Parallel Weight 2 Points on Hilbert Modular Eigenvarieties and the Parity Conjecture

Abstract

Let $F$ be a totally real field and let $p$ be an odd prime which is totally split in $F$ . We define and study one-dimensional ‘partial’ eigenvarieties interpolating Hilbert modular forms over $F$ with weight varying only at a single place $v$ above $p$ . For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch–Kato conjecture for finite slope Hilbert modular forms with trivial central character (with a technical assumption if $[F:\mathbb{Q}]$ is odd), by reducing to the case of parallel weight $2$ . As another consequence of our results on partial eigenvarieties, we show, still under the assumption that $p$ is totally split in $F$ , that the ‘full’ (dimension $1+[F:\mathbb{Q}]$ ) cuspidal Hilbert modular eigenvariety has the property that many (all, if $[F:\mathbb{Q}]$ is even) irreducible components contain a classical point with noncritical slopes and parallel weight $2$ (with some character at $p$ whose conductor can be explicitly bounded), or any other algebraic weight.

Parabolic eigenvarieties via overconvergent cohomology

“…This utility is illustrated in the example of GL 2 over a number field F, where special cases of the above theory have appeared repeatedly: -In the case where F is totally real, partial p-adic families were used in [4,26], with applications to the trivial zero and parity conjectures respectively. -For more general F, versions of Theorem A have been proved and used to construct Stark-Heegner points on elliptic curves [19,20,41], and when F is imaginary quadratic, to construct conjectural Stark-Heegner cycles attached to Bianchi modular forms [43].…”

Classical forms of weight one in ordinary families

“…Partially ordinary families are also discussed briefly in [28], again working cohomologically with quaternionic automorphic forms. We should mention that similar families have been constructed independently by Yamagami in [31] and by Johansson-Newton in [19]. Both Yamagami and Johansson-Newton work with more general v-finite-slope rather than v-ordinary families, though one can think of the ord p (U v ) = 0 locus in the v-finite-slope eigenvariety as being the v-ordinary families that we study.…”

2‐adic slopes of Hilbert modular forms over Q(5)

“…If , then and by making certain restrictions on the weight space one can study the slopes of . If is totally split, work of Newton–Johansson [9] shows that the methods of [10] can be used to describe the slopes of , which for general is not the case. Moreover, they construct partial eigenvarieties and prove that over a boundary annulus these partial eigenvarieties decompose as a union of components which are finite over weight space.…”

“…The above theorem together with the computational evidence suggest that, in this example, the slopes tend to zero as one approaches the boundary of weight space. In particular, this would imply, via similar methods to those in [9], that over the boundary annulus the corresponding eigenvariety is a disjoint union of rigid spaces which are finite flat over this boundary annulus. Moreover, we expect that methods similar to those of [9] would also prove the parity part of the Bloch–Kato conjecture for this specific example (if the scaling behaviour of the slopes were proven).…”