Power Savings for Counting (Twisted) Abelian Extensions of Number Fields
Brandon Alberts
备注：
46 pages, v2 has been updated to include new references, correct various typos, and include a short appendix on alternate orderings and restricted local conditions
We prove significant power savings for the error term when counting abelian extensions of number fields (as well as the twisted version of these results for nontrivial Galois modules). In some cases over $\mathbb{Q}$, these results reveal lower order terms following the same structure as the main term that were not previously known. Assuming the generalized Lindelöf hypothesis for Hecke $L$-functions, we prove square root power savings for the error compared to the order of the main term.
We show how the Selberg $Λ^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, we give such an error term for the counting function of $S_5$-quintic fields.
Let $P$ be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions $n\leq N$ to $n! = P(x)$ which yields a power saving over the trivial bound. In particular, this applies to a century-old problem of Brocard and Ramanujan. The previous best result was that the number of solutions is $o(N)$. The proof uses techniques of Diophantine and Padé approximation.
We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9% of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and power-saving bounds for sums of products of Kloosterman sums where the length of the sum is below the square-root threshold of the modulus. Appli…
We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of $q$, and we use the theory of Deligne and Katz to estimate certain complete exponential sums in several variables and prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of $q$. When at least one of the forms $f$ an…
We prove an asymptotic formula for the number of multi-quadratic number fields of bounded discriminant with a power-saving error term. Furthermore, we explicitly calculate the leading coefficient and extend our result to totally real multi-quadratic number fields.
In this note, we prove a power-saving remainder term for the function counting $S_3$-sextic number fields. We also give a prediction on the second main term. We also present numerical data on counting functions for $S_3$-sextic number fields. Our data indicates that our prediction is likely to be correct, and it also suggests the existence of additional lower order terms which we have not yet been able to explain.
Following Sen[arXiv:0911.1563], we study the counting of (`twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with N=4 supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, M_{24}, playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states. The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, the …
We prove an estimate on the number of rational points on the Grassmannian variety of bounded twisted height, refining the classical results of Schmidt ([12]) and Thunder ([20]) over the rational field: most importantly, our formula counts all points. Among the consequences are a couple of new implications on the classical subject of counting rational points on flag varieties.
We estimate the growth rate of the function which counts the number of torsion points of order at most $T$ on an algebraic subvariety of the algebraic torus $\mathbb G_m^n$ over some algebraically closed field. We prove a general upper bound which is sharp, and characterize the subvarieties for which the growth rate is maximal. For all other subvarieties there is a better bound which is power saving compared to the general one. Our result includes asymptotic formulas in characteristic zero where we use Laurent's Theorem, the Manin-Mumford Conjecture. However, we also obtain new upper bounds fo…
Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree $d$ and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over $\mathbb{Z}$) in a homogeneously expanding star body in $\mathbb{R}^{d+1}$. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials -- this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which a…
Manin's conjecture predicts the distribution of rational points on Fano varieties. Using explicit parameterizations of rational points by integral points on universal torsors and lattice-point-counting techniques, it was proved for several specific varieties over $\mathbb{Q}$, in particular del Pezzo surfaces. We show how this method can be implemented over arbitrary number fields $K$, by proving Manin's conjecture for a singular quartic del Pezzo surface of type $\mathbf{A}_3+\mathbf{A}_1$. The parameterization step is treated in high generality with the help of twisted integral models of uni…
The asymptotic formula of the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups is given by the average of the product of the number of local solutions twisted by the Brauer-Manin obstruction. The similar result is also true for homogeneous spaces of reductive groups with some restriction. As application, we will give the explicit asymptotic formulae of the number of integral points of certain norm equations and explain that the asymptotic formula of the number of integral points in Theorem 1.1 of \cite{EMS} is equal to the p…
In previous work, the authors introduced "soft" methods to prove the effective (i.e. with power savings error) equidistribution of "shears" in cusped hyperbolic surfaces. In this paper, we study the same problem but now allow full use of the spectral theory of automorphic forms to produce explicit exponents, and uniformity in parameters. We give applications to counting square values of quadratic forms.
In this paper we prove an explicit version of a function field analogue of a classical result of Odoni about norms in number fields in the case of a cyclic Galois extensions. In the particular case of a quadratic extension, we recover the result of Bary-Soroker, Smilanski, and Wolf which deals with finding asymptotics for a function field version on sums of two squares, improved upon by Gorodetsky , and reproved by the author in his Ph.D thesis using the method of this paper. The main tool is a twisted Grothendieck Lefschetz trace formula, inspired by the work of Church, Farb and Ellenberg on …
We extend the dictionary between the BPS spectrum of Heterotic strings and the one of F-/M-theory compactifications on $K3$ fibered Calabi-Yau 3-folds to cases with higher rank non-Abelian gauge groups and in particular to dual pairs between Heterotic CHL orbifolds and compactifications on Calabi-Yau 3-folds with a compatible genus one fibration. We show how to obtain the new supersymmetric index purely from the Calabi-Yau geometry by reconstructing the Noether-Lefschetz generators, which are vector-valued modular forms. There is an isomorphism between the latter objects and vector-valued latt…
The purpose of the present paper is to develop the enumerative geometry of dormant $G$-opers for a semisimple algebraic group $G$. In the present paper, we construct a compact moduli stack admitting a perfect obstruction theory by introducing the notion of a dormant faithful twisted $G$-oper (or a "$G$-do'per", for short). The resulting virtual fundamental class induces a semisimple $2$d TQFT (= $2$-dimensional topological quantum field theory) counting the number of $G$-do'pers. This $2$d TQFT gives an analogue of the Witten-Kontsevich theorem describing the intersection numbers of psi classe…
