A deformation problem for Galois representations over imaginary quadratic fields

We prove the modularity of certain residually reducible p-adic Galois representations of an imaginary quadratic field assuming the uniqueness of the residual representation. We obtain an R = T theorem using a new commutative algebra criterion that might be of independent interest. To apply the criterion, one needs to show that the quotient of the universal deformation ring R by its ideal of reducibility is cyclic Artinian of order no greater than the order of the congruence module T/J, where J is an Eisenstein ideal in the local Hecke algebra T. The inequality is proven by applying the Main conjecture of Iwasawa Theory for Hecke characters and using a result of Berger [Compos Math 145(3):603–632, 2009]. This strengthens our previous result [Berger and Klosin, J Inst Math Jussieu 8(4):669–692, 2009] to include the cases of an arbitrary p-adic valuation of the L-value, in particular, cases when R is not a discrete valuation ring. As a consequence we show that the Eisenstein ideal is principal and that T is a complete intersection.

Deformations and the rigidity method

We study compatible families of three-dimensional Galois representations constructed in the étale cohomology of a smooth projective variety. We prove a theorem asserting that the residual images will be generically large if certain easy-to-check conditions are satisfied. We only consider representations with coefficients in an imaginary quadratic field. For primes inert in this field, the residual representations (when irreducible) are unitary. We apply our result to an example constructed by van Geemen and Top, obtaining a family of special linear groups and one of special unitary groups as Galois groups over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. We also consider the case of cohomological modular forms for a congruence subgroup of SL [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. Assuming Clozel's conjecture stating that a geometric family of three-dimensional Galois representations can be attached to them, we verify for three examples the conditions guaranteeing generically large images and explicitly bound the finite set of primes with non-maximal image. We also discuss what intrinsic conditions a modular form should verify to guarantee that the images of the attached Galois representations will be generically large.

We introduce a new method of proof for R = T theorems in the residually reducible case. We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation ρ 0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents ρ 1 and ρ 2. Under some assumptions on Selmer groups associated with ρ 1 and ρ 2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaïche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Using a new commutative algebra criterion we show that given enough information on the Hecke side one gets an R = T-theorem. We then apply the technique to modularity problems for 2-dimensional representations over an imaginary quadratic field and a 4-dimensional representation over Q.

Starting from the analytic class number formula involving its L -function, we first give an expression for the class number of an imaginary quadratic field which, in the case of large discriminants, provides us with a much more powerful numerical technique than that of counting the number of reduced definite positive binary quadratic forms, as has been used by Buell in order to compute his class number tables. Then, using class field theory, we will construct a periodic character χ \chi , defined on the ring of integers of a field K that is a quadratic extension of a principal imaginary quadratic field k , such that the zeta function of K is the product of the zeta function of k and of the L -function L ( s , χ ) L(s,\chi ) . We will then determine an integral representation of this L -function that enables us to calculate the class number of K numerically, as soon as its regulator is known. It will also provide us with an upper bound for these class numbers, showing that Hua’s bound for the class numbers of imaginary and real quadratic fields is not the best that one could expect. We give statistical results concerning the class numbers of the first 50000 quadratic extensions of Q ( i ) {\mathbf {Q}}(i) with prime relative discriminant (and with K / Q a non-Galois quartic extension). Our analytic calculation improves the algebraic calculation used by Lakein in the same way as the analytic calculation of the class numbers of real quadratic fields made by Williams and Broere improved the algebraic calculation consisting in counting the number of cycles of reduced ideals. Finally, we give upper bounds for class numbers of K that is a quadratic extension of an imaginary quadratic field k which is no longer assumed to be of class number one.

Let K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

The coefficient space is a kind of resolution of singularities of the universal flat deformation space for a given Galois representation of some local field. It parametrizes (in some sense) the finite flat models for the Galois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.

The goal of this article is to give an explicit classification of the\npossible $p$-adic Galois representations that are attached to elliptic curves\n$E$ with CM defined over $\\mathbb{Q}(j(E))$. More precisely, let $K$ be an\nimaginary quadratic field, and let $\\mathcal{O}_{K,f}$ be an order in $K$ of\nconductor $f\\geq 1$. Let $E$ be an elliptic curve with CM by\n$\\mathcal{O}_{K,f}$, such that $E$ is defined by a model over\n$\\mathbb{Q}(j(E))$. Let $p\\geq 2$ be a prime, let $G_{\\mathbb{Q}(j(E))}$ be the\nabsolute Galois group of $\\mathbb{Q}(j(E))$, and let $\\rho_{E,p^\\infty}\\colon\nG_{\\mathbb{Q}(j(E))}\\to \\operatorname{GL}(2,\\mathbb{Z}_p)$ be the Galois\nrepresentation associated to the Galois action on the Tate module $T_p(E)$. The\ngoal is then to describe, explicitly, the groups of\n$\\operatorname{GL}(2,\\mathbb{Z}_p)$ that can occur as images of\n$\\rho_{E,p^\\infty}$, up to conjugation, for an arbitrary order\n$\\mathcal{O}_{K,f}$.\n

In this paper we study deformations of mod $p$ Galois representations $\\tau$\n(over an imaginary quadratic field $F$) of dimension $2$ whose\nsemi-simplification is the direct sum of two characters $\\tau_1$ and $\\tau_2$.\nAs opposed to our previous work we do not impose any restrictions on the\ndimension of the crystalline Selmer group $H^1_{\\Sigma}(F, {\\rm Hom}(\\tau_2,\n\\tau_1)) \\subset {\\rm Ext}^1(\\tau_2, \\tau_1)$. We establish that there exists a\nbasis $\\mathcal{B}$ of $H^1_{\\Sigma}(F, {\\rm Hom}(\\tau_2, \\tau_1))$ arising\nfrom automorphic representations over $F$ (Theorem 8.1). Assuming among other\nthings that the elements of $\\mathcal{B}$ admit only finitely many crystalline\ncharacteristic 0 deformations we prove a modularity lifting theorem asserting\nthat if $\\tau$ itself is modular then so is its every crystalline\ncharacteristic zero deformation (Theorems 8.2 and 8.5).\n

Hodge structures of type (n, 0, . . . , 0, n) Burt Totaro Completing earlier work by Albert, Shimura found all the possible endomor- phism algebras (tensored with the rationals) for complex abelian varieties of a given dimension [12, Theorem 5]. In five exceptional cases, every abelian variety on which a certain algebra acts has “extra endomorphisms”, so that the full endomorphism algebra is bigger than expected. Complex abelian varieties X up to isogeny are equivalent to polarizable Q-Hodge structures of weight 1, with Hodge numbers (n, n) (where n is the dimension of X). In this paper, we generalize Shimura’s classification to determine all the possible endomorphism algebras for polarizable Q-Hodge structures with Hodge numbers (n, 0, . . . , 0, n). For Hodge structures of odd weight, the answer is the same as for abelian varieties. For Hodge structures of even weight, the answer is similar but not identical. The proof combines ideas from Shimura with Green-Griffiths-Kerr’s approach to computing Mumford-Tate groups [4, Proposition VI.A.5]. As with abelian varieties, the most interesting feature of the classification is that in certain cases, every Hodge structure on which a given algebra acts must have extra endomorphisms. (Throughout this discussion, we only consider polarizable Hodge structures.) One known case (pointed out to me by Beauville) is that every Q- Hodge structure with Hodge numbers (1, 0, 1) has endomorphisms by an imaginary quadratic field and hence is of complex multiplication (CM) type, meaning that its Mumford-Tate group is commutative. More generally, every Q-Hodge structure with Hodge numbers (n, 0, n) that has endomorphisms by a totally real field F of degree n has endomorphisms by a totally imaginary quadratic extension field of F , and hence is of CM type. Another case, which seems to be new, is that a Q-Hodge structure V with Hodge numbers (2, 0, 2) that has endomorphisms by an imaginary quadratic field F 0 must have endomorphisms by a quaternion algebra over Q. In this case, V need not be of CM type; there is a period space isomorphic to CP 1 of Hodge structures of this type, whereas there are only countably many Hodge structures of CM type. To motivate the results of this paper on endomorphism algebras, consider the geometric origin of Hodge structures. A Hodge structure comes from geometry if it is a summand of the cohomology of a smooth complex projective variety defined by an algebraic correspondence. Griffiths found (“Griffiths transversality”) that a family of Hodge structures coming from geometry can vary only in certain directions, expressed by the notion of a variation of Hodge structures [15, Theorem 10.2]. In particular, any variation of Hodge structures of weight at least 2 with Hodge numbers (n, 0, . . . , 0, n) (so there is at least one 0) is locally constant; more generally, this holds whenever there are no two adjacent nonzero Hodge numbers. This has the remarkable consequence that only countably many Hodge structures of weight at least 2 with Hodge numbers (n, 0, . . . , 0, n) come from geometry. Very little is

We prove that the j -invariant of an elliptic curve defined over an imaginary quadratic number field having good reduction everywhere satisfies certain Diophantine equations under some hypothesis on the arithmetic of the quadratic field. By solving the Diophantine equations explicitly in the rings of quadratic integers, we show the non-existence of such elliptic curve for certain imaginary quadratic fields. This extends the results due to Setzer and Stroeker.

We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston–Graham–Pintz–Yildirim (Proc Lond Math Soc 98:741–774, 2009), and Maynard (Ann Math 181:383–413, 2015). An important consequence of our main theorem is existence of infinitely many pairs alpha _1, alpha _2 which are product of two primes in the imaginary quadratic field K such that |sigma (alpha _1-alpha _2)|le 2 for all embeddings sigma of K if the class number of K is one and |sigma (alpha _1-alpha _2)|le 8 for all embeddings sigma of K if the class number of K is two.

Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-$p$extensions of imaginary quadratic number fields for$p$an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of$\mathbb{F}_{q}(t)$, the Galois groups of the maximal unramified pro-$p$extensions, as$q\rightarrow \infty$, have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

Introduction The aim of this paper is to rework the material in Chapter III of Gross and Zagier's “Heegner points and derivatives of L -series” —see [GZ] in the list of references—based on more systematic deformation-theoretic methods, so as to treat all imaginary quadratic fields, all residue characteristics, and all j -invariants on an equal footing. This leads to more conceptual arguments in several places and interpretations for some quantities which appear to otherwise arise out of thin air in [GZ, Ch. III]. For example, the sum in [GZ, Ch. III, Lemma 8.2] arises for us in (9−6), where it is given a deformation-theoretic meaning. Provided the analytic results in [GZ] are proven for even discriminants, the main results in [GZ] would be valid without parity restriction on the discriminant of the imaginary quadratic field. Our order of development of the basic results follows [GZ, Ch. III], but the methods of proof are usually quite different, making much less use of the “numerology” of modular curves. Here is a summary of the contents. In Section 2 we consider some background issues related to maps among elliptic curves over various bases and horizontal divisors on relative curves over a discrete valuation ring. In Section 3 we provide a brief survey of the Serre–Tate theorem and the Grothendieck existence theorem, since these form the backbone of the deformation-theoretic methods which underlie all subsequent arguments.

A superelliptic curve over a discrete valuation ring $\mathscr{O}$ of residual characteristic p is a curve given by an equation $\mathscr{C}\;:\; y^n=\,f(x)$ , with $\textrm{Disc}(\,f)\neq 0$ . The purpose of this article is to describe the Galois representation attached to such a curve under the hypothesis that f(x) has all its roots in the fraction field of $\mathscr{O}$ and that $p \nmid n$ . Our results are inspired on the algorithm given in Bouw and WewersGlasg (Math. J.59(1) (2017), 77–108.) but our description is given in terms of a cluster picture as defined in Dokchitser et al. (Algebraic curves and their applications, Contemporary Mathematics, vol. 724 (American Mathematical Society, Providence, RI, 2019), 73–135.).