We improve upon the local bound in the depth aspect for sup-norms of newforms on D^×, where D is an indefinite quaternion division algebra over Q. Our sup-norm bound implies a depth-aspect subconvexity bound for L(1/2,f×θ_χ), where f is a (varying) newform on D^× of level p^n, and θ_χ is an (essentially fixed) automorphic form on GL_2 obtained as the theta lift of a Hecke character χ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via p-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

We construct the Frobenius structure on a rigid connection Be_{\hat G} on G_m for a split reductive group \hat G introduced by Frenkel-Gross. These data form a \hat G-valued overconvergent F-isocrystal Be^†_{\hat G} on G_{m,F_p}, which is the p-adic companion of the Kloosterman \hat G-local system Kl_{\hat G} constructed by Heinloth-Ngô-Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of Be^†_{\hat G} when \hat G is almost simple (which recovers the calculation of monodromy group of Kl_{\hat G} due to Katz and Heinloth–Ngô–Yun), and prove a conjecture of Heinloth-Ngô-Yun on the functoriality between different Kloosterman \hat G-local systems. We show that the Frobenius Newton polygons of Kl_{\hat G} are generically ordinary for every \hat G and are everywhere ordinary on |G_{m,F_p}| when \hat G is classical or G_2.

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We present an integral representation for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. Our construction is uniform over all classical groups, and is applicable to all cuspidal representations; it does not require genericity. The main new ideas of the construction are the use of generalized Speh representations as inducing data for the Eisenstein series and the introduction of a new (global and local) model, which generalizes the Whittaker model. Here we consider linear groups, but our construction also extends to arbitrary degree metaplectic coverings; this will be the topic of an upcoming work.

In this paper we prove the \pm-main conjecture formulated by Kobayashi for elliptic curves with supersingular reduction at p such that a_p = 0. We also prove as a corollary the p-part of the BSD formula at supersingular primes when the analytic rank is 0.