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We study the generalized Fermat equation
$x^{2}+y^{3}=z^{p}$
, to be solved in coprime integers, where
$p\geqslant 7$
is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve
$X(p)$
. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic
$p$
-torsion modules. Using these criteria we produce the minimal list of twists of
$X(p)$
that have to be considered, based on local information at 2 and 3; this list depends on
$p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$
. We solve the equation completely when
$p=11$
, which previously was the smallest unresolved
$p$
. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on
$X_{0}(11)$
defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case
$p=13$
. The source code for the various computations is supplied as supplementary material with the online version of this article.
Fix an odd prime p. Let
$\mathcal{D}_n$
denote a non-abelian extension of a number field K such that
$K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $
and whose Galois group has the form
$ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $
where g > 0 and
$0 \lt n'\leq n$
. Given a modular Galois representation
$\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$
which is p-ordinary and also p-distinguished, we shall write
$\mathcal{H}(\overline{\rho})$
for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant
\begin{equation}
\lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of }
\text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big)
\end{equation}
at all
$f\in\mathcal{H}(\overline{\rho})$
, under the assumption
$\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$
for at least one form f0. We can then easily deduce that
$\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$
is constant along branches of
$\mathcal{H}(\overline{\rho})$
, generalising a theorem of Emerton, Pollack and Weston for
$\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$
.
For example, if
$\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$
has the structure of a p-adic Lie extension then our formulae include the cases where: either (i)
$\mathcal{D}_{\infty}/K$
is a g-fold false Tate tower, or (ii)
$\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$
has dimension ≤ 3 and is a pro-p-group.
Let
$A$
be the product of an abelian variety and a torus defined over a number field
$K$
. Fix some prime number
$\ell$
. If
$\unicode[STIX]{x1D6FC}\in A(K)$
is a point of infinite order, we consider the set of primes
$\mathfrak{p}$
of
$K$
such that the reduction
$(\unicode[STIX]{x1D6FC}\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{p})$
is well-defined and has order coprime to
$\ell$
. This set admits a natural density. By refining the method of Jones and Rouse [Galois theory of iterated endomorphisms, Proc. Lond. Math. Soc. (3)100(3) (2010), 763–794. Appendix A by Jeffrey D. Achter], we can express the density as an
$\ell$
-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of
$\ell$
) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.
Let
$p$
be a prime, let
$K$
be a complete discrete valuation field of characteristic
$0$
with a perfect residue field of characteristic
$p$
, and let
$G_{K}$
be the Galois group. Let
$\unicode[STIX]{x1D70B}$
be a fixed uniformizer of
$K$
, let
$K_{\infty }$
be the extension by adjoining to
$K$
a system of compatible
$p^{n}$
th roots of
$\unicode[STIX]{x1D70B}$
for all
$n$
, and let
$L$
be the Galois closure of
$K_{\infty }$
. Using these field extensions, Caruso constructs the
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$
-modules, which classify
$p$
-adic Galois representations of
$G_{K}$
. In this paper, we study locally analytic vectors in some period rings with respect to the
$p$
-adic Lie group
$\operatorname{Gal}(L/K)$
, in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules, we can establish the overconvergence property of the
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$
-modules.
We show that the Galois cohomology groups of
$p$
-adic representations of a direct power of
$\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$
can be computed via the generalization of Herr’s complex to multivariable
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules. Using Tate duality and a pairing for multivariable
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules we extend this to analogues of the Iwasawa cohomology. We show that all
$p$
-adic representations of a direct power of
$\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$
are overconvergent and, moreover, passing to overconvergent multivariable
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.
We consider families of Siegel eigenforms of genus
$2$
and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call the Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zero locus defined by the Galois level of the family admits an automorphic description: it is the locus of points that arise from overconvergent eigenforms for
$\operatorname{GL}_{2}$
, via a
$p$
-adic Langlands lift attached to the symmetric cube representation. Our proof goes via the comparison of the Galois level with a ‘fortuitous’ congruence ideal. Some of the
$p$
-adic lifts are interpolated by a morphism of rigid analytic spaces from an eigencurve for
$\operatorname{GL}_{2}$
to an eigenvariety for
$\operatorname{GSp}_{4}$
, while the remainder appear as isolated points on the eigenvariety.
J. Bellaïche and M. Dimitrov showed that the
$p$
-adic eigencurve is smooth but not étale over the weight space at
$p$
-regular theta series attached to a character of a real quadratic field
$F$
in which
$p$
splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over
$F$
at the overconvergent cuspidal Eisenstein points, being the base change lift for
$\text{GL}(2)_{/F}$
of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.
Darmon, Lauder, and Rotger conjectured that the relative tangent space of an eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of
$p$
-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.
We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group
$\unicode[STIX]{x1D6E4}$
. Under favorable conditions, the cohomology is freely generated in a single degree over this graded Hecke algebra.
From this construction we extract an action of certain
$p$
-adic Galois cohomology groups on
$H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q}_{p})$
, and formulate the central conjecture: the motivic
$\mathbf{Q}$
-lattice inside these Galois cohomology groups preserves
$H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q})$
.
We construct, over any CM field, compatible systems of
$l$
-adic Galois representations that appear in the cohomology of algebraic varieties and have (for all
$l$
) algebraic monodromy groups equal to the exceptional group of type
$E_{6}$
.
We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over
$\mathbb{Q}$
of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use
$R=T$
theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from
$\text{GSp}(4)$
to
$U(4)$
. In that case we obtain a corollary for abelian surfaces.
Let
$p$
and
$\ell$
be distinct primes, and let
$\overline{\unicode[STIX]{x1D70C}}$
be an orthogonal or symplectic representation of the absolute Galois group of an
$\ell$
-adic field over a finite field of characteristic
$p$
. We define and study a liftable deformation condition of lifts of
$\overline{\unicode[STIX]{x1D70C}}$
‘ramified no worse than
$\overline{\unicode[STIX]{x1D70C}}$
’, generalizing the minimally ramified deformation condition for
$\operatorname{GL}_{n}$
studied in Clozel et al. [Automorphy for some
$l$
-adic lifts of automorphic mod
$l$
Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields.
Let F be a number field, let N ≥ 3 be an integer, and let k be a finite field of characteristic ℓ. We show that if ρ:GF → GLN(k) is a continuous representation with image of ρ containing SLN(k) then, under moderate conditions at primes dividing ℓ∞, there is a continuous representation ρ:GF → GLN(W(k)) unramified outside finitely many primes with ρ ~ρ mod ℓ. Stronger results are presented for ρ:Gℚ → GL3(k).
Let
$G$
be a semisimple Lie group with associated symmetric space
$D$
, and let
$\unicode[STIX]{x1D6E4}\subset G$
be a cocompact arithmetic group. Let
$\mathscr{L}$
be a lattice inside a
$\mathbb{Z}\unicode[STIX]{x1D6E4}$
-module arising from a rational finite-dimensional complex representation of
$G$
. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup
$H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}$
as
$\unicode[STIX]{x1D6E4}_{k}$
ranges over a tower of congruence subgroups of
$\unicode[STIX]{x1D6E4}$
. In particular, they conjectured that the ratio
$\log |H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}|/[\unicode[STIX]{x1D6E4}:\unicode[STIX]{x1D6E4}_{k}]$
should tend to a nonzero limit if and only if
$i=(\dim (D)-1)/2$
and
$G$
is a group of deficiency
$1$
. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including
$\operatorname{GL}_{n}(\mathbb{Z})$
for
$n=3,4,5$
and
$\operatorname{GL}_{2}(\mathscr{O})$
for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron–Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron–Venkatesh conjecture.
We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.
The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over
$\overline{\mathbb{F}}_{p}$
of parallel weight 1 and level prime to
$p$
is unramified above
$p$
. This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight 1. The proof is based on the observation that parallel weight 1 forms in characteristic
$p$
embed into the ordinary part of parallel weight
$p$
forms in two different ways per prime dividing
$p$
, namely via ‘partial’ Frobenius operators.
Modular curves like X0(N) and X1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here by Xnsp(p) and Xnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures of p-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc. (3) 77(1) (1998), 1–38; J. Algebra231(1) (2000), 414–448).
Let
$K$
be the field of fractions of a local Henselian discrete valuation ring
${\mathcal{O}}_{K}$
of characteristic zero with perfect residue field
$k$
. Assuming potential semi-stable reduction, we show that an unramified Galois action on the second
$\ell$
-adic cohomology group of a K3 surface over
$K$
implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga–Satake Abelian varieties. On our way, we settle existence and termination of certain flops in mixed characteristic, and study group actions and their quotients on models of varieties.
Suppose that
$F/F^{+}$
is a CM extension of number fields in which the prime
$p$
splits completely and every other prime is unramified. Fix a place
$w|p$
of
$F$
. Suppose that
$\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$
is a continuous irreducible Galois representation such that
$\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$
is upper-triangular, maximally non-split, and generic. If
$\overline{r}$
is automorphic, and some suitable technical conditions hold, we show that
$\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$
can be recovered from the
$\text{GL}_{3}(F_{w})$
-action on a space of mod
$p$
automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for
$\overline{r}$
, show the existence of an ordinary lifting of
$\overline{r}$
, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations
$\overline{r}$
to which our main theorem applies.
Let
$A\rightarrow B$
be a morphism of Artin local rings with the same embedding dimension. We prove that any
$A$
-flat
$B$
-module is
$B$
-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond’s criterion [The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391, Theorem 2.1]. We also prove that if there is a nonzero
$A$
-flat
$B$
-module, then
$A\rightarrow B$
is flat and is a relative complete intersection. Then we explain how this result allows one to simplify Wiles’s proof of Fermat’s last theorem: we do not need the so-called ‘Taylor–Wiles systems’ any more.