Studies in automorphic representations and the Fargues-Fontaine curve

This thesis consists of two parts.

In the first part, we study the relationship between special values of L-functions and congruences of automorphic representations. We relate L(1,π,Ad◦) to the congruence ideals for cohomological cuspidal automorphic representations π of GL_n over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint L-functions. For CM fields, we apply the result, together with local-global compatibility for Galois representations and Galois deformation theory, to obtain a lower bound on the cardinality of certain Selmer groups in terms of L(1, π, Ad◦), giving partial progress toward the Bloch-Kato conjecture. The main method is to relate automorphic representations to the cohomology of locally symmetric spaces for GL_n. To implement this strategy, we analyze the relations between several cohomology theories. Cohomology plays a central role because cup products correspond to Rankin-Selberg convolutions and hence to adjoint L-functions. This part of the essay has been published in [Fon26].

In the second part, we introduce a notion of Higgs bundles on the Fargues-Fontaine curve, motivated by the trace formula and the geometric ideas underlying the fundamental lemma. We show that there is an Artin v-stack parametrising Higgs bundles on the Fargues-Fontaine curve. Then we prove three abelianisation theorems, such as a version of the BNR correspondence, which relates Higgs bundles to line bundles on suitable spectral curves. We further describe an action of a Picard stack on the moduli stack of Higgs bundles and show that, modulo this action, there is a natural injective map of etale stacks from the product of B_dR+ -affine Springer fibers to the Hitchin fiber that induces an equivalence of categories on every geometric point. Finally, we discuss connections with number-theoretic objects.