Series Of Conjectures Research Articles

New Results About Some Chain Conditions in Serre Conjecture Rings

Let φ:T→T/I be the natural homomorphism, where I is a nonzero ideal of an integral domain T. We define R:=φ−1(D), where D is a subring of T/I. This paper aims to investigate the conditions under which the Serre conjecture ring R⟨m⟩ is a strong S-domain. Several examples are constructed to demonstrate both the scope and limitations of the results.

Graded Lie algebras, compactified Jacobians and arithmetic statistics

A simply laced Dynkin diagram gives rise to a family of curves over \mathbb{Q} and a coregular representation, using deformations of simple singularities and Vinberg theory, respectively. Thorne conjectured and partially proved a strong link between the arithmetic of these curves and the rational orbits of these representations. In this paper, we complete Thorne’s picture and show that 2-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang. The main innovations are an analysis of torsors on affine spaces using results of Colliot-Thélène and the Grothendieck–Serre conjecture, a study of geometric properties of compactified Jacobians using the Białynicki-Birula decomposition, and a general construction of integral orbit representatives.

The Grothendieck–Serre conjecture over valuation rings

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.

The multivariate Serre conjecture ring

It is well-known that for any integral domain R, the Serre conjecture ring R〈X〉, i.e., the localization of the univariate polynomial ring R[X] at monic polynomials, is a Bézout domain of Krull dimension ≤1 if R is a Bézout domain of Krull dimension ≤1. Consequently, defining by induction R〈X1,…,Xn〉:=(R〈X1,…,Xn−1〉)〈Xn〉, the ring R〈X1,…,Xn〉 is a Bézout domain of Krull dimension ≤1 if so is R. The fact that R〈X1,…,Xn〉 is a Bézout domain when R is a valuation domain of Krull dimension ≤1 was the cornerstone of Brewer and Costa's theorem stating that if R is a one-dimensional arithmetical ring then finitely generated projective R[X1,…,Xn]-modules are extended. It is also the key of the proof of the Gröbner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain R of Krull dimension ≤1, any n∈N>0, and any finitely generated ideal I of R[X1,…,Xn], the ideal LT(I) generated by the leading terms of the elements of I with respect to the lexicographic monomial order is finitely generated. Since the ring R〈X1,…,Xn〉 can also be defined directly as the localization of the multivariate polynomial ring R[X1,…,Xn] at polynomials whose leading coefficients according to the lexicographic monomial order with X1<X2<⋯<Xn is 1, we propose to generalize the fact that R〈X1,…,Xn〉 is a Bézout domain of Krull dimension ≤1 when R is a Bézout domain of Krull dimension ≤1 to any rational monomial order, bolstering the evidence for the Gröbner Ring Conjecture in the rational case. We give an example showing that this result is no more true in the irrational case.

Serre weight conjectures for p-adic unitary groups of rank 2

We prove a version of the weight part of Serre's conjecture for mod $p$ Galois representations attached to automorphic forms on rank 2 unitary groups which are non-split at $p$. More precisely, let $F/F^+$ denote a CM extension of a totally real field such that every place of $F^+$ above $p$ is unramified and inert in $F$, and let $\overline{r}: \textrm{Gal}(\overline{F^+}/F^+) \longrightarrow {}^C\mathbf{U}_2(\overline{\mathbb{F}}_p)$ be a Galois parameter valued in the $C$-group of a rank 2 unitary group attached to $F/F^+$. We assume that $\overline{r}$ is semisimple and sufficiently generic at all places above $p$. Using base change techniques and (a strengthened version of) the Taylor-Wiles-Kisin conditions, we prove that the set of Serre weights in which $\overline{r}$ is modular agrees with the set of Serre weights predicted by Gee-Herzig-Savitt.

A Serre weight conjecture for geometric Hilbert modular forms in characteristic $p$

Let p be a prime and F a totally real field in which p is unramified. We consider mod p Hilbert modular forms for F , defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to p in characteristic p . For a mod p Hilbert modular Hecke eigenform of arbitrary weight (without parity hypotheses), we associate a two-dimensional representation of the absolute Galois group of F , and we give a conjectural description of the set of weights of all eigenforms from which it arises. This conjecture can be viewed as a “geometric” variant of the “algebraic” Serre weight conjecture of Buzzard–Diamond–Jarvis, in the spirit of Edixhoven's variant of Serre's original conjecture in the case F = \mathbb{Q} . We develop techniques for studying the set of weights giving rise to a fixed Galois representation, and prove results in support of the conjecture, including cases of partial weight 1.

On the Grothendieck–Serre conjecture for classical groups

We prove some new cases of the Grothendieck-Serre conjecture for classical groups. This is based on a new construction of the Gersten-Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, with explicit second residue maps; the complex is shown to be exact when the ring is of dimension $\le 2$ (or $\le 4$, with additional hypotheses on the algebra with involution). Note that we do not assume that the ring contains a field.

On the Gersten–Witt complex of an Azumaya algebra with involution

Let (A,σ) be an Azumaya algebra with involution over a regular ring R. We prove that the Gersten–Witt complex of (A,σ) defined by Gille is isomorphic to the Gersten–Witt complex of (A,σ) defined by Bayer-Fluckiger, Parimala and the author. Advantages of both constructions are used to show that the Gersten–Witt complex is exact when dim⁡R≤3, indA≤2 and σ is orthogonal or symplectic. This means that the Grothendieck–Serre conjecture holds for the group R-scheme of σ-unitary elements in A under the same hypotheses; R is not required to contain a field.

An Unconditional Explicit Bound on the Error Term in the Sato–tate Conjecture

AbstractLet $f(z) = \sum_{n=1}^\infty a_f(n)q^n$ be a holomorphic cuspidal newform with even integral weight $k\geq 2$, level N, trivial nebentypus and no complex multiplication. For all primes p, we may define $\theta_p\in [0,\pi]$ such that $a_f(p) = 2p^{(k-1)/2}\cos \theta_p$. The Sato–Tate conjecture states that the angles θp are equidistributed with respect to the probability measure $\mu_{\textrm{ST}}(I) = \frac{2}{\pi}\int_I \sin^2 \theta \; d\theta$, where $I\subseteq [0,\pi]$. Using recent results on the automorphy of symmetric power L-functions due to Newton and Thorne, we explicitly bound the error term in the Sato–Tate conjecture when f corresponds to an elliptic curve over $\mathbb{Q}$ of arbitrary conductor or when f has square-free level. In these cases, if $\pi_{f,I}(x) := \#\{p \leq x : p \nmid N, \theta_p\in I\}$ and $\pi(x) := \# \{p \leq x \}$, we prove the following bound: $$ \left| \frac{\pi_{f,I}(x)}{\pi(x)} - \mu_{\textrm{ST}}(I)\right| \leq 58.1\frac{\log((k-1)N \log{x})}{\sqrt{\log{x}}} \qquad \text{for} \quad x \geq 3. $$ As an application, we give an explicit bound for the number of primes up to x that violate the Atkin–Serre conjecture for f.

Density of rational points near/on compact manifolds with certain curvature conditions

In this article we establish an asymptotic formula for the number of rational points, with bounded denominators, within a given distance to a compact submanifold M of RM with a certain curvature condition. Our result generalises earlier work of Huang for hypersurfaces (J.-J. Huang (2020) [18]), as our curvature condition reduces to Gaussian curvature being bounded away from 0 when M−dim⁡M=1. An interesting feature of our result is that the asymptotic formula holds beyond the conjectured range of the distance to M. Furthermore, we obtain an upper bound for the number of rational points on M with additional power saving to the bound in the analogue of Serre's dimension growth conjecture for compact submanifolds of RM when M−dim⁡M>1.

An 8-periodic exact sequence of Witt groups of Azumaya algebras with involution

Given an Azumaya algebra with involution $$(A,\sigma )$$ over a commutative ring R and some auxiliary data, we construct an 8-periodic chain complex involving the Witt groups of $$(A,\sigma )$$ and other algebras with involution, and prove it is exact when R is semilocal. When R is a field, this recovers an 8-periodic exact sequence of Witt groups of Grenier-Boley and Mahmoudi, which in turn generalizes exact sequences of Parimala–Sridharan–Suresh and Lewis. We apply this result in several ways: We establish the Grothendieck–Serre conjecture on principal homogeneous bundles and the local purity conjecture for certain outer forms of $${\mathbf {GL}}_n$$ and $${\mathbf {Sp}}_{2n}$$ , provided some assumptions on R. We show that a 1-hermitian form over a quadratic étale or quaternion Azumaya algebra over a semilocal ring R is isotropic if and only if its trace (a quadratic form over R) is isotropic, generalizing a result of Jacobson. We also apply it to characterize the kernel of the restriction map $$W(R)\rightarrow W(S)$$ when R is a (non-semilocal) 2-dimensional regular domain and S is a quadratic étale R-algebra, generalizing a theorem of Pfister. In the process, we establish many fundamental results concerning Azumaya algebras with involution and hermitian forms over them.

Grothendieck–Serre in the quasi-split unramified case

AbstractThe Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group schemeGover a regular local ringRis trivial. We settle it in the case whenGis quasi-split andRis unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.

Differential operators mod p : analytic continuation and consequences

This paper concerns certain $\mod p$ differential operators that act on automorphic forms over Shimura varieties of type A or C. We show that, over the ordinary locus, these operators agree with the $\mod p$ reduction of the $p$-adic theta operators previously studied by some of the authors. In the characteristic $0$, $p$-adic case, there is an obstruction that makes it impossible to extend the theta operators to the whole Shimura variety. On the other hand, our $\mod p$ operators extend ("analytically continue", in the language of de Shalit and Goren) to the whole Shimura variety. As a consequence, motivated by their use by Edixhoven and Jochnowitz in the case of modular forms for proving the weight part of Serre's conjecture, we discuss some effects of these operators on Galois representations. Our focus and techniques differ from those in the literature. Our intrinsic, coordinate-free approach removes difficulties that arise from working with $q$-expansions and works in settings where earlier techniques, which rely on explicit calculations, are not applicable. In contrast with previous constructions and analytic continuation results, these techniques work for any totally real base field, any weight, and all signatures and ranks of groups at once, recovering prior results on analytic continuation as special cases.

Modularity of PGL2(𝔽p)-representations over totally real fields

We study an analog of Serre's modularity conjecture for projective representations [Formula: see text], where K is a totally real number field. We prove cases of this conjecture when [Formula: see text].

On the Grothendieck–Serre conjecture on principal bundles in mixed characteristic

Let $R$ be a regular local ring. Let $\mathbf {G}$ be a reductive $R$-group scheme. A conjecture of Grothendieck and Serre predicts that a principal ${\mathbf {G}}$-bundle over $R$ is trivial if it is trivial over the quotient field of $R$. The conjecture is known when $R$ contains a field. We prove the conjecture for a large class of regular local rings

Potentially diagonalizable modular lifts of large weight

We prove that for a Hecke cuspform f∈Sk(Γ0(N),χ) and a prime l>max⁡{k,6} such that l∤N, there exists an infinite family {kr}r≥1⊆Z such that for each kr, there is a cusp form fkr∈Skr(Γ0(N),χ) such that the Deligne representation ρfkr,l is a crystalline and potentially diagonalizable lift of ρ‾f,l. When f is l-ordinary, we base our proof on the theory of Hida families, while in the non-ordinary case, we adapt a local-to-global argument due to Khare and Wintenberger in the setting of their proof of Serre's modularity conjecture, together with a result on existence of lifts with prescribed local conditions over CM fields, a flatness result due to Böckle and a local dimension result by Kisin. We discuss the motivation and tentative future applications of our result in ongoing research on the automorphy of GL2n-type representations in the higher level case.

Almost all primes satisfy the Atkin–Serre conjecture and are not extremal

Let \(f(z)=\sum _{n=1}^{\infty } a_f(n)e^{2\pi i n z}\) be a non-CM holomorphic cuspidal newform of trivial nebentypus and even integral weight \(k\ge 2\). Deligne’s proof of the Weil conjectures shows that \(|a_f(p)|\le 2p^{\frac{k-1}{2}}\) for all primes p. We prove for 100% of primes p that \( 2p^{\frac{k-1}{2}}{\log \log p}/{\sqrt{\log p}}<|a_f(p)|<\lfloor 2p^{\frac{k-1}{2}}\rfloor .\) Our proof gives an effective upper bound for the size of the exceptional set. The lower bound shows that the Atkin–Serre conjecture is satisfied for 100% of primes, and the upper bound shows that \(|a_f(p)|\) is as large as possible (i.e., p is extremal for f) for 0% of primes. Our proofs use the effective form of the Sato–Tate conjecture proved by the second author, which relies on the recent proof of the automorphy of the symmetric powers of f due to Newton and Thorne.

Notes on a Grothendieck–Serre Conjecture in Mixed Characteristic Case

Let R be a discrete valuation ring with infinite residue field and X a smooth projective curve over R. Let G be a simple simply-connected group scheme over R and E a principal G-bundle over X. It is proved that E is trivial locally for the Zariski topology on X providing E is trivial over the generic point of X. The main aim of the present paper is to develop a method rather than to get a very strong concrete result.

Bounds for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function

In the paper, the author presents bounds for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function. This result partially confirms one in a series of conjectures on completely monotonic degrees of remainders of asymptotic expansions for the logarithm of the gamma function and for polygamma functions.

Potentially diagonalisable lifts with controlled Hodge-Tate weights

Motivated by the weight part of Serre's conjecture we consider the following question. Let K/\mathbb{Q}_p be a finite extension and suppose \overline{\rho} : G_K \rightarrow \operatorname{GL}_n(\overline{\mathbb{F}}_p) admits a crystalline lift with Hodge-Tate weights contained in the range [0,p] . Does \overline{\rho} admit a potentially diagonalisable crystalline lift of the same Hodge-Tate weights? We answer this question in the affirmative when K = \mathbb{Q}_p and n \leq 5 , and \overline{\rho} satisfies a mild 'cyclotomic-free' condition. We also prove partial results when K/\mathbb{Q}_p is unramified and n is arbitrary.