We generalize our method for subconvex bounds for GL 2 × GL 1 to the setting of the Waldspurger's formula for compact torical integrals. We address the two major difficulties: one is the lack of split places with small norm, the other is the test vector problem. The final bound is valid with arbitrary high probability and is better than the known bounds for a non-empty interval.

In the 80s, Zagier and Jacquet-Zagier tried to derive the Selberg trace formula by applying the Rankin-Selberg method to the automorphic kernel function. Their derivation was incomplete due to a puzzle of the computation of a residue. We solve this puzzle and complete the derivation. The main input is an extension of the theory of regularized integrals invented by Zagier, which is of independent interest.Contents 1 1 The setting in [5] deals with arbitrary central character, but the Jacquet-Zagier puzzle we are discussing concerns only the case of trivial central character.2 For our convention, PGL 2 does not mean the quotient of GL 2 by its center in the category of algebraic groups, but PGL 2 (k) = k × \GL 2 (k) in the group theoretic sense for any field k. The former is in general a larger group. 3 Our convention is to take s as the spectral parameter for the induced representation πs = Ind GL 2 (A)

We generalize our previous method on subconvexity problem for GL 2 × GL 1 with cuspidal representations to Eisenstein series, and deduce a Burgess-like subconvex bound for Hecke characters, i.e., the bound |L(1/2, χ)| ≪ F,ǫ C(χ) 1/4−(1−2θ)/16+ǫ for varying Hecke characters χ over a number field F with analytic conductor C(χ). As a main tool, we apply the extended theory of regularized integral due to Zagier developed in a previous paper to obtain the relevant triple product formulas of Eisenstein series.

We report an integrated compact technique that can "spin" and "twist" light on a silicon photonics platform, with the generated light beams possessing both spin angular momentum (SAM) and orbital angular momentum (OAM). It demonstrates the potential of SAM/OAM optics for on-chip integration.