (p^n)-th roots of a fixed uniformizer, and G_\infty its absolute Galois group. In this article, we define a class of p-adic torsion representations of G_\infty, named quasi-semi-stable. We prove that these representations are explicitly described by a certain category of linear algebra objects. The results of this note should be consider as a first step in the understanding of the structure of quotients of two lattices in a crystalline (resp. semi-stable) Galois representation.

Let K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field [Formula: see text], p-ring spaces Vp(c) modulo c are introduced by defining a morphism ψ : f ↦ Vp(f) from the divisor lattice ℕ of positive integers to the lattice 𝒮 of subspaces of the direct product Vpof the p-elementary class group 𝒞/𝒞pand unit group U/Upof K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group Gal (N | ℚ) and sharing a common discriminant dNand conductor c over K. The number mp(d, c) of these extensions is given by a formula in terms of positions of p-ring spaces in 𝒮, whose complexity increases with the dimension of the vector space Vpover the finite field 𝔽p, called the modified p-class rank σpof K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with 0 ≤ σp≤ 1 only. Here, the results are extended to σp= 2, underpinned by concrete numerical examples.

For a quadratic field $K$, we investigate continuous mod $p$ representations of $Gal(\overline {K}/K)$ that are unramified away from $\{p,\infty \}$. We prove that for certain $(K,p)$ there are no such irreducible representations. We also list some imaginary quadratic fields for which such irreducible representations exist. As an application, we look at elliptic curves with good reduction away from $2$ over quadratic fields.

where A(co) stands for the discriminant of the complex lattice generated by 1 and to and g2(u) is the Weierstrass invariant of this lattice. If ti=Q(\/ — D) is an imaginary quadratic number field, it is well known from the theory of class fields with complex multiplication that the singular values j(a), where a£u, la >0, generate algebraic number fields which are abelian over fi, namely the so-called ring class fields. Detailed references for the literature on the theory of ring class fields may be found in the report of Deuring [l]. In the rather extensive theory of ring class fields it is shown that ®Uia))/Q iS a normal extension with its Galois group © being an extension of the abelian Galois group s of Q(j(a))/Cl, completely determined by the relations

Let X X be a complete smooth variety defined over a number field K K and let i i be an integer. The absolute Galois group G a l K \mathrm {Gal}_K of K K acts on the i i th étale cohomology group H e ´ t i ( X K ¯ , Q ℓ ) H^i_{\mathrm {\acute {e}t}}(X_{\bar K},\mathbb {Q}_\ell ) for all primes ℓ \ell , producing a system of ℓ \ell -adic representations { Φ ℓ } ℓ \{\Phi _\ell \}_\ell . The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φ ℓ \Phi _\ell admits a reductive Q \mathbb {Q} -form that is independent of ℓ \ell if X X is projective. Denote by Γ ℓ \Gamma _\ell and G ℓ \mathbf {G}_\ell respectively the monodromy group and the algebraic monodromy group of Φ ℓ s s \Phi _\ell ^{\mathrm {ss}} , the semisimplification of Φ ℓ \Phi _\ell . Assuming that G ℓ 0 \mathbf {G}_{\ell _0} satisfies some group theoretic conditions for some prime ℓ 0 \ell _0 , we construct a connected quasi-split Q \mathbb {Q} -reductive group G Q \mathbf {G}_{\mathbb {Q}} which is a common Q \mathbb {Q} -form of G ℓ ∘ \mathbf {G}_\ell ^\circ for all sufficiently large ℓ \ell . Let G Q s c \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}} be the universal cover of the derived group of G Q \mathbf {G}_{\mathbb {Q}} . As an application, we prove that the monodromy group Γ ℓ \Gamma _\ell is big in the sense that Γ ℓ s c ≅ G Q s c ( Z ℓ ) \Gamma _\ell ^{\mathrm {sc}}\cong \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}}(\mathbb {Z}_\ell ) for all sufficiently large ℓ \ell .

Let K be a number field and $$K_\mathrm{ur}$$ be the maximal extension of K that is unramified at all places. In a previous article (Kim, J Number Theory 166:235–249, 2016), the first author found three real quadratic fields K such that $$\mathrm {Gal}(K_\mathrm{ur}/K)$$ is finite and non-abelian simple under the assumption of the generalized Riemann hypothesis (GRH). In this article, we extend the methods of Kim (2016) and identify more quadratic number fields K such that $$\mathrm {Gal}(K_\mathrm{ur}/K)$$ is a finite nonsolvable group and also explicitly calculate their Galois groups under the assumption of the GRH. In particular, we find the first imaginary quadratic field with this property.

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\bar{\unicode[STIX]{x1D70C}}$ be a continuous, absolutely irreducible representation of its absolute Galois group with values in a finite field of characteristic $p$. We prove that the Galois representations that become crystalline of a fixed regular weight after an abelian extension are Zariski-dense in the generic fiber of the universal deformation ring of $\bar{\unicode[STIX]{x1D70C}}$. In fact we deduce this from a similar density result for the space of trianguline representations. This uses an embedding of eigenvarieties for unitary groups into the spaces of trianguline representations as well as the corresponding density claim for eigenvarieties as a global input.

Let K be a number field. We prove that its ray class group modulo p 2 (resp. 8 ) if p > 2 (resp. p = 2 ) characterizes its p -rationality. Then we give two short and very fast PARI Programs (Sections 3.1, 3.2) testing if K (defined by an irreducible monic polynomial) is p -rational or not. For quadratic fields we verify some densities of 3 -rational fields related to Cohen–Lenstra–Martinet ones and analyse Greenberg’s conjecture on the existence of p -rational fields with Galois groups ( ℤ / 2 ℤ ) t needed for the construction of some Galois representations with open image. We give examples for p = 3 , t = 5 and t = 6 (Sections 5.1, 5.2) and illustrate other approaches (Pitoun–Varescon, Barbulescu–Ray). We conclude about the existence of imaginary quadratic fields, p -rational for all p ≥ 2 (Angelakis–Stevenhagen study on the concept of “minimal absolute abelian Galois group”) which may enlighten a conjecture of p -rationality (Hajir–Maire) giving large Iwasawa μ -invariants of some uniform pro- p -groups. All programs (in “verbatim”) can be used by the reader by simply copied and pasted.

Unramified extensions of quadratic fields

Let $K$ be a complete, discretely valued field with finite residue field and $G_K$ its absolute Galois group. The subject of this note is the study of the set of positive integers $d$ for which there exists an absolutely irreducible $\ell$-adic representation of $G_K$ of dimension $d$ with rational traces on inertia. Our main result is that non-Sophie Germain primes are not in this set when the residue characteristic of $K$ is $> 3$. The result stated in the title is a special case.