We prove R = T theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin on whether certain Hecke algebras T are discrete valuation rings. In order to prove these results we determine (using the theory of Breuil modules) when two finite flat group schemes G and H of order p over an arbitrarily tamely ramified discrete valuation ring admit an extension not killed by p.
We prove the equality of several $τ$-recurrent sequences, which were first considered by Pellarin, and which have close connections to Drinfeld vectorial modular forms. Our result has several consequences: an $A$-expansion for the $l^\text{th}$ power ($1 \leq l \leq q$) of the deformation of the weight 2 Eisenstein series; relations between Drinfeld modular forms with $A$-expansions; a new proof of relations between special values of Pellarin $L$-series.
One of the many number theoretic topics investigated by the ancient Greeks was perfect numbers, which are positive integers equal to the sum of their proper positive integral divisors. Mathematicians from Euclid to Euler investigated these mysterious numbers. We present results on perfect numbers in the ring of Eisenstein integers.
We generalise the sum-of-divisors-function $σ$ and evenness to the rings of integers of certain algebraic number fields. In particular, we present necessary and sufficient conditions for even Eisenstein integers to be (norm-)perfect based on the work of McDaniel [McD] on Gaussian integers. Furthermore, some results concerning odd norm-perfect Eisenstein integers and the rings of integers of other cyclotomic fields are proven.
There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.
We study nearly holomorphic Siegel Eisenstein series of general levels and characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of $\mathbb{Q}_p$ for all sufficiently large primes $p$. We also prove that the pullbacks of these Eisenstein series to $\mathbb{H}_n \times \mathbb{H}_n$ are cuspidal under certain assumptions.
We prove that the cuspidal eigencurve $C_{\mathrm{cusp}}$ is étale over the weight space at any classical weight $1$ Eisenstein point $f$ and meets two Eisenstein components of the eigencurve $C$ transversally at $f$. Further, we prove that the local ring of $C$ at $f$ is Cohen--Macaulay but not Gorenstein and compute the Fourier coefficients of a basis of overconvergent weight $1$ modular forms lying in the same generalised eigenspace as $f$. In addition, we prove an $R=T$ theorem for the local ring at $f$ of the closed subspace of $C$ given by the union of $C_{\mathrm{cusp}}$ and one Eisenst…
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.
Let p > 2 be a prime, let n > m > 0. Let pi_n be the norm of zeta_{p^n} - 1 under C_{p-1}, so that Z_(p)[pi_n] | Z_(p) is a purely ramified extension of discrete valuation rings of degree p^{n-1}. The minimal polynomial of pi_n over Q(pi_m) is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at pi_m. The function field analogue, as introduced by Carlitz and Hayes, is studied as well.
We use pseudodeformation theory to study the analogue of Mazur's Eisenstein ideal with certain squarefree levels. Given a prime number $p>3$ and a squarefree number $N$ satisfying certain conditions, we study the Eisenstein part of the $p$-adic Hecke algebra for $Γ_0(N)$, and show that it is a local complete intersection and isomorphic to a pseudodeformation ring. We also show that in certain cases, the Eisenstein ideal is not principal and that the cuspidal quotient of the Hecke algebra is not Gorenstein. As a corollary, we prove that "multiplicity one" fails for the modular Jacobian in these…
The Euler phi function on a given integer $n$ yields the number of positive integers less than $n$ that are relatively prime to $n$. Equivalently, it gives the order of the group of units in the quotient ring $\mathbb{Z}/(n)$. We generalize the Euler phi function to the Eisenstein integer ring $\mathbb{Z}[ρ]$ where $ρ$ is the primitive third root of unity $e^{2πi/3}$ by finding the order of the group of units in the ring $\mathbb{Z}[ρ]/(θ)$ for any given Eisenstein integer $θ$. As one application we investigate a sufficiency criterion for when certain unit groups $\left(\mathbb{Z}[ρ]/(γ^n)\rig…
Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight $k$ and level $Γ_0(\ell)$ at the maximal Eisenstein ideal containing $p$. We give a necessary and sufficient condition for the $\mathbb{Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for …
Let H=O(n)xO(1) be an anisotropic subgroup of G=O(n, 1) and let A be the adele ring of k=Q. Consider the periods of an Eisenstein series on G against a form F on H. Relying on a variant of Levi-Sobolev spaces, we describe certain Poincaré series as fundamental solutions for the laplacian, and use them to establish a spectral identity concerning the second moments (in F-aspect) of the Eisenstein series.
We generalize a result of Galatius and Venkatesh which relates the graded module of cohomology of locally symmetric spaces to the graded homotopy ring of the derived Galois deformation rings, by removing certain assumptions, and in particular by allowing congruences inside the localized Hecke algebra.
We prove that the universal unramified deformation ring $R^{\mathrm{unr}}$ of a continuous Galois representation $\overlineρ: G_{F^{+}} \rightarrow \mathrm{GL}_n(k)$ (for a totally real field $F^{+}$ and finite field $k$) is finite over $\mathcal{O} = W(k)$ in many cases. We also prove (under similar hypotheses) that the universal deformation ring $R^{\mathrm{univ}}$ is finite over the local deformation ring $R^{\mathrm{loc}}$.
We give a different approach to the well-known modularity lifting results of Wiles and Taylor. Instead of Taylor-Wiles systems we use a Galois cohomological discovery of R. Ramakrishna. This paper is 2 years older than math.NT/0210296 where another approach to modularity lifting results was presented.
We study the potentially semi-stable deformation rings for Galois representations taking their values in $PGL_n$, by comparing them to the deformation rings for $GL_n$. As an application, we state an analogue of the Breuil-Mézard conjecture, and we show that the case of $PGL_n$ follows from the case of $GL_n$.
We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring $\mathcal{R}$ classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group $π_0 \mathcal{R}$ recovers Mazur's deformation ring. We give evidence that these rings $\mathcal{R}$ occur in the wild: For suitable Galois representations, the Langlands program predicts that $π_0 \mathcal{R}$ should act on the homology of an arithmetic group. We explain how the Taylor--Wiles method can be used to upgrade such an action to a graded action of $π_* \mathcal{R…
Let $K$ be a finite extension of $\mathbb{Q}_p$, and choose a uniformizer $π\in K$, and put $K_\infty:=K(\sqrt[p^\infty]π)$. We introduce a new technique using restriction to $\Gal(\ol K/K_\infty)$ to study flat deformation rings. We show the existence of deformation rings for $\Gal(\ol K/K_\infty)$-representations ``of height $\leqslant h$'' for any positive integer $h$, and we use them to give a variant of Kisin's proof of connected component analysis of a certain flat deformation rings, which was used to prove Kisin's modularity lifting theorem for potentially Barsotti-Tate representations.…
We extend Kisin's results on the structure of characteristic $0$ Galois deformation rings to deformation rings of Galois representations valued in arbitrary connected reductive groups $G$. In particular, we show that such Galois deformation rings are complete intersection. In addition, we study explicitly the structure of the moduli space $X_{φ,N}$ of (framed) $(φ,N)$-modules when $G=\GL_n$. We show that when $G=\GL_3$ and $K_0=\Q_p$, $X_{φ,N}$ has a singular component, and we construct a moduli-theoretic resolution of singularities.
We define deformation rings for potentially semi-stable deformations of fixed discrete series inertial type in dimension $2$. In the case of representations of the Galois group of $\mathbf{Q}_p$, we prove an analogue of the Breuil-Mézard conjecture for these rings. As an application, we give some results on the existence of congruences modulo $p$ for newforms in $S_k(Γ_0(p))$.
We study the Zariski closure of points in local deformation rings corresponding to potential semi-stable representations with certain prescribed $p$-adic Hodge theoretic properties. We show in favourable cases that the closure is equal to a union of irreducible components of the deformation space. We also study an analogous question for global Hecke algebras.
We compute the universal deformation ring of an odd Galois two dimensional representation of Gal$(M/Q)$ with an upper triangular image, where $M$ is the maximal abelian pro-$p$-extension of $F_{\infty}$ unramified outside a finite set of places S, $F_{\infty}$ being a free pro-$p$-extension of a subextension $F$ of the field $K$ fixed by the kernel of the representation. We establish a link between the latter universal deformation ring and the universal deformation ring of the representation of Gal$(K_S/Q)$, where $K_S$ is the maximal pro-$p$-extension of $K$ unramified outside $S$. We then gi…
We prove the weight part of Serre's conjecture in generic situations for forms of $U(3)$ which are compact at infinity and split at places dividing $p$ as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights $(2,1,0)$ for $K/\mathbb{Q}_p$ unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings.
We compute presentations of crystalline framed deformation rings of a two dimensional representation $\barρ$ of the absolute Galois group of $\mathbb{Q}_p$, when $\barρ$ has scalar semi-simplification, the Hodge-Tate weights are small and $p>2$. In the non-trivial cases, we show that the special fibre is geometrically irreducible, generically reduced and the Hilbert-Samuel multiplicity is either $1$, $2$ or $4$ depending on $\barρ$. We show that in the last two cases the deformation ring is not Cohen-Macaulay.
We study G-valued Galois deformation rings with prescribed properties, where G is an arbitrary (not necessarily connected) reductive group over an extension of Z_l for some prime l. In particular, for the Galois groups of p-adic local fields (with p possibly equal to l) we prove that these rings are generically smooth, compute their dimensions, and show that functorial operations on Galois representations give rise to well-defined maps between the sets of irreducible components of the corresponding deformation rings. We use these local results to prove lower bounds on the dimension of global d…
A well known result of B. Mazur gives a lower bound for the Krull dimension of the universal deformation ring associated to an absolutely irreducible residual representation in terms of the group cohomology of the adjoint representation. The question about equality - at least in the Galois case - also goes back to B. Mazur. In the general case the question about equality is the subject of Gouvêa's "Dimension conjecture". In this note we provide a counterexample to this conjecture. More precisely, we construct an absolutely irreducible residual representation with smooth universal deformation r…
We compute the deformation rings of two dimensional mod l representations of Gal(Fbar/F) with fixed inertial type, for l an odd prime, p a prime distinct from p and F/Q_p a finite extension. We show that in this setting (when p is also odd) an analogue of the Breuil-Mézard conjecture holds, relating the special fibres of these deformation rings to the mod l reduction of certain irreducible representations of GL_2(O_F).
We study the inverse problem for the versal deformation rings $R(Γ,V)$ of finite dimensional representations $V$ of a finite group $Γ$ over a field $k$ of positive characteristic $p$. This problem is to determine which complete local commutative Noetherian rings with residue field $k$ can arise up to isomorphism as such $R(Γ,V)$. We show that for all integers $n \ge 1$ and all complete local commutative Noetherian rings $\mathcal{W}$ with residue field $k$, the ring $\mathcal{W}[[t]]/(p^n t,t^2)$ arises in this way. This ring is not a local complete intersection if $p^n\mathcal{W}\neq\{0\}$, s…
We prove the finiteness of Selmer groups attached to lifts of certain 2-dimensional mod p representations of the absolute Galois group of Q. The mod p representation can be either even or odd. The lifts considered are the ones that were proven to exist by the seond named author.
Let $\mathcal{W}$ be a complete local commutative Noetherian ring with residue field $k$ of positive characteristic $p$. We study the inverse problem for the versal deformation rings $R_{\mathcal{W}}(Γ,V)$ relative to $\mathcal{W}$ of finite dimensional representations $V$ of a profinite group $Γ$ over $k$. We show that for all $p$ and $n \ge 1$, the ring $\mathcal{W}[[t]]/(p^n t,t^2)$ arises as a universal deformation ring. This ring is not a complete intersection if $p^n\mathcal{W}\neq\{0\}$, so we obtain an answer to a question of M. Flach in all characteristics. We also study the `inverse …
We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_K$ for a finite extension $K/\mathbb{Q}_p$. This is done by considering a moduli space of Breuil--Kisin modules, satisfying an additional Galois condition, over the universal deformation ring. For $K$ unramified over $\mathbb{Q}_p$ and Hodge--Tate weights in $[0,p]$, we study the geometry of this space. As a consequence we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_p$, with Hodge--Tate weights in $[0,p]$, are potentiall…
Given a continuous, odd, semi-simple $2$-dimensional representation of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$ and a prime $\ell$ not dividing $Np$, we study the relation between the universal deformation rings of the corresponding pseudo-representation for the groups $G_{\mathbb{Q},N\ell p}$ and $G_{\mathbb{Q},Np}$. As a related problem, we investigate when the universal pseudo-representation arises from an actual representation over the universal deformation ring. Under some hypotheses, we prove analogues of theorems of Boston and Böckle for the reduced pseudo-defor…
We study potentially crystalline deformation rings for a residual, ordinary Galois representation $\overlineρ: G_{\mathbb{Q}_p}\rightarrow \mathrm{GL}_3(\mathbb{F}_p)$. We consider deformations with Hodge-Tate weights $(0,1,2)$ and inertial type chosen to contain exactly one Fontaine-Laffaille modular weight for $\overlineρ$. We show that, in this setting, the potentially crystalline deformation space is formally smooth over $\mathbb{Z}_p$ and any potentially crystalline lift is ordinary. The proof requires an understanding of the condition imposed by the monodromy operator on Breuil modules w…
For a smooth group scheme $G$ over an extension of $\mathbf{Z}_p$ such that the generic fiber of $G$ is reductive, we study the generic fiber of the Galois deformation ring for a $G$-valued mod $p$ representation of the absolute Galois group of a finite extension of $\mathbf{Q}_\ell$ with $\ell \neq p$. In particular, we show it admits a regular dense open locus, and that it is equidimensional of dimension $\dim G$.
One of the basic questions in number theory is to determine semi-simple l-adic representations of the absolute Galois group of a number field. In this paper, we discuss the question for two dimensional representations over a totally real number field.
Let $F$ be a CM field and let $(\overline{r}_{π,λ})_λ$ be the compatible system of residual $\mathcal{G}_n$-valued representations of $\operatorname{Gal}_{F}$ attached to a RACSDC automorphic representation $π$ of $\operatorname{GL}_n(\mathbb{A})$, as studied by Clozel, Harris and Taylor and others. Under mild assumptions, we prove that the fixed-determinant universal deformation rings attached to $\overline{r}_{π,λ}$ are unobstructed for all places $λ$ in a subset of Dirichlet density $1$, continuing the investigations of Mazur, Weston and Gamzon. During the proof, we develop a general framew…
Let R^univ be the universal deformation ring of a residual representation of a local Galois group. Kisin showed that many loci in MaxSpec(R^univ[1/p]) of interest are Zariski closed, and gave a way to study the generic fiber of the corresponding quotient of R^univ. However, his method gives little information about the quotient ring before inverting p. We give a method for studying this quotient in certain cases, and carry it out in the simplest non-trivial case. Precisely, suppose that V_0 is the trivial two dimensional representation and let R be the unique Z_p-flat and reduced quotient of R…
Using $p$-adic local Langlands correspondence for $\operatorname{GL}_2(\mathbb{Q}_2)$ and an ordinary $R = \mathbb{T}$ theorem, we prove that the support of patched modules for quaternionic forms meet every irreducible component of the potentially semi-stable deformation ring. This gives a new proof of the Breuil-Mézard conjecture for 2-dimensional representations of the absolute Galois group of $\mathbb{Q}_2$, which is new in the case $\overline{r}$ a twist of an extension of the trivial character by itself. As a consequence, a local restriction in Paškūnas' proof of Fontaine-Mazur conjecture…
Using $p$-adic local Langlands correspondence for $\operatorname{GL}_2(\mathbb{Q}_p)$, we prove that the support of patched modules constructed by Caraiani, Emerton, Gee, Geraghty, Paskunas, and Shin meet every irreducible component of the potentially semistable deformation ring. This gives a new proof of the Breuil-Mézard conjecture for 2-dimensional representations of the absolute Galois group of $\mathbb{Q}_p$ when $p > 2$, which is new in the case $p = 3$ and $\bar{r}$ a twist of an extension of the trivial character by the mod p cyclotomic character. As a consequence, a local restriction …
Given a continuous, odd, reducible and semi-simple $2$-dimensional representation $\barρ_0$ of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$, we study the relation between the universal deformation ring of the pseudo-representation corresponding to $\barρ_0$ (pseudo-deformation ring) and the big $p$-adic Hecke algebra to prove that the maximal reduced quotient of the pseudo-deformation ring is isomorphic to the local component of the big $p$-adic Hecke algebra corresponding to $\barρ_0$ if a certain global Galois cohomology group has dimension $1$. This partially extends th…
For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, Böckle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouvêa-Mazur. We generalize Böckle's result to the context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under …
Let k denote a perfect field of characteristic 5. We show that the versal deformation ring of an element of order 5 and Hasse conductor 2 as automorphism of a ring of formal power series k[[t]] computed by Bertin and Mezard, is in fact universal. This provides the first example of a non trivial universal deformation ring in higher conductor.
On the dimension conjecture for global Galois deformation rings in characteristic two
Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究院）
Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group, and B is a block of kG with dihedral defect group D which is Morita equivalent to the principal 2-modular block of a finite simple group. We determine the universal deformation ring R(G,V) for every kG-module V which belongs to B and has stable endomorphism ring k. It follows that R(G,V) is always isomorphic to a subquotient ring of WD. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersection…
