Eichler cohomology and zeros of polynomials associated to derivatives of L-functions

In recent years, a number of papers have been devoted to the study of zeros of period polynomials of modular forms. In the present paper, we study cohomological analogues of the Eichler–Shimura period polynomials corresponding to higher L-derivatives. We state a general conjecture about the locations of the zeros of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative “period polynomials” in the case of cusp forms. The unimodularity of the roots seems to be a very subtle property which is special to our “period polynomials”. This is suggested by numerical experiments on families of perturbed “period polynomials” (Section 5.3) suggested by Zagier. We prove a special case of our conjecture in the case of Eisenstein series. Although not much is currently known about derivatives higher than first order ones for general modular forms, celebrated recent work of Yun and Zhang established the analogues of the Gross–Zagier formula for higher L-derivatives in the function field case. A critical role in their work was played by a notion of “super-positivity”, which, as recently shown by Goldfeld and Huang, holds in infinitely many cases for classical modular forms. As will be discussed, this is similar to properties which were required by Jin, Ma, Ono, and Soundararajan in their proof of the Riemann Hypothesis for Period Polynomials, thus suggesting a connection between the analytic nature of our conjectures here and the framework of Yun and Zhang.

Finiteness of Frobenius Traces of a Sheaf on a Flat Arithmetic Scheme

For a lisse $\ell $-adic sheaf on a scheme flat and of finite type over $\mathbb{Z}$, we consider the field generated over $ \mathbb{Q}$ by Frobenius traces of the sheaf at closed points. Assuming conjectural properties of geometric Galois representations of number fields and the Generalized Riemann Hypothesis, we prove that the field is finite over $\mathbb{Q}$ when the sheaf is de Rham at $\ell $ pointwise. This is a number field analog of Deligne’s finiteness result about Frobenius traces in the function field case.

Eventually stable rational functions

For a field [Formula: see text], rational function [Formula: see text] of degree at least two, and [Formula: see text], we study the polynomials in [Formula: see text] whose roots are given by the solutions in [Formula: see text] to [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th iterate of [Formula: see text]. When the number of irreducible factors of these polynomials stabilizes as [Formula: see text] grows, the pair [Formula: see text] is called eventually stable over [Formula: see text]. We conjecture that [Formula: see text] is eventually stable over [Formula: see text] when [Formula: see text] is any global field and [Formula: see text] is any point not periodic under [Formula: see text] (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when [Formula: see text] has a discrete valuation for which (1) [Formula: see text] has good reduction and (2) [Formula: see text] acts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of Sookdeo on the finiteness of [Formula: see text]-integral points in backward orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.

Squarefree doubly primitive divisors in dynamical sequences

LetKbe a number field or a function field of characteristic 0, letφ∈K(z) with deg(φ) ⩾ 2, and letα∈ ℙ1(K). LetSbe a finite set of places ofKcontaining all the archimedean ones and the primes whereφhas bad reduction. After excluding all the natural counterexamples, we define a subsetA(φ,α) of ℤ⩾0× ℤ>0and show that for all but finitely many (m,n) ∈A(φ,α) there is a prime 𝔭 ∉Ssuch that ord𝔭(φm+n(α)−φm(α)) = 1 andαhas portrait (m,n) under the action ofφmodulo 𝔭. This latter condition implies ord𝔭(φu+v(α)−φu(α)) ⩽ 0 for (u,v) ∈ ℤ⩾0× ℤ>0satisfyingu<morv<n. Our proof assumes a conjecture of Vojta for ℙ1× ℙ1in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram–Silverman, Faber–Granville and the authors.

Low-lying zeros of quadratic Dirichlet -functions: lower order terms for extended support

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.

Connected components of affine Deligne–Lusztig varieties in mixed characteristic

We determine the set of connected components of minuscule affine Deligne–Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne–Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport–Zink spaces.