The Trace Formula for BunG(X)

This chapter aims to prove Theorem 1.4.4.1, which is formulated as follows: Theorem 5.0.0.3, let X be an algebraic curve over F q and let G be a smooth affine group scheme over X. Suppose that the fibers of G are connected and that the generic fiber of G is semisimple. Then the moduli stack BunG(X) satisfies the Grothendieck–Lefschetz trace formula. However, Theorem 5.0.0.3 cannot be deduced directly from the Grothendieck–Lefschetz trace formula for global quotient stacks because the moduli stack BunG(X) is usually not quasi-compact. The strategy instead will be to decompose BunG (X) into locally closed substacks BunG(X)[P,ν‎] which are more directly amenable to analysis.

Cubic Twin Prime Polynomials are Counted by a Modular Form

We present the geometry behind counting twin prime polynomials in $\mathbb{F}_{q}[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^{3}=Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder.The formula we get in degree 3 is compatible with the Hardy–Littlewood heuristic on average, agrees with the prediction for $q\equiv 2$ (mod 3), but shows anomalies for $q\equiv 1$ (mod 3).

On the Lefschetz trace formula for Lubin–Tate spaces

We give a new proof of the Lefschetz trace formula for Lubin-Tate spaces. Our proof is based on the locally finite cell decompositions of these spaces and on Mieda’s version of the Lefschetz trace formula for certain open adic spaces. This proof is rather different from the proofs of Strauch and Mieda, and it might be generalized to other Rapoport-Zink spaces as soon as there exist suitable cell decompositions. For example, in another paper we have proved a Lefschetz trace formula for some unitary group Rapoport-Zink spaces by using similar ideas as here.