Let X be a smooth, projective, geometrically connected curve over a finite field Fq, and let G be a split semisimple algebraic group over Fq. Its dual group Gˆ is a split reductive group over Z. Conjecturally, any l-adic Gˆ-local system on X (equivalently, any conjugacy class of continuous homomorphisms π1(X)→Gˆ(Q¯¯¯¯l)) should be associated with an everywhere unramified automorphic representation of the group G. We show that for any homomorphism π1(X)→Gˆ(Q¯¯¯¯l) of Zariski dense image, there exists a finite Galois cover Y→X over which the associated local system becomes automorphic.

We prove a new upper bound for the dimension of the space of cohomological automorphic forms of fixed level and growing parallel weight on $\mathrm{GL}_2$ over a number field which is not totally real, improving the one obtained by Marshall. The main tool of the proof is the mod $p$ representation theory of $\mathrm{GL}_2(\mathbb{Q}_p)$ as started by Barthel-Livne and Breuil, and developed by Paskunas.

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Abstract We present a relative trace formula approach to the Gross–Zagier formula and its generalization to higher-dimensional unitary Shimura vari- eties. As a crucial ingredient, we formulate a conjectural arithmetic funda- mental lemma for unitary Rapoport–Zink spaces. We prove the conjecture when the Rapoport–Zink space is associated to a unitary group in two or three variables.

Multizeta values in positive characteristic were first introduced and studied by Thakur. He and Lara Rodr\'{\i}guez \cite{LRT} discovered and conjectured certain zeta-like families. Kuan, Lin and Yu stated more conjectures about zeta-like multizeta values in \cite{LK}. In the present paper we study and give the transparent formula for certain Anderson-Thakur polynomials. This enables us to confirm the conjectured zeta-like families.