Computational aspects of modular forms and Galois representations
Bas Edixhoven, Jean-Marc Couveignes, Robin de Jong, Franz Merkl, Johan Bosman
备注：
438 pages. This book is to appear in the series 'Annals of Mathematics Studies' of Princeton University Press, after reviewing and more typesetting. Comments are very welcome. Main changes: we have added deterministic algorithms based on computations with complex numbers, we treat forms of level one and general weight.
This is a book about computational aspects of modular forms and the Galois representations attached to them. The main result is the following: Galois representations over finite fields attached to modular forms of level one can, in almost all cases, be computed in polynomial time in the weight and the size of the finite field. As a consequence, coefficients of modular forms can be computed fast via congruences, as in Schoof's algorithm for the number of points of elliptic curves over finite fields. The most important feature of the proof of the main result is that exact computations involving systems of polynomial equations in many variables are avoided by approximations and height bounds, i.e., bounds for the accuracy that is necessary to derive exact values from the approximations.
We outline the proof of a theorem of M. Baker and A. Tamagawa that gives a complete description of the torsion points on a modular curve embedded in its Jacobian using the notion of an `almost rational torsion point.'
We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the Fontaine-Mazur conjectures, specifically, the modularity of certain potentially Barsotti-Tate Galois representations. The proof follows the template of Wiles, Taylor-Wiles, and Breuil-Conrad-Diamond-Taylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the integers of a local field. This makes crucial use of the filtered ϕ_1-modules of C. Breuil.
We discuss the following two problems: 1) The properties of the multiple zeta-values and their generalizations, multiple polylogarithms at N-th roots of unity; 2) The action of the absolute Galois group on the pro-l-completion of the fundamental group of the projective line without zero, infinity, and all N-th roots of unity; and a surprising connection of these problems with the geometry and topology of modular varieties for GL_m.
We show that any two-dimensional odd dihedral representation ρover a finite field of characteristic p>0 of the absolute Galois group of the rational numbers can be obtained from a Katz modular form of level N, character εand weight k, where N is the conductor, εis the prime-to-p part of the determinant and k is the so-called minimal weight of ρ. In particular, k=1 if and only if ρis unramified at p. Direct arguments are used in the exceptional cases, where general results on weight and level lowering are not available.
The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let's mention : (1) the control of the image of the Galois representation modulo $p$, (2) Hida's congruence criterion outside an explicit set of primes $p$, and (3) the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras. We study the arithmetic of the Hilbert modular forms by studying their modulo $p$ Galois representations and our main…
We expand on Nekovář's construction of the plectic half transfer to define a plectic Galois action on Hilbert modular varieties. More precisely, we study in a unifying fashion Shimura varieties associated to groups that differ only in the centre from $R_{F/\mathbb{Q}}{\rm GL}_2$. We define plectic Galois actions on the CM points and on the set of connected components of these Shimura varieties, and show that these two actions are compatible. This extends the plectic conjecture of Nekovář--Scholl.
For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some recent results. It focusses mostly on the 2-dimensional case and underlines in particular the importance of understanding coefficient fields.
In this paper, we propose an improved algorithm for computing mod $\ell$ Galois representations associated to eigenforms of arbitrary levels prime to $\ell$. Precisely, we present a method to find the Jacobians of modular curves which have the smallest possible dimensions in a well-defined sense to realize the modular Galois representations. This algorithm also works without the assumption $\ell \ge k-1$, where $k$ are the weights of the eigenforms.
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.
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.
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 introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We exhibit some examples and provide an infinite Galois family of non-liftable weight one Katz modular eigenforms over an algebraic closure of F_p for p in {3,5,7,11}.
We give a direct approach to recover some of the results of Wiles and Tayor on modularity of certain 2-dimensional p-adic representations of the absolute Galois group of Q.
In this work we provide a level raising theorem for $\mod λ^n$ modular Galois representations. It allows one to see such a Galois representation that is modular of level $N$, weight 2 and trivial Nebentypus as one that is modular of level $Np$, for a prime $p$ coprime to $N$, when a certain local condition at $p$ is satisfied. It is a generalization of a result of Ribet concerning $\mod \ell$ Galois representations.
We establish a relation between Galois reducibility and Endoscopy for genus 2 Siegel cusp forms which have rational eigenvalues and are unramified at 3
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.
We construct plane models of the modular curve $X_H(\ell)$, and use their explicit equations to compute Galois representations associated to modular forms for values of $\ell$ that are significantly higher than in prior works.
We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic group. We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.
We address the problem of the determination of the images of three-dimensional geometric and modular Galois representations. In the modular case the existence of these representations is only conjectural. We give conditions to ensure that the images are as large as possible for almost every prime. We compute some examples and stress the applications to inverse Galois theory.
This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented.
This article computes the Galois groups of congruence covers arising in the context of certain hyperbolic triangle groups. As a consequence of this computation, the genera of the respective curves are deduced.
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.
We formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.
In this paper we describe an algorithm for computing mod $\ell$ Galois representations associated to modular forms of weight $k$ when $\ell <k-1$. As applications, we use this algorithm to explicitly compute the cases with $Δ_{k}$ for $k=16,20, 22, 26$ and all the unexceptional primes $\ell$ with $\ell <k-1$.
In this letter we show that for certain infinite families of modular forms of growing level it is possible to have a control result for the exceptional primes of the attached Galois representations. As an application, a uniform version of a result of Wiese on Galois realizations of linear groups over finite fields of arbitrarily large exponent is proved.
We prove new cases of the inverse Galois problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight $3$ newforms will show that there are Galois extensions of $\mathbb{Q}$ with Galois group $PSL_2(\mathbb{F}_p)$ for all primes $p$ and $PSL_2(\mathbb{F}_{p^3})$ for all odd primes $p \equiv \pm 2, \pm 3, \pm 4, \pm 6 \pmod{13}$.
In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Pa…
Let $F$ be a totally real number field and let $p$ be a prime unramified in $F$. We prove the existence of Galois pseudo-representations attached to mod $p^m$ Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties for $F$ of level prime to $p$.
This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences.
We prove modularity for any irreducible crystalline $\ell$-adic odd 2-dimensional Galois representation (with finite ramification set) unramified at 3 verifying an "ordinarity at 3" easy to check condition, with Hodge-Tate weights $\{0, w \}$ such that $2 w < \ell$ (and $\ell > 3$) and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p \}) = \Q $. This result applies in particular to any motivic compatible family of odd two-dimensional Galois representations of $\Gal(\bar{\Q}/\Q)$ if the motive has rational coefficients, good reduction at 3, and the "ordinarity at 3" conditi…
In this paper we explicitly compute mod-l Galois representations associated to modular forms. To be precise, we look at cases with l<=23 and the modular forms considered will be cusp forms of level 1 and weight up to 22. We present the result in terms of polynomials associated to the projectivised representations. As an application, we will improve a known result on Lehmer's non-vanishing conjecture for Ramanujan's tau function.
We describe the effect of the differential operators defined by Boecherer-Nagaoka, Flander-Ghitza and Yamauchi on the Galois representations (conjecturally) attached to Siegel modular eigenforms.
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 the book edited by B. Edixhoven and J.-M. Couveignes. 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 eff…
This paper gives an expository account of our experiments concerning relations between modular forms for congruence subgroups of SL(3,Z) and three dimensional Galois representations. The main new result presented here is a calculation of the variations of the Hodge structure corresponding to the motives we consider in realizing the Galois representations. It turns out that the period spaces for the Hodge structures are four dimensional, while the geometric realizations of such Hodge structures can appear in subspaces of dimension at most one.
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 of modular curves points and dispenses us of the need for equations of modular curves and for q-expansion computations. The resulting algorithm compares very favourably to the complex-analytic method.
We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing $ζ_5$. Applied to imaginary quadratic fields, this proportion is at least $2/5$. Applied to cyclotomic fields $\mathbb{Q}(ζ_n)$ with $5\nmid n$, this proportion is at least $1-\varepsilon$ with only finitely many exceptions of $n$, for any choice of $\varepsilon > 0$.
We prove modularity of some two dimensional, 2-adic Galois representations over totally real fields that are nearly ordinary and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the 2-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above 2.
We study modular Galois representations mod $p^m$. We show that there are three progressively weaker notions of modularity for a Galois representation mod $p^m$: we have named these `strongly', `weakly', and `dc-weakly' modular. Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These notions of modularity are relative to a fixed level $M$. Using results of Hida we display a `stripping-of-powers of $p$ away from the level' type of result: A mod $p^m$ strongly modular representation of some level $Np^r$ is always dc-weakly modular of level $N$ (here, $N$ is a natural numb…
In this paper, we consider Galois representations of the absolute Galois group $\text{Gal}(\overline {\mathbb Q}/\mathbb Q)$ attached to modular forms for noncongruence subgroups of $\text{SL}_2(\mathbb Z)$. When the underlying modular curves have a model over $\mathbb Q$, these representations are constructed by Scholl and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over $\mathbb Q$. As Scholl re…
We address the problem of the determination of the images of the Galois representations attached to genus 2 Siegel cusp forms of level 1 having multiplicity one. These representations are symplectic. We prove that the images are as large as possible for almost every prime, if the Siegel cusp form is not a Maass spezialform and verifies two easy to check conditions.
In this paper we discuss applications of our earlier work in studying certain Galois groups and splitting fields of rational functions in $\mathbb Q\left(X_0(N)\right)$ using Hilbert's irreducibility theorem and modular forms. We also consider computational aspect of the problem using MAGMA and SAGE.
This paper continues the study of certain two-dimensional Galois representations attached to modular forms (mod p) via a construction due to Deligne. In particular, we prove a criterion for determining when the representation is split when restricted to a decomposition group at p.While this problem was already well understood for "nonexceptional" modular forms, here we specifically address the "exceptional" case. Although the problem is not solved in all generality, it is reasonable to hope that this proof will in fact provide the framework of a proof in the more general case.
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 that the groups PSL(r,q^d) can be realized F_q(T)-regularly as Galois groups over the purely transcendental field F_q(T)(t_1,...,t_{r-1}) if r is even and r/2 is coprime to q^d-1. The method is to use twisted moduli schemes of Drinfeld modules (a la Shih). If one has sufficient control over modular forms, it can be made into an algorithm to find explicit equations. We do this for PSL(2,q^2) using Drinfeld modular forms.
In a previous article, the second author proved that the image of the Galois representation mod $λ$ attached to a Hilbert modular newform is large or all but finitely many primes $λ$, if the form is not a theta series. In this brief note, we give an explicit bound for this exceptional finite set of primes and determine the images in three different examples. Our examples are of Hilbert newforms on real quadratic fields, of parallel or non-parallel weight and of different levels.
Building on ideas of Vatsal, Cornut proved a conjecture of Mazur asserting the generic nonvanishing of Heegner points on an elliptic curve E as one ascends the anticyclotomic Z_p-extension of a quadratic imaginary extension K/Q. In the present article Cornut's result is extended by replacing the elliptic curve E with the Galois cohomology of Deligne's 2-dimensional l-adic representation attached to a modular form of weight 2k>2, and replacing the family of Heegner points with an analogous family of special cohomology classes.
We show that the mod p Galois representations attached to a Q-curve E of degree d over an imaginary quadratic number field K are surjective for all p larger than some constant M_{K,d}, if E has potentially multiplicative reduction at any prime not dividing 6. The proof uses Mazur's formal immersion method, a result of Darmon and Merel on non-split modular curves, and an analytic argument showing that Jacobians of certain twisted modular curves admit quotients with Mordell-Weil rank 0. In combination with a previous theorem of the author and Skinner on modularity of Q-curves, the surjectivity r…
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.
The ghost conjecture, formulated by this article's authors, predicts the list of p-adic valuations of the non-zero p-th eigenvalues ("slopes") for overconvergent p-adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the "ghost series"). The prediction is restricted to eigenforms whose Galois representation modulo p is reducible on a decomposition group at p. It has been discovered, however, that the conjecture is not formulated correctly. Here we explain the issue and propose a salvage.
We determine the stable reduction at $p$ of all three point covers of the projective line with Galois group ${\rm SL}_2(p)$. As a special case, we recover the results of Deligne and Rapoport on the reduction of the modular curves $X_0(p)$ and $X_1(p)$. Our method does not use the fact that modular curves are moduli spaces. Instead, we rely on results of Raynaud and the authors which describe the stable reduction of three point covers whose Galois group is strictly divisible by $p$.
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.
