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In an influential 2008 paper, Baker proposed a number of conjectures relating the Brill-Noether theory of algebraic curves with a divisor theory on finite graphs. In this note, we examine Baker's Brill-Noether existence conjecture for special divisors. For g ≤ 5 and ρ(g, r, d) non-negative, every graph of genus g is shown to admit a divisor of rank r and degree at most d. As further evidence, the conjecture is shown to hold in rank 1 for a number families of highly connected combinatorial types of graphs. In the relevant genera, our arguments give the first combinatorial proof of Brill-Noether existence theorem for metric graphs, giving a partial answer to a related question of Baker. Divisor theory on finite graphsThe main reference for this section is the original paper of Baker and Norine [4]. A graph G will mean a finite connected graph possibly with loops and multiple edges. The vertex and edge sets of G will be denoted V (G) and E(G) respectively. The genus of G, denoted g(G), is defined to be g(G) := |E(G)| − |V (G)| + 1.

We show that the Taylor-Wiles method can be applied to the cohomology of a Shimura variety S of PEL type attached to a unitary similitude group G, with coefficients in the coherent sheaf attached to an automorphic vector bundle F , when S has a smooth model over a p-adic integer ring. This generalizes the main results of the article [25], which treated the case when S is compact. As in the previous article, the starting point is a theorem of Lan and Suh that proves the vanishing of torsion in the cohomology under certain conditions on the parameters of the bundle F and the prime p. Most of the additional difficulty in the non-compact case is related to showing that the contributions of boundary cohomology are all of Eisenstein type. We also need to show that the coverings giving rise to the diamond operators can be extended to étale coverings of appropriate toroidal compactifications.The result is applied to show that, when the Taylor-Wiles method applies, the congruence ideal attached to a coherent cohomological realization of an automorphic Galois representation is independent of the signatures of the hermitian form to which G is attached. We also show that the Gorenstein hypothesis used to construct p-adic L-functions in [13] is valid under rather general hypotheses.