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.
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 show that an infinite family of odd complex 2-dimensional Galois representations ramified at 5 having nonsolvable projective image are modular, thereby verifying Artin's conjecture for a new case of examples. Such a family contains the original example studied by Buhler. In the process, we prove that an infinite family of residually modular Galois representations are modular by studying $Λ$-adic Hecke algebras.
Let p>2 be prime. We complete the proof of the weight part of Serre's conjecture for rank two unitary groups for mod p representations in the totally ramified case, by proving that any weight which occurs is a predicted weight. Our methods are a mixture of local and global techniques, and in the course of the proof we establish some purely local results on crystalline extension classes.
We generalize the main result of arXiv:1206.6631 [math.NT] to all totally real fields. In other words, for $p>2$ prime, we prove (under a mild Taylor-Wiles hypothesis) that if a modular representation is unramified and $p$-distinguished at all places above $p$, then it arises from a mod $p$ Hilbert modular form of parallel weight one. This (mostly) resolves the weight one part of Serre's conjecture for totally real fields.
In this survey paper we present recent results obtained by Khare, Wintenberger and the author that have led to a proof of Serre's conjecture, such as existence of compatible families, modular upper bounds for universal deformation rings and existence of minimal lifts, prime switching and modularity propagation, weight reduction (via existence of conjugates) and (iterated) killing ramification. The main tools used in the proof of these results are modularity lifting theorems a la Wiles and a result of potential modularity due to R. Taylor.
In this article we give a proof of Serre's conjecture for the case of odd level and arbitrary weight. Our proof does not depend on any generalization of Kisin's modularity lifting results to characteristic 2 (moreover, we will not consider at all characteristic 2 representations at any step of our proof). The key tool in the proof is a very general modularity lifting result of Kisin, which is combined with the methods and results of previous articles on Serre's conjecture by Khare, Wintenberger, and the author, and modularity results of Schoof for semistable abelian varieties of small conducto…
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 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.
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…
In the present article, we extend previous results of the author and we show that when $K$ is any quadratic imaginary field of class number one, Fermat's equation $a^p+b^p+c^p=0$ does not have integral coprime solutions $a,b,c \in K \setminus \{ 0 \}$ such that $2 \mid abc$ and $p \geq 19$ is prime. The results are conjectural upon the veracity of a natural generalisation of Serre's modularity conjecture.
The purpose of this short note is to present a simplified proof of Serre's modularity conjecture using the strong modularity lifting results currently available. This second version includes extra details on definitions and proofs than the previous one.
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…
We explain in this letter how using a recent Modularity Lifting Theorem proved by Lue Pan the proofs of Serre's Modularity Conjecture over $\mathbb{Q}$ given by Khare-Wintenberger and the author can be greatly simplified. The main difference with the previous proofs is that neither the process of "weight reduction" developed by Khare in his proof of the level 1 case nor the alternative method of "weight reduction via Galois conjugation" developed by the author are required: weight reduction can now be obtained just with two applications of Pan's result, together with a Lemma that guarantees th…
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…
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 …
We give a refinement of the local class field theory of Serre and Hazewinkel. This refinement allows the theory to treat extensions that are not necessarily totally ramified. Such a refinement was obtained and used in the authors' paper on Fontaine's property (Pm), where the explanation had to be rather brief. In this paper, we give a complete account, from necessary knowledge of an appropriate Grothendieck site to the details of the proof. We start by reviewing the local class field theory of Serre and Hazewinkel.
We study the possible weights of an irreducible two-dimensional mod p representation of the absolute Galois group of F which is modular in the sense of that it comes from an automorphic form on a definite quaternion algebra with centre F which is ramified at all places dividing p, where F is a totally real field. 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.
We construct infinite families of irreducible supersingular mod $p$ representations of $\mathrm{GL}_2(F)$ with $\mathrm{GL}_2(\mathcal{O}_F)$-socle compatible with Serre's modularity conjecture, where $F / \mathbb{Q}_p$ is any finite extension with residue field $\mathbb{F}_{p^2}$ and ramification degree $e \leq (p-1)/2$. These are the first such examples for ramified $F / \mathbb{Q}_p$.
For p = 3 and p = 5, we exhibit a finite nonsolvable extension of the rational numbers which is ramified only at p via explicit computations with Hilbert modular forms.
This brief note only contains a modest contribution: we just fix some inaccuracies in the proof of the prime level weight 2 case of Serre's conjecture given in Khare's preprint "On Serre's modularity conjecture for 2-dimensional mod p representations of G_Q unramified outside p", for the case of trivial character. More precisely, the modularity lifting result needed at a crucial step is the one for the case of a deformation corresponding to a $p$-adic semistable (in the sense of Fontaine) Galois representation attached to a semistable abelian variety, but in Khare's preprint it is applied a le…
If $f(z)$ is a modular form of weight $k$, then the differential operator $\vartheta_k$ defined by $\vartheta_k(f) = \frac{1}{2πi} \frac{d}{dz}f(z) - \frac{k}{12} E_2(z) f(z)$ (known as the Ramanujan-Serre derivative map) is a modular form of weight $k+2$. In this paper, we obtain a simple extension of this map and use it to get a general method to derive certain convolution sums of the divisor functions (using the theory of modular forms). Explicit expressions are given for four types of convolution sums and we provide many examples for all these types of sums.
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…
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.
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 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 extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We use these results to prove many new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent p-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in arXiv:1111.2804.
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…
We give an intrinsic parametrisation of the set of tamely ramified extensions of a local field with finite residue field and bring to the fore the role played by group cohomology. We show that two natural definitions of the cohomology class of a tamely ramified finite galoisian extension coincide, and can be recovered from the parameter. We also give an elementary proof of Serre's mass formula in the tame case and in the simplest wild case, and we classify tame galoisian extensions of degree the cube of a prime.
Serre proved that any holomorphic cusp form of weight one for $Γ_1(N)$ is lacunary while a holomorphic modular form for $Γ_1(N)$ of higher integer weight is lacunary if and only if it is a linear combination of cusp forms of CM-type (see Serre, subsections 7.6 and 7.7). In this paper, we show that when a non-zero modular function of arbitrary real weight for any finite index subgroup of the modular group ${\SL}_2(\Z)$ is lacunary, it is necessarily holomorphic on the upper-half plane, finite at the cusps and has non-negative weight.
Using deformed or twisted Eisenstein Series, we construct a Jacobi-Serre derivative on even-weight Jacobi forms that generalizes the classical Serre derivative on modular forms. As an application, we obtain Ramanujan equations for the index $1$ Eisenstein series $E_{4,1}, E_{6,1}$ and a newly defined $E_{2,1}$. Finally, we relate the deformed Eisenstein Series directly to the classical first Jacobi theta function.
Serre and Stark found a basis for the space of modular forms of weight 1/2 in terms of theta series. In this paper, we generalize their result - under certain mild restrictions on the level and character - to the case of weight 1/2 Hilbert modular forms over a totally real field of narrow class number 1. The methods broadly follow those of Serre-Stark; however we are forced to overcome technical difficulties which arise when we move out of Q.
We determine the conditions under which singular values of multiple $η$-quotients of square-free level, not necessarily prime to~6, yield class invariants, that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. We show that the singular values lie in subfields of the ring class fields of index $2^{k' - 1}$ when $k' \geq 2$ primes dividing the level are ramified in the imaginary-quadratic field, which leads to faster computations of elliptic curves with prescribed complex multiplication. The result is generalised to singular values of modular functions on $X_0^+ (…
Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(\mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on $Γ_{0}(4N)$ for $N=1,2,4$. A proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications we obtain congruences of Borcherds exponents, congruences of quotient of Eisentein series and congruences of values of $L$-functions at a certain point are also studied. Furthermore, the congruences of the Fourier coeff…
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 study an analog of Serre's conjecture over imaginary quadratic fields. In particular, we ask whether the weight recipe of Buzzard, Diamond and Jarvis will hold in this setting. Using a program written by the author, we provide computational evidence that this is in fact the case. In order to justify the method used in the program, we prove that a modular symbols method will work for arbitrary weights over imaginary quadratic fields.
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…
The proof of Serre's conjecture on Galois representations over finite fields allows us to show, using a method due to Serre himself, that all rigid Calabi-Yau threefolds defined over Q are modular.
We use relative trace formula to prove a non-vanishing result and a subconvexity result for the twisted base change $L$-functions associated to Hilbert modular forms whose local components at ramified places are some supercuspidal representations. This generalizes the work of Feigon and Whitehouse.
In this paper, we show that an odd Galois representation rhobar: Gal(Qbar/Q) --> GL_2(F_9) satisfying certain local conditions at 3 and 5 is modular. Our main tool is an idea of Taylor, which reduces the problem to that of exhibiting points on a Hilbert modular surface which are defined over a solvable extension of Q, and which satisfy certain reduction properties. As a corollary, we show that Hilbert-Blumenthal abelian surfaces over Q with good ordinary reduction at 3 and 5 are modular.
Iwasawa theory of modular forms over anticyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields has been studied by several authors, starting from the works of Bertolini-Darmon and Iovita-Spiess, under the crucial assumption that the prime $p$ is unramified in $K$. We start in this article the systematic study of anticyclotomic $p$-adic $L$-functions when $p$ is ramified in $K$. In particular, when $f$ is a weight $2$ modular form attached to an elliptic curve $E/\mathbb{Q}$ having multiplicative reduction at $p$, and $p$ is ramified in $K$, we show an analogue of the exceptional z…
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 construct families of Hilbert modular newforms without exceptional primes. This is achieved by generalizing the notion of good-dihedral primes, introduced by Khare and Wintenberger in their proof of Serre's modularity conjecture, to totally real fields.
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 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".
We show that Siegel modular forms of level Γ_0(p^m) are p-adic modular forms. Moreover we show that derivatives of such Siegel modular forms are p-adic. Parts of our results are also valid for vector-valued modular forms. In our approach to p-adic Siegel modular forms we follow Serre closely; his proofs however do not generalize to the Siegel case or need some modifications.
Let $p>5$ be a prime integer and $K/\mathbb{Q}_p$ a finite ramified extension with ring of integers $\mathcal{O}$ and uniformizer $π$. Let $n>1$ be a positive integer and $ρ_n:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O}/π^n)$ be a continuous Galois representation. In this article we prove that under some technical hypotheses the representation $ρ_n$ can be lifted to a representation $ρ:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O})$. Furthermore, we can pick the lift restriction to inertia at any finite set of primes (at the cost of allowing some extra ramification) and get a deformation problem whose…
Each finite $p$-perfect group $G$ ($p$ a prime) has a universal central $p$-extension. For a perfect group these central extensions come from its {\sl Schur multiplier}. Serre gave a Stiefel-Whitney class approach to analyzing spin covers of alternating groups ($p=2$) aimed at geometric covering space problems. This included the regular version of the Inverse Galois Problem. Every finite simple group with order divisible by $p$ has an infinite string of perfect centerless group covers exhibiting nontrivial Schur multipliers for the prime $p$. Sequences of moduli spaces of curves attached to $G…
