Quadratic Extension Of Number Fields Research Articles

Distinction inside L-packets of SL(n)

If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $\mathrm{SL}_n(F)$-distinguished representations inside a distinguished unitary L-packet of $\mathrm{SL}_n(E)$ are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of $N(E)/N(F)$. Then we establish a global analogue of this result. For this, let $E/F$ be a quadratic extension of number fields and let $\pi$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished square integrable automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Let $(\sigma,d)$ be the unique pair associated to $\pi$, where $\sigma$ is a cuspidal representation of $\mathrm{GL}_r(\mathbb{A}_E)$ with $n=dr$. Using an unfolding argument, we prove that an element of the L-packet of $\pi$ is distinguished with respect to $\mathrm{SL}_n(\mathbb{A}_F)$ if and only if it has a degenerate Whittaker model for a degenerate character $\psi$ of type $r^d:=(r,\dots,r)$ of $N_n(\mathbb{A}_E)$ which is trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $\mathrm{SL}_n$. As a first application, under the assumptions that $E/F$ splits at infinity and $r$ is odd, we establish a local-global principle for $\mathrm{SL}_n(\mathbb{A}_F)$-distinction inside the L-packet of $\pi$. As a second application we construct examples of distinguished cuspidal automorphic representations $\pi$ of $\mathrm{SL}_n(\mathbb{A}_E)$ such that the period integral vanishes on some canonical copy of $\pi$, and of everywhere locally distinguished representations of $\mathrm{SL}_n(\mathbb{A}_E)$ such that their L-packets do not contain any distinguished representation.

Unramified extensions of quadratic number fields with certain perfect Galois groups

In this paper, we provide infinite families of quadratic number fields with everywhere unramified Galois extensions of Galois group [Formula: see text] and [Formula: see text], respectively. To my knowledge, these are the first instances of infinitely many such realizations for perfect groups which are not generated by involutions, a property which makes them difficult to approach for the problem in question and leads to somewhat delicate local–global problems in inverse Galois theory.

On quaternion algebras over some extensions of quadratic number fields

Let p and q be two positive primes, let \(\ell\) be an odd positive prime and let F be a quadratic number field. Let K be an extension of F of degree \(\ell\) such that K is a dihedral extension of \({\mathbb {Q}}\), or else let K be an abelian \(\ell\)-extension of F unramified over F whenever \(\ell\) divides the class number of F. In this paper, we provide a complete characterization of division quaternion algebras \(H_{K}(p, q)\) over K.

The $$\mathrm {PGSp}_{1,1}({\mathbb {A}})$$-distinguished representations

In this paper, we use the regularized quaternionic Siegel–Weil formula to deal with the $$\mathrm {PGSp}_{1,1}({\mathbb {A}})$$ -period problem for a cuspidal representation $$\tau $$ of $$\mathrm {PGSp}_4({\mathbb {A}}_E)$$ , where E/F is a quadratic extension of number fields and E is contained in the 4-dimensional quaternion division algebra D of F with $$s_D=2$$ .

On L-functions for U2n+1 × ResE/FGLm (m > n)

We present the basics of the local theory, which arises from global Rankin-Selberg integrals, attached to pairs of irreducible globally generic cuspidal automorphic representations of the quasi-split unitary group U2n+1 and ResE/FGLm, for a quadratic extension of number fields E/F, when m>n.

Quaternion distinguished representations and unstable base change for unitary groups

Let E/F be a quadratic extension of number fields and D a quaternion algebra over F which E embeds. Flicker and Rallis conjectured that a cuspidal automorphic representation π of GL(2n,E) is the unstable base change lift of a generic cuspidal automorphic representation σ of the quasi-split unitary group U(2n) if and only if it is distinguished by GL(2n,F). We conjecture that π is distinguished by GL(n,D) if and only if σ is generic with respect to certain non-degenerate character attached to D. We use the relative trace formula to prove the n=1 case of our conjecture.

Local root numbers, Bessel models, and a conjecture of Guo and Jacquet

Let E/F be a quadratic extension of number fields and D a quaternion algebra over F containing E. Let πD be a cuspidal automorphic representation of GL(n,D) and π its Jacquet–Langlands transfer to GL(2n). Guo and Jacquet conjectured that if πD is distinguished by GL(n,E), then π is symplectic and L(1/2,πE)≠0, where πE is the base change of π to E. When n is odd, Guo and Jacquet also conjectured a converse. The converse does not always hold when n is even, but we conjecture it holds if and only if certain local root number conditions are satisfied, which is if and only if the corresponding generic representation of the split special orthogonal group SO(2n+1) has a special E-Bessel model. We use the theta correspondence to relate E-Bessel periods on SO(5) with GL(2,E)-periods on GL(2,D), and deduce part of our conjecture when n=2.

On representations distinguished by unitary groups

Let E/F be a quadratic extension of number fields. We study periods and regularized periods of cusp forms and Eisenstein series on \(\operatorname {GL}_{n}( \mathbf {A}_{E})\) over a unitary group of a Hermitian form with respect to E/F. We provide factorization for these periods into locally defined functionals, express these factors in terms of suitably defined local periods and characterize global distinction. We also study in detail the analogous local question and analyze the space of invariant linear forms under a unitary group.

Determination of cusp forms on GL(2) by coefficients restricted to quadratic subfields (with an appendix by Dipendra Prasad and Dinakar Ramakrishnan)

Given E/F a quadratic extension of number fields and a cuspidal representation π of GL2(AE), we give a full description of the fibers of the Asai transfer of π. We then determine the extent to which Fourier coefficients defined by integral ideals of F determine the representation π.

Parity-induced Selmer growth for symplectic, ordinary families

Let p be an odd prime, and let K / K 0 be a quadratic extension of number fields. Denote by K ± the maximal Z p -power extensions of K that are Galois over K 0 , with K + abelian over K 0 and K − dihedral over K 0 . In this paper we show that for a Galois representation over K 0 satisfying certain hypotheses, if it has odd Selmer rank over K then for one of K ± its Selmer rank over L is bounded below by [ L : K ] for L ranging over the finite subextensions of K in K ± . Our method of proof generalizes a method of Mazur and Rubin, building upon results of Nekovář, and applies to abelian varieties of arbitrary dimension, (self-dual twists of) modular forms of even weight, and (twisted) Hida families.

On the 2-Rank of the Hilbert Kernel of Number Fields

Abstract. Let $E/F$ be a quadratic extension of number fields. In this paper, we show that the genus formula for Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the 2-rank of the Hilbert kernel of $E$ provided that the 2-primary Hilbert kernel of $F$ is trivial. However, since the original genus formula is not explicit enough in a very particular case, we first develop a refinement of this formula in order to employ it in the calculation of the 2-rank of $E$ whenever $F$ is totally real with trivial 2-primary Hilbert kernel. Finally, we apply our results to quadratic, bi-quadratic, and tri-quadratic fields which include a complete 2-rank formula for the family of fields $\mathbb{Q}(\sqrt{2},\sqrt{\delta )}$ where $\delta $ is a squarefree integer.

Hecke’s Integral Formula for Relative Quadratic Extensions of Algebraic Number Fields

AbstractLet K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker’s type which relates the 0-th Laurent coefficients at s = 1 of zeta functions of K and F.

Finding large Selmer rank via an arithmetic theory of local constants

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let κ - denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(κ-/K) by inversion). We prove (under mild hypotheses on p) that if the Zp-rank of the pro-p Selmer group Sp(E/K) is odd, then rank Zp S p (E/F) > [F: K] for every finite extension F of K in κ - .

Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields

AbstractWe obtain Hauptmoduls of genus zero congruence subgroups of the type (p) := Γ0(p) + wp, where p is a prime and wp is the Atkin–Lehner involution. We then use the Hauptmoduls, along with modular functions on Γ1(p) to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.

On the SL (2) period integral

Let E/F be a quadratic extension of number fields. For a cuspidal representation π of SL 2 (A E ), we study in this paper the integral of functions in π on SL 2 ( F )\SL 2 (A F ). We characterize the nonvanishing of these integrals, called period integrals, in terms of π having a Whittaker model with respect to characters of E \A E which are trivial on A F . We show that the period integral in general is not a product of local invariant functionals, and find a necessary and sufficient condition when it is. We exhibit cuspidal representations of SL 2 (A E ) whose period integral vanishes identically while each local constituent admits an SL2-invariant linear functional. Finally, we construct an automorphic representation π on SL 2 (A E ) which is abstractly SL 2 (A F ) distinguished but for which none of the elements in the global L -packet determined by it is distinguished by SL 2 (A F ).

Kloosterman identities over a quadratic extension II

We prove certain identities between Kloosterman integrals. They constitute the fundamental lemma of a relative trace formula for Hecke functions. The main application of the trace formula in question is the following result. Let E/F be a quadratic extension of number fields. A cuspidal automorphic representation of GL(n,EA) is distinguished by some unitary group if and only if it is the base change of an automorphic cuspidal representation of GL(n,FA).

A SIMPLE EXAMPLE OF EXTENSIONS WITHOUT AN INTEGRAL BASIS

Several examples of quadratic extensions of number fields without an integral basis are given. Bibliography: 5 titles.

Stable Bi-Period Summation Formula and Transfer Factors

AbstractThis paper starts by introducing a bi-periodic summation formula for automorphic forms on a group G(E), with periods by a subgroup G(F), where E/F is a quadratic extension of number fields. The split case, where E = F ⊕ F, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups H which occur in the case of standard conjugacy.The spectral side of the bi-period summation formula involves periods, namely integrals over the group of F-adele points of G, of cusp forms on the group of E-adele points on the group G. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to “periodic” automorphic forms, are related to analogous bi-period distributions associated to “periodic” automorphic forms on the endoscopic symmetric spaces H(E)/H(F). This offers a sharpening of the theory of liftings, where periods play a key role.The stabilization depends on the “fundamental lemma”, which conjectures that the unit elements of the Hecke algebras on G and H have matching orbital integrals. Even in stating this conjecture, one needs to introduce a “transfer factor”. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case.Finally, the fundamental lemma is verified for SL(2).

On quadratic extensions of number fields and Iwasawa invariants for basic Z3-extensions

Let Z3 be the ring of 3-adic integers. For each number field F, let F∞,3 denote the basic Z3-extension over F; let λ3(F) and μ3(F) denote respectively the Iwasawa λ- and μ-invariants of F∞,3/F. Here a number field means a finite extension over the rational field Q contained in the complex field C;F⊂C,[F:Q]<∞. Now let k be a number field. Let L-denote the infinite set of totally imaginary quadratic extensions in C over k(so that L-coincides with the set L- in the text when k is totally real); let L+ denote the infinite set of quadratic extensions in C over k in which every infinite place of k splits (so that L+ coincides with the set L+ in the text when k is totally real). After studying the distribution of certain quadratic extensions over k, that of certain cubic extensions over k, and the relation between the two distributions, this paper proves that, if k is totally real, then a subset of {K∈L-|λ3(K)=λ3(k),μ3(K)=μ3(k)} has an explicit positive density in L-. The paper also proves that a subset of {L∈L+|λ3(L)=μ3(L)=0} has an explicit positive density in L+ if 3 does not divide the class number of k but is divided by only one prime ideal of k. Some consequences of the above results are added in the last part of the paper.

Critical values of the twisted tensor L-function in the imaginary quadratic case

The twisted tensor L-function of f , which we denote by G(s, f), is a certain Dirichlet series associated to a quadratic extension of number fields K/F , and a cuspidal automorphic function f over K. It was introduced in [1] by Asai, following previous work of Shimura, in the case when f is a Hilbert modular cusp form over a real quadratic extension K of Q. In the past twenty odd years, this L-function has been considered more generally: for instance [11] and [12] deal with quadratic extensions of totally real fields, [17] with imaginary quadratic extensions of Q, and [3], [4] and [14] with general quadratic extensions of number fields. All these papers have been primarily concerned with establishing analytic properties of G(s, f) analogous to those in [1], such as meromorphic continuation to the entire complex plane, location and finiteness of the number of poles, and functional equation. The aim of this paper is to prove a rationality result for G(s, f) in the imaginary quadratic setting. If K is an imaginary quadratic field, and f a cusp form associated to K, we establish that there is a ‘period’ Ωj(f) such that