L-invariants and congruences for Galois representations of dimension 3, 4, and 8

Abstract

The arithmetic of Galois representations plays a central role in modern number theory. In this thesis we consider representations arising from tensor products of the two-dimensional representations attached to modular forms by Deligne. In particular, we shall study the Iwasawa theory of the adjoint representation, as well as certain double and triple products of Deligne's representations.

In the first half we will undertake a computational study of L-invariants attached to symmetric squares of modular forms. Let ƒ be a primitive modular form of weight 𝘬 and level 𝘕, and 𝘱 ∤ 𝘕 a prime greater than two for which the attached representation is ordinary. The 𝘱-adic 𝘓-function Sym²ƒ always vanishes at s=1, even though the complex 𝘓-function does not have a zero there. The L-invariant itself appears on the right-hand side of the formula

\frac{\text{\rm d}\;}{\text{\rm d}s}\mathbf{L}_p(\text{\rm Sym}^2f,s) \Bigg|_{s=k-1}= \mathcal{L}_p(\text{\rm Sym}^2f) \times (1-\alpha_p^{-2}p^{k-2})(1-\alpha_p^{-2}p^{k-1}) \times \frac{L_{\infty}(\text{\rm Sym}^2f,k-1)}{\pi^{k-1}\langle f,f\rangle_N}

where X²-aₚ(ƒ)𝘟 + p = (𝘟-αₚ)(𝘟-βₚ) with αₚ ∈ Zˣₚ.

Now let 𝘌 be an elliptic curve over Q with associated modular form ƒᴇ, and 𝘱 ≠ 2 a prime of good ordinary reduction. We devise a method to calculate Lₚ(Sym²ƒᴇ) effectively, then show it is non-trivial for almost all pairs of elliptic curves 𝘌 of conductor 𝘕ᴇ ≤ 300, with 4|𝘕ᴇ, and ordinary primes 𝘱 < 17. Hence, in these cases at least, the order of the zero in Lₚ(Sym²ƒᴇ, 𝘴) at 𝘴 = 1 is exactly one. We also generalise this method to compute symmetric square L-invariants for modular forms of weight 𝘬 > 2.

In the second half we will establish congruences between 𝘱-adic 𝘓-functions. In the late 1990s, Vatsal showed that a congruence modulo 𝘱 ᵛ between two newforms implied a congruence between their respective 𝘱-adic 𝘓-functions. We shall prove an analogous statement for both the double product and triple product 𝘱-adic 𝘓-functions, Lₚ(f⊗g) and Lₚ (f⊗g⊗h): the former is cyclotomic in its nature, while the latter is over the weight space.

As a corollary, we derive transition formulae relating analytic λ-invariants for pairs of congruent Galois representations for 𝘝𝒻 ⊗ 𝘝𝓰, and for 𝘝𝒻 ⊗ 𝘝𝓰⊗ 𝘝ₕ.