Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one obtains compatible families of mod p representations. While the representations that occur in such a p-adic or mod p family are strongly correlated, in a sense each member of the family reveals a new face of the motive. In recent celebrated work of Wiles playing off a pair of Galois representations in different characteristics has been crucial.
In this paper we investigate when a pair of mod p and mod q representations of the absolute Galois group of a number field K simultaneously arises from an {\it automorphic motive}: we do this in the 1-dimensional (Section 2) and 2-dimensional (Section 3: this time assuming $K={\mathbb Q}$) cases. In Section 3 we formulate a mod pq version of Serre's conjecture refining in part a question of Barry Mazur and Ken Ribet.
The conjecture of Serre referred in the title is the one about modularity of odd Galois representations into GL(2,F) where F is a finite field of characteristic p. We present an analogous conjecture where GL(2) is replaced by GL(n). We explain the analogue of "oddness." We then present some theoretical and experimental evidence for the conjecture, primarily when n = 3. Our conjecture does not require the Galois representation to be irreducible. Our most interesting examples involve the sum of an irreducible even 2-dimensional representation and a 1-dimensional representation.
We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre's conjectures in low levels and weights. For instance we prove that there are no irreducible (p,p) type group schemes over the rational integers. We prove that a Galois representation as above of Artin cond…
Given a totally real number field $F$ and a mod $p$ Galois representation $ρ\colon G_F\to \mathrm{GL}_2(\bar{\mathbf{F}}_p)$, we propose an explicit definition of the set of Serre weights $W(ρ)$ attached to $ρ$. We prove that our explicit definition is equivalent to previous definitions available in the literature. As a consequence we obtain an explicit Serre's modularity conjecture for Hilbert modular forms over totally real number fields. Our work generalises previous work of Dembélé--Diamond--Roberts and Calegari--Emerton--Gee--Mavrides which together give explicit and equivalent sets of we…
We formulate a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for n = 4 using automorphic inductions of Hecke characters.
We prove the explicit version of the Buzzard--Diamond--Jarvis conjecture formulated by Diamond--Dembélé--Roberts. More precisely, we prove that it is equivalent to the original Buzzard--Diamond--Jarvis conjecture, which was proved for odd primes (under a mild Taylor--Wiles hypothesis) in earlier work of the third author and coauthors.
Let F be a totally real field, and v a place of F dividing an odd prime p. We study the weight part of Serre's conjecture for continuous, totally odd, two-dimensional mod p representations rhobar of the absolute Galois group of F that are reducible locally at v. Let W be the set of predicted Serre weights for the semisimplification of rhobar restricted to the decomposition group at v. We prove that when the local representation is generic, the Serre weights in W for which rhobar is modular are exactly the ones that are predicted (assuming that rhobar is modular). We also determine precisely wh…
A generalization of Serre's Conjecture asserts that if $F$ is a totally real field, then certain characteristic $p$ representations of Galois groups over $F$ arise from Hilbert modular forms. Moreover it predicts the set of weights of such forms in terms of the local behavior of the Galois representation at primes over $p$. This characterization of the weights, which is formulated using $p$-adic Hodge theory, is known under mild technical hypotheses if $p > 2$. In this paper we give, under the assumption that $p$ is unramified in $F$, a conjectural alternative description for the set of weight…
In 1987 Serre conjectured that any mod l ("ell", not "1") two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where l is unramified. The hard work is in formulating an analogue of the "weight" part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a "mod l Langlands philosophy". Using ideas of Emerton and Vigneras, we formulate a …
We prove that a selfdual $GL_3$-Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to $3$-dimensional Galois representations with the ground field not equal to $\mathbb{Q}$. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.
Suppose $K/\mathbb{Q}_p$ is finite and $\overline{r}\colon G_K\to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ is a reducible Galois representation. In this paper we prove that we can use the results by the author in [Ste22] to obtain a decomposition of the set of Serre weights $W(\overline{r})$ into a disjoint union of at most $(e+1)^f$ 'packets' of weights (where $f$ is the residue degree and $e$ the ramification degree of $K$) under the assumption that $\overline{r}$ is weakly generic. Thereby, we improve on results of Diamond--Savitt in [DS15] which give a similar decomposition, by rather diffe…
We give an effective version of a result reported by Serre asserting that the images of the Galois representations attached to an abelian surface with $\End(A)= \mathbb{Z}$ are as large as possible for almost every prime. Our algorithm depends on the truth of Serre's conjecture for two dimensional odd irreducible Galois representations. Assuming this conjecture we determine the finite set of primes with exceptional image.
We present a Serre-type conjecture on the modularity of four-dimensional symplectic mod p Galois representations. We assume that the Galois representation is irreducible and odd (in the symplectic sense). The modularity condition is formulated using the etale and the algebraic de Rham cohomology of Siegel modular varieties of level prime to p. We concentrate on the case when the Galois representation is ordinary at p and we give a corresponding list of Serre weights. When the representation is moreover tamely ramified at p, we conjecture that all weights of this list are modular, otherwise we …
Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$ without complex multiplication and with conductor $N$. For each positive integer $m$, the action of the absolute Galois group $G_{\mathbb{Q}}=\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the torsion points over $\bar{\mathbb{Q}}$ gives rise to a representation of $G_\mathbb{Q}$. A celebrated paper of Serre shows that this representation is surjective for all sufficiently large primes; the other primes are termed \emph{exceptional}. Serre conjectures that there are no exceptional primes $\ell>37$ for any non CM elliptic curv…
We study the weight part of (a generalisation of) Serre's conjecture for mod l Galois representations associated to automorphic representations on unitary groups of rank n for odd primes l. Given a modular Galois representation, we use automorphy lifting theorems to prove that it is modular in many other weights. We make no assumptions on the ramification or inertial degrees of l. We give an explicit strengthened result when n = 3 and l splits completely in the underlying CM field.
We study the weight part of (a generalisation of) Serre's conjecture for mod l Galois representations associated to automorphic representations on rank two unitary groups for odd primes l. We propose a conjectural set of Serre weights, agreeing with all conjectures in the literature, and under a mild assumption on the image of the mod l Galois representation we are able to show that any modular representation is modular of each conjectured weight. We make no assumptions on the ramification or inertial degrees of l. Our main innovation is to make use of the lifting techniques introduced in our …
We prove existence of conjugate Galois representations, and we use it to derive a simple method of weight reduction. As a consequence, an alternative proof of the level 1 case of Serre's conjecture follows.
In this paper, we introduce a condition $\mathrm{(F}_m'\mathrm{)}$ on a field $K$, for a positive integer $m$, that generalizes Serre's condition (F) and which still implies the finiteness of the Galois cohomology of finite Galois modules annihilated by $m$ and algebraic $K$-tori that split over an extension of degree dividing $m$, as well as certain groups of étale and unramified cohomology. Various examples of fields satisfying $\mathrm{(F}_m'\mathrm{)}$, including those that do not satisfy (F), are given.
In this note we prove the non existence of certain irreducible two dimensional representations of the the absolute Galois group. Such results are predicted by Serre's conjecture and we use Fontaine's methods to verify these predictions in a small number of cases. We also use a methods of Faltings to study certain finiteness conjectures for Galois representations. Key words: Serre's conjecture, Fontaine-Mazur conjectures, Galois representations.
We gives an explicit genus 3 curve over Q such that the Galois action on the torsion points of its Jacobian is a large as possible. That such curves exist is a consequence of a theorem of D. Zureick-Brown and the author; however, those methods do not produce explicit examples. We shall apply the general strategies of Hall and Serre in their open image theorems. We also make use of Serre's conjecture to show that the modulo ell Galois actions are irreducible. While we computationally focus on a single curve, the methods of this paper can be applied to a large family of genus 3 curves.
We survey some recent progress on generalizations of conjectures of Serre concerning the cohomology of arithmetic groups, focusing primarily on the "weight" aspect. This is intimately related to (generalizations of) a conjecture of Breuil and Mézard relating the geometry of potentially semistable deformation rings to modular representation theory. Recently, B. Levin, S. Morra, and the authors established these conjectures in tame generic contexts by constructing projective varieties (local models) in mixed characteristic whose singularities model, in generic cases, those of tamely potentially …
Let E be an elliptic curve over the rationals without complex multiplication. The absolute Galois group of Q acts on the group of torsion points of E, and this action can be expressed in terms of a Galois representation rho_E:Gal(Qbar/Q) \to GL_2(Zhat). A renowned theorem of Serre says that the image of rho_E is open, and hence has finite index, in GL_2(Zhat). We give the first general bounds of this index in terms of basic invariants of E. For example, the index can be bounded by a polynomial function of the logarithmic height of the j-invariant of E. As an application of our bounds, we settl…
In this paper, we call strongly modular those reducible semi-simple odd mod $l$ Galois representations for which the conclusion of the strongest form of Serre's original modularity conjecture holds. Under the assumption that the Serre weight $k$ satisfies $l\textgreater{}k+1$, we give a precise characterization of strongly modular representations, hence generalizing a classical theorem of Ribet pertaining to the case of conductor $1$.When the representation $ρ$ is not strongly modular, we give a necessary and sufficient condition on the primes $p$ not dividing $Nl$ for which it arises in level…
We prove some new cases of weight part of Serre's conjectures for mod $p$ Galois representations associated to automorphic representations on unitary groups $U(d)$. The approach is a generalization of the work of Gee-Liu-Savitt, namely, we study reductions of certain crystalline representations, as well as crystalline lifts of these reductions.
In this article, I discuss material which is related to the recent proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of Taniyama--Shimura.
In this paper, we consider mod $\ell$ Galois representations of $\mathbb{Q}$. In particular, we obtain an effective criterion to distinguish two semisimple 2-dimensional, odd mod $\ell$ Galois representations up to isomorphism. Serre's conjecture (Khare-Wintenberger's theorem), Sturm's theorem, and its modification by Kohnen are used in our proof.
We prove a version of the weight part of Serre's conjecture for mod $p$ Galois representations attached to automorphic forms on rank 2 unitary groups which are non-split at $p$. More precisely, let $F/F^+$ denote a CM extension of a totally real field such that every place of $F^+$ above $p$ is unramified and inert in $F$, and let $\overline{r}: \textrm{Gal}(\overline{F^+}/F^+) \longrightarrow {}^C\mathbf{U}_2(\overline{\mathbb{F}}_p)$ be a Galois parameter valued in the $C$-group of a rank 2 unitary group attached to $F/F^+$. We assume that $\overline{r}$ is semisimple and sufficiently generi…
Let p be a prime and F a totally real field in which p is unramified. We consider mod p Hilbert modular forms for F, defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to p in characteristic p. For a mod p Hilbert modular Hecke eigenform of arbitrary weight (without parity hypotheses), we associate a two-dimensional representation of the absolute Galois group of F, and we give a conjectural description of the set of weights of all eigenforms from which it arises. This conjecture can be viewed as a "geometric" variant of the "algebraic" Serre weight conj…
Using the link between mod $p$ Galois representations of $\qu$ and mod $p$ modular forms established by Serre's Conjecture, we compute, for every prime $p\leq 1999$, a lower bound for the number of isomorphism classes of continuous Galois representation of $\qu$ on a two--dimensional vector space over $\fbar$ which are irreducible, odd, and unramified outside $p$.
We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, and relate their geometry to the weight part of Serre's conjecture for GL(2).
These are the lecture notes from a five-hour mini-course given at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. Their aim is to give an overview of Serre's modularity conjecture and of its proof by Khare, Wintenberger, and Kisin, as well as of the results of other mathematicians that played an important role in the proof. Along the way we remark on some recent (as of 2012) work concerning generalizations of the conjecture.
We prove that there exists an integer p_0 such that X_split(p)(Q) is made of cusps and CM-points for any prime p>p_0. Equivalently, for any non-CM elliptic curve E over Q and any prime p>p_0 the image of the Galois representation induced by the Galois action on the p-division points of E is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an old question of Serre.
For each of the groups PSL2(F25), PSL2(F32), PSL2(F49), PGL2(F25), and PGL2(F27), we display the first explicitly known polynomials over Q having that group as Galois group. Each polynomial is related to a Galois representation associated to a modular form. We indicate how computations with modular Galois representations were used to obtain these polynomials. For each polynomial, we also indicate how to use Serre's conjectures to determine the modular form giving rise to the related Galois representation.
We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of $U(n)$ which are compact at infinity and split at places dividing $p$ in generic situations. That is, we show that all modular weights for a mod $p$ Galois representation are contained in the set predicted by Herzig. Under some additional hypotheses, we also show modularity of all the "obvious" weights.
Let $E$ be an elliptic curve over $\mathbb{Q}$ such that $\mathrm{End}_{\bar{\mathbb{Q}}}(E)=\mathbb{Z}$ and which admits a non-trivial cyclic $\mathbb{Q}$-isogeny. We prove that, for $p>37$, the residual mod $p$ Galois representation $\barρ_{E,p}:G_{\mathbb{Q}}\rightarrow\mathrm{GL}_2(\mathbb{F}_p)$ is surjective.
Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states that as long as $E$ has no Complex Multiplication (CM), the map $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$ is surjective for all but finitely many $\ell$. We say that a prime number $\ell$ is exceptional (relative to the pair $(E,K)$) if this map is not surjective. Here we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of $E$. We show in particular…
Let $E$ be an elliptic curve over the rationals that does not have complex multiplication. For each prime $\ell$, the action of the absolute Galois group on the $\ell$-torsion points of $E$ can be given in terms of a Galois representation $ρ_{E,\ell}\colon \operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to GL_2(\mathbb{F}_\ell)$. An important theorem of Serre says that $ρ_{E,\ell}$ is surjective for all sufficiently large $\ell$. In this paper, we describe a simple algorithm based on Serre's proof that can quickly determine the finite set of primes $\ell$ for which $ρ_{E,\ell}$ is not surject…
We prove the following uniformity principle: if one of the Galois representations in the family attached to a genus two Siegel cusp form of weight $k>3$, "semistable" and with multiplicity one, is reducible (for an odd prime $p$),then all the representations in the family are reducible. This, combined with Serre's conjecture (which is now a theorem) gives a proof of the Endoscopy Conjecture.
We prove that for forms of U(3) which are compact at infinity and split at places dividing a prime p, in generic situations the Serre weights of a mod p modular Galois representation which is irreducible when restricted to each decomposition group above p are exactly those previously predicted by the third author. We do this by combining explicit computations in p-adic Hodge theory, based on a formalism of strongly divisible modules and Breuil modules with descent data which we develop in the paper, with a technique that we call "weight cycling".
Serre's strong conjecture, now a theorem of Khare and Wintenberger, states that every two-dimensional continuous, odd, irreducible mod $p$ Galois representation $ρ$ arises from a modular form of a specific minimal weight $k(ρ)$, level $N(ρ)$ and character $ε(ρ)$. In this short paper we show that the minimal weight $k(ρ)$ is equal to a notion of minimal weight inspired by the recipe for weights introduced by Buzzard, Diamond and Jarvis. Moreover, using the Breuil-Mézard conjecture we show that both weight recipes are equal to the smallest $k \geq 2$ such that $ρ$ has a crystalline lift of Hodge…
There are two famous Abel Theorems. Most well-known is his description of abelian (analytic) functions on a one dimensional compact complex torus. The other collects together those complex tori, with their prime degree isogenies, into one space. Riemann's generalization of the first features his famous theta functions. His deepest work aimed at extending Abel's second theorem; he died before he fulfilled this. That extension is often pictured on complex higher dimension torii. For Riemann, though, it was to spaces of Jacobians of compact Riemann surfaces, W, of genus g, toward studying functio…
Let $p>2$ be a prime. Let $K$ be a tamely ramified finite extension over $\mathbb Q_p$ with ramification index $e$, and let $G_K$ be the Galois group. We study Kisin modules attached to crystalline representations of $G_K$ whose labeled Hodge-Tate weights are relatively small (a sort of "$er\le p$" condition where $r$ is the maximal Hodge-Tate weight). In particular, we show that these Kisin modules admit "adapted bases". We then apply these results in the special case $e=2$ to study reductions and liftings of certain crystalline representations. As a consequence, we establish some new cases o…
Let F be a totally real field and p an odd prime. If r is a continuous, semisimple, totally odd mod p representation of the absolute Galois group of F which is tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which r is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, which supposed that p was unramified in F. We also prove a theorem towards the conjecture and provide some computational evidence.
We study the possible weights of an irreducible 2-dimensional modular mod p representation of the absolute Galois group of F, where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the prime above p. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.
Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[\ell^n])$, and when $A$ is an elliptic curve we also describe the full Galois group of $K(A_{\text{tors}})/K$. This makes explicit previous results of Serre and Ribet, and strengthens a theorem of Banaszak, Gajda and Krasoń. Our bounds are especially sharp in case the CM type of $A$ is nondegenerate.
We give a parametrization of the possible Serre invariants $(N,k,ν)$ of modular mod $\ell$ Galois representations of the exceptional types $A_4$, $S_4$, $A_5$, in terms of local data attached to the fields cut out by the associated projective representations. We show how this result combined with certain global considerations leads to an effective procedure that will determine for a given eigenform $f$ and prime $\ell$ whether a mod $\ell$ representation attached to $f$ is exceptional. We illustrate with numerical examples.
To each Drinfeld module over a finitely generated field with generic characteristic, one can associate a Galois representation arising from the Galois action on its torsion points. Recent work of Pink and Rütsche has described the image of this representation up to commensurability. Their theorem is qualitative, and the objective of this paper is to complement this theory with a worked out example. In particular, we give examples of Drinfeld modules of rank 2 for which the Galois action on their torsion points is as large as possible. We will follow the approach that Serre used to give explici…
Using Serre's proposed complement to Shih's Theorem, we obtain PSL_2(F_p) as a Galois group over Q for at least 614 new primes p. Under the assumption that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for 3/8 of the primes not covered by previous results; it would also suffice to assume a certain (plausible, and perhaps tractable) conjecture concerning class numbers of quadratic fields. The key issue is to understand rational points on Atkin-Lehner twists of X_0(N). In an appendix, we explore the existence of local points on these curves.
Let $ρ_1$ and $ρ_2$ be a pair of residual, odd, absolutely irreducible two-dimensional Galois representations of a totally real number field $F$. In this article we propose a conjecture asserting existence of "safe" chains of compatible systems of Galois representations linking $ρ_1$ to $ρ_2$. Such conjecture implies the generalized Serre's conjecture and is equivalent to Serre's conjecture under a modular version of it. We prove a weak version of the modular variant using the connectedness of certain Hecke algebras, and we comment on possible applications of these results to establish some ca…
We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(\FF_{\ell})$ are Galois groups over $\QQ$ for large enough primes $\ell$.
