Modular Galois Representations Research Articles

On the mod p cohomology for GL2

Let p be a prime number and F a totally real number field unramified at places above p. Let r‾:Gal(F‾/F)→GL2(Fp‾) be a modular Galois representation which satisfies the Taylor-Wiles hypothesis and some technical genericity assumptions. For v a fixed place of F above p, we prove that many of the admissible smooth representations of GL2(Fv) over Fp‾ associated to r‾ in the corresponding Hecke-eigenspaces of the mod p cohomology have Gelfand–Kirillov dimension [Fv:Qp]. This builds on and extends the work of Breuil-Herzig-Hu-Morra-Schraen in [2] and Hu-Wang in [12], giving a unified proof in all cases (r‾ either semisimple or not at v).

$$\lambda $$-Invariant stability in families of modular Galois representations

$$\lambda $$-Invariant stability in families of modular Galois representations

Congruence relations for r-colored partitions

Let ℓ≥5 be prime. For the partition function p(n) and 5≤ℓ≤31, Atkin found a number of examples of primes Q≥5 such that there exist congruences of the form p(ℓQ3n+β)≡0(modℓ). Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every ℓ. In this paper, for a wide range of c∈Fℓ, we prove congruences of the form p(ℓQ3n+β0)≡c⋅p(ℓQn+β1)(modℓ) for infinitely many primes Q. For a positive integer r, let pr(n) be the r-colored partition function. Our methods yield similar congruences for pr(n). In particular, if r is an odd positive integer for which ℓ>5r+19 and 2r+2≢2±1(modℓ), then we show that there are infinitely many congruences of the form pr(ℓQ3n+β)≡0(modℓ). Our methods involve the theory of modular Galois representations.

Ideal Class Groups of Number Fields Associated to Modular Galois Representations

Ideal Class Groups of Number Fields Associated to Modular Galois Representations

Congruences like Atkin’s for the partition function

Let p ( n ) p(n) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p ( Q 3 ℓ n + β ) ≡ 0 ( mod ℓ ) p( Q^3 \ell n+\beta )\equiv 0\pmod \ell where ℓ \ell and Q Q are prime and 5 ≤ ℓ ≤ 31 5\leq \ell \leq 31 ; these lie in two natural families distinguished by the square class of 1 − 24 β ( mod ℓ ) 1-24\beta \pmod \ell . In recent decades much work has been done to understand congruences of the form p ( Q m ℓ n + β ) ≡ 0 ( mod ℓ ) p(Q^m\ell n+\beta )\equiv 0\pmod \ell . It is now known that there are many such congruences when m ≥ 4 m\geq 4 , that such congruences are scarce (if they exist at all) when m = 1 , 2 m=1, 2 , and that for m = 0 m=0 such congruences exist only when ℓ = 5 , 7 , 11 \ell =5, 7, 11 . For congruences like Atkin’s (when m = 3 m=3 ), more examples have been found for 5 ≤ ℓ ≤ 31 5\leq \ell \leq 31 but little else seems to be known. Here we use the theory of modular Galois representations to prove that for every prime ℓ ≥ 5 \ell \geq 5 , there are infinitely many congruences like Atkin’s in the first natural family which he discovered and that for at least 17 / 24 17/24 of the primes ℓ \ell there are infinitely many congruences in the second family.

Moduli-friendly Eisenstein series over the p-adics and the computation of modular Galois representations

We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of “moduli-friendly” Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic points of modular curves and dispenses us of the need for equations of modular curves and for q-expansion computations in the construction of models of modular Jacobians. The resulting algorithm compares very favourably to our complex-analytic method.

Integral period relations and congruences

Under relatively mild and natural conditions, we establish an integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the {\it congruence number} controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch-Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modular form, the cotangent complex of the corresponding Hecke algebra and the archimedean regulator attached to some Beilinson-Flach element.

Realization of modular Galois representations in the Jacobians of modular curves

In Tian (Acta Arith. 164:399–412, 2014), the author improved the algorithm proposed by Edixhoven and Couveignes for computing mod \(\ell \) Galois representations associated to eigenforms f for the cases that \(\ell \ge k-1\) and f has level one, where k is the weight of f. In this paper, we generalize the results of Tian (Acta Arith. 164:399–412, 2014) and present a method to find the Jacobians of modular curves of minimal dimensions to realize the modular Galois representations. Our method works for the cases that \(\ell \ge 5\) may be any prime without the assumption \(\ell \ge k-1\) and the eigenforms f have arbitrary levels prime to \(\ell \). Moreover, if \(k>2\), we give criteria for realizing the mod \(\ell \) Galois representations in the Jacobians \(J_0\) of \(X_0\).

On Euler systems for adjoint Hilbert modular Galois representations

We prove the existence of Euler systems for adjoint modular Galois representations using deformations of Galois representations coming from Hilbert modular forms and relate them to $p$-adic $L$-functions under a conjectural formula for the Fitting ideals of some equivariant congruence modules for abelian base change.

Selmer groups of symmetric powers of ordinary modular Galois representations

Let $p$ be a fixed odd prime number, $\mu$ be a Hida family over the Iwasawa algebra of one variable, $\rho_{\mu}$ its Galois representation, $\Bbb{Q}_\infty/\Bbb{Q}$ the $p$-cyclotomic tower and $S$ the variable of the cyclotomic Iwasawa algebra. We compare, for $n\leq 4$ and under certain assumptions, the characteristic power series $L(S)$ of the dual of Selmer groups $\textrm{Sel}(\Bbb{Q}_{\infty},\textrm{Sym}^{2n}\otimes\textrm{det}^{-n} \rho_{\mu})$ to certain congruence ideals (the case $n=1$ has been treated by H. Hida). In particular, we express the first term of the Taylor expansion at the trivial zero $S=0$ of $L(S)$ in terms of an $\scr{L}$-invariant and a congruence number. We conjecture the non-vanishing of this $\scr{L}$-invariant; this implies therefore that these Selmer groups are cotorsion. We also show that our $\scr{L}$-invariants coincide with Greenberg's $\scr{L}$-invariants calculated by R. Harron and A. Jorza.

MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL

AbstractLet $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We describe the $\mathfrak{m}$-torsion in the $\text{mod}\,p$ cohomology of Shimura curves with full congruence level at $v$ as a $\text{GL}_{2}(k_{v})$-representation. In particular, it only depends on $\overline{r}|_{I_{F_{v}}}$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $\text{GL}_{2}(\mathbf{F}_{q})$-projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math. 200(1) (2015), 1–96].

Multiplicity one for wildly ramified representations

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor-Wiles hypotheses and is generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. Then the $\mathfrak{m}$-torsion in the mod $p$ cohomology of Shimura curves with full congruence level at $v$ coincides with the $\mathrm{GL}_2(k_v)$-representation $D_0(\overline{r}|_{G_{F_v}})$ constructed by Breuil and Pa\v{s}k\={u}nas. In particular, it depends only on the local representation $\overline{r}|_{G_{F_v}}$, and its Jordan-H\"older factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and independently of Hu-Wang, which proved these results when $\overline{r}|_{G_{F_v}}$ was additionally assumed to be tamely ramified. The main new tool is a method for computing Taylor-Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti-Tate deformation rings and their intersection theory.

Integral models of Hilbert modular varieties in the ramified case, deformations of modular Galois representations, and weight one forms

We prove the strong Artin conjecture for continuous, totally odd, two-dimensional representations of the absolute Galois group of a totally real field F.

Stable lattices in modular Galois representations and Hida deformation

In this paper, we discuss the variation of the numbers of the isomorphic classes of stable lattices when the weight and the level vary in a Hida deformation by using the Kubota–Leopoldt p-adic L-function. Then in Corollary 1.7, we give a sufficient condition for the numbers of the isomorphic classes of stable lattices in Hida deformation to be infinite.

Certification of modular Galois representations

We show how the output of the algorithm to compute modular Galois representations described in our previous article can be certified. We have used this process to compute certified tables of such Galois representations obtained thanks to an improved version of this algorithm, including representations modulo primes up to 31 and representations attached to a newform with non-rational (but of course algebraic) coefficients, which had never been done before. These computations take place in the Jacobian of modular curves of genus up to 26. The resulting data are available on the author's webpage.

Serre weights for U⁢(n){U(n)}

Abstract 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 {n=3} and l splits completely in the underlying CM field.

Local torsion on abelian surfaces with real multiplication by ${\bf Q}(\sqrt{5})$

Fix an integer d ≥ 1. In 2008, David and Weston showed that, on average, an elliptic curve over Q picks up a nontrivial p-torsion point defined over a finite extension K of the p-adics of degree at most d for only finitely many primes p. This paper proves an analogous averaging result for principally polarized abelian surfaces A over Q with real multiplication by [Formula: see text] and a level-[Formula: see text] structure. Furthermore, we indicate how the result on abelian surfaces with real multiplication relates to the deformation theory of modular Galois representations.

Local Indecomposability of Hilbert Modular Galois Representations

We prove the indecomposability of Galois representation restricted to the p-decomposition group attached to a non CM nearly p-ordinary weight two Hilbert modular form under mild conditions.

Computing modular Galois representations

We compute modular Galois representations associated with a newform \(f\), and study the related problem of computing the coefficients of \(f\) modulo a small prime \(\ell \). To this end, we design a practical variant of the complex approximations method presented in Edixhoven and Couveignes (Ann. of Math. Stud., vol. 176, Princeton University Press, Princeton, 2011). Its efficiency stems from several new ingredients. For instance, we use fast exponentiation in the modular jacobian instead of analytic continuation, which greatly reduces the need to compute abelian integrals, since most of the computation handles divisors. Also, we introduce an efficient way to compute arithmetically well-behaved functions on jacobians, a method to expand cuspforms in quasi-linear time, and a trick making the computation of the image of a Frobenius element by a modular Galois representation more effective. We illustrate our method on the newforms \(\Delta \) and \(E_4 \cdot \Delta \), and manage to compute for the first time the associated faithful representations modulo \(\ell \) and the values modulo \(\ell \) of Ramanujan’s \(\tau \) function at huge primes for \(\ell \in \{ 11,13,17,19,29\}\). In particular, we get rid of the sign ambiguity stemming from the use of a projective representation as in Bosman (On the computation of Galois representations associated to level one modular forms. arxiv.org/abs/0710.1237, 2007). As a consequence, we can compute the values of \(\tau (p)~\mathrm{mod}~2^{11} \times 3^6 \times 5^3 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 691 \approx 2.8 \times 10^{19}\) for huge primes \(p\). The representations we computed lie in the jacobian of modular curves of genus up to \(22\).

Weight cycling and Serre-type conjectures for unitary groups

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".