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# Theoretical and algorithmic aspects of congruences between modular Galois representations

In this thesis we examine a number of cases in which the Questions posed above can be answered. In the whole work (except just in some definitions in Chapter 1) we will deal with normalized newforms of weight 2 without nebentypus. Whenever we work with congruences between two modular forms f and g of respective levels Nf and Ng (Nf ≥ Ng), we will assume Ng | Nf . Such assumption is not too restrictive, since for g minimal and irreducible in a prime ℓ, and f and g congruent modulo λ (with λ | ℓ), Ribet’s lowering the level provides a modular form of level dividing both Nf and Ng which is congruent modulo λ with f and g. In Chapter 1 we give some basic background and tools, to be used in the following chapters. We give the basic definitions of modular forms, Hecke algebras, representations and, to every newform, we attach different kinds of representations. Then, we define the concept of congruent representations and finally state Serre’s conjecture, which will allow us to apply the work developed in the next chapters to all Abelian varieties of GL2-type. In Chapter 2 we develop some algorithms to give an answer to Question 0.1.2 above. This chapter served as an inspiration for a joint work with Gabor Wiese ([TW09]), and it owes some of the results to it. All our algorithms will be based on computations with the characteristic polynomials Qp of the eigenvalues of the Hecke operators of the newforms to be compared (§2.6). Thus, our very first task is merely to compute a huge database containing all these polynomials up to some prescribed order— namely we compute the polynomials for all newforms of level N ≤ 2000 and all primes p < 1000. Given two different newforms, in §2.4 we will use the resultant of the pre-computed polynomials to compute a finite set {ℓn1 1 , . . . , ℓns s } containing all possible congruences between f and g (i.e. if ℓn ∤ ℓn1 1 · . . . · ℓns s , then there exists no congruence between f and g modulo λn, for any λ | ℓ): Lemma 0.2.1. If f and g are congruent modulo λn, then ℓn divides the resultant of every couple Pf,p and Pg,p, p ∤ ℓNf . This result is not optimal, nevertheless it suggests the idea to define the local congruence number (§2.5), which will provide an algorithm to get a better upper bound L+ than the one computed with the resultants (§§2.7–2.9). Our next algorithm will find an upper bound (in sense described above) for congruences between a given newform f and its conjugates σ(f) (§2.10). Next step is to find an algorithm to determine a lower bound for congruences between modular forms (§2.13). In other words, we will get a number L− such that if ℓn divides L−, f and g are congruent modulo ℓn. To develop this algorithm we will have to introduce first two results: applying the Hecke bound (§2.11) and an idea of Gabor Wiese (§2.12). We finish this chapter giving some examples that compare upper bound algorithms with the lower bound one and we see that in many cases we obtain that L− = L+ and thence our algorithms do determine all congruences between the tested modular forms. Chapter 3 is based on a joint work with Luis Dieulefait ([DT09]). The main result (§3.1) answers Question 0.1.3 in some cases. Theorem 0.2.2. Let ℓ, p ∤ Ng, ℓ > 2 be two different prime numbers. Let f be in S2(pkNg), k ≥ 1, and let g ∈ S2(Ng) be minimal with respect to λ. Let ρf,λn be the representation modulo λn attached to f. Suppose that ρf,λ ∼ ρg,λ and that they are irreducible, and assume that for any other h ∈ S2(Ng), ρg,λ ≁ ρh,λ. If ℓ = 3, let L = Q(√−3) and suppose that ρg,λ|GL is irreducible. Then, m := min{n ∈ N : ρf,λn ≁ ρg,λn} = min{n ∈ N : ρf,λn|Ip ≁ ρg,λn|Ip}. In the case of this theorem, then, we can assert that the reason for f and g not to be congruent any more is that the p-inertia modulo λm does not vanish as it did for λm−1. This can be reread also as a generalization of Ribet’s Lowering the Level. Using Theorem 0.2.2 we can give two corollaries which tell how the image of the p-inertia of the representation on f must look like. These results can be applied to determine images of Galois representations. In the next section (§3.2) we introduce the necessary terminology about deformation theory to prove this theorem, and it follows (§3.3) the proof of Theorem 0.2.2. Finally we will give some examples—computed with the algorithms from Chapter 2—of couples of newforms satisfying the conditions of the theorem as well as ones of the corollaries. Due to the nature of the first chapter, the knowledge of its content is indispensable to understand the following parts of the work. Chapter 2 and 3 can be read independently but they complement each other being the algorithms of the former an easy way to find examples for the latter. The last chapter is a brief description of possible expansions and improvements of the work developed in this thesis. In appendices A-C we give lists of some of the most interesting results obtained with ours algorithms. The complete lists of results and the codes of the algorithms can be found in http://www.iem.uni-due.de/~xavier/thesis.

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