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For each parabolic subgroup \mathrm {P} of the general linear group \operatorname {GL}_n(\mathbb {F}_q), a conjecture due to Lewis, Reiner and Stanton [Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), pp. 831–873] predicts a formula for the Hilbert series of the space of invariants \mathcal {Q}_m(n)^\mathrm {P} where \mathcal {Q}_m(n) is the quotient ring \mathbb {F}_q[x_1,\ldots,x_n]/(x_1^{q^m},\ldots,x_n^{q^m}). In this paper, we prove the conjecture for the Borel subgroup \mathrm {B} by constructing a linear basis for \mathcal {Q}_m(n)^\mathrm {B}. The construction is based on an operator \delta which produces new invariants from old invariants of lower ranks. We also upgrade the conjecture of Lewis, Reiner and Stanton by proposing an explicit basis for the space of invariants for each parabolic subgroup. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence and uniqueness of bounded solutions of the degenerate complex Monge-Ampère type equations on compact Hermitian manifolds and obtain the asymptotics of these solutions. As applications, we give partial answers to the Tosatti-Weinkove conjecture and Demailly-Păun conjecture. [ABSTRACT FROM AUTHOR]

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in \mathbb {P}^5, and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known. In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in \mathbb {P}^4 and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces. [ABSTRACT FROM AUTHOR]

It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the q-Morris constant term identity[J. Combin. Theory Ser. A 81 (1998), pp. 69–87]. This conjecture was proved and extended by Károlyi, Nagy, Petrov, and Volkov (KNPV) in 2015 [Adv. Math. 277 (2015), pp. 252–282]. In this paper, we obtain two symmetric function generalizations of the q-Baker–Forrester ex-conjecture. These include: (i) a q-Baker–Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's result. [ABSTRACT FROM AUTHOR]

For a natural unit interval order P, we describe proper colorings of the incomparability graph of P in the language of heaps. We also introduce a combinatorial operation, called a local flip , on the heaps. This operation defines an equivalence relation on the proper colorings, and the equivalence relation refines the ascent statistic introduced by Shareshian and Wachs. In addition, we define an analogue of noncommutative symmetric functions introduced by Fomin and Greene, with respect to P. We establish a duality between the chromatic quasisymmetric function of P and these noncommutative symmetric functions. This duality leads us to positive expansions of the chromatic quasisymmetric functions into several symmetric function bases. In particular, we present some partial results for the e-positivity conjecture. [ABSTRACT FROM AUTHOR]

Let \sigma and \tau denote a pair of absolutely irreducible p-ordinary and p-distinguished Galois representations into \operatorname {GL}_2(\overline {\mathbb {F}}_p). Given two primitive forms (f,g) such that \operatorname {wt}(f)>\operatorname {wt}(g)> 1 and where \overline {\rho }_f\cong \sigma and \overline {\rho }_g\cong \tau, we show that the Iwasawa Main Conjecture for the double product \rho _f\otimes \rho _g depends only on the residual Galois representation \sigma \otimes \tau : G_{\mathbb {Q}}\rightarrow \operatorname {GL}_4(\overline {\mathbb {F}}_p). More precisely, if IMC(f\otimes g) is true for one pair (f,g) with \overline {\rho }_f \cong \sigma and \overline {\rho }_g\cong \tau and whose \mu-invariant equals zero, then it is true for every congruent pair too. [ABSTRACT FROM AUTHOR]

A long standing conjecture in Hamiltonian Dynamics states that every contact form on the standard contact sphere S^{2n+1} has at least n+1 simple periodic Reeb orbits. In this work, we consider a refinement of this problem when the contact form has a suitable symmetry and we ask if there are at least n+1 simple symmetric periodic orbits. We show that there is at least one symmetric periodic orbit for any contact form and at least two symmetric closed orbits whenever the contact form is dynamically convex. [ABSTRACT FROM AUTHOR]

A separating path system for a graph G is a collection \mathcal {P} of paths in G such that for every two edges e and f, there is a path in \mathcal {P} that contains e but not f. We show that every n-vertex graph has a separating path system of size O(n \log ^* n). This improves upon the previous best upper bound of O(n \log n), and makes progress towards a conjecture of Falgas-Ravry–Kittipassorn–Korándi–Letzter–Narayanan and Balogh–Csaba–Martin–Pluhár, according to which an O(n) bound should hold. [ABSTRACT FROM AUTHOR]

Let K/\mathbb {Q}_p be a finite unramified extension, \overline {\rho }:\mathrm {Gal}(\overline {\mathbb {Q}}_p/K)\rightarrow \mathrm {GL}_n(\overline {\mathbb {F}}_p) a continuous representation, and \tau a tame inertial type of dimension n. We explicitly determine, under mild regularity conditions on \tau, the potentially crystalline deformation ring R^{\eta,\tau }_{\overline {\rho }} in parallel Hodge–Tate weights \eta =(n-1,\cdots,1,0) and inertial type \tau when the shape of \overline {\rho } with respect to \tau has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre's conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150]. [ABSTRACT FROM AUTHOR]

We study the structure of Jacobians of geometrically reduced curves over arbitrary (i.e., not necessarily perfect) fields. We show that, while such a group scheme cannot in general be decomposed into an affine and an Abelian part as over perfect fields, several important structural results for these group schemes nevertheless have close analoga over imperfect fields. We apply our results to prove two conjectures due to Bosch-Lütkebohmert-Raynaud about the existence of Néron models and Néron lft-models over excellent Dedekind schemes in the special case of Jacobians of geometrically reduced curves. Finally, we prove some existence results for semi-factorial models and related objects for general geometrically integral curves in the local case. [ABSTRACT FROM AUTHOR]

We show a truncated second main theorem of level one with explicit exceptional sets for analytic maps into \mathbb P^2 intersecting the coordinate lines with sufficiently high multiplicities. The proof is based on a greatest common divisor theorem for an analytic map f:\mathbb C\mapsto \mathbb P^n and two homogeneous polynomials in n+1 variables with coefficients which are meromorphic functions of the same growth as the analytic map f. As applications, we study some cases of Campana's orbifold conjecture for \mathbb P^2 and finite ramified covers of \mathbb P^2 with three components admitting sufficiently large multiplicities. In addition, we explicitly determine the exceptional sets. Consequently, it implies the strong Green-Griffiths-Lang conjecture for finite ramified covers of \mathbb G_m^2. [ABSTRACT FROM AUTHOR]

Given an odd prime number p and a p-stabilized Artin representation \rho over \mathbb {Q}, we introduce a family of p-adic Stark regulators and we formulate an Iwasawa-Greenberg main conjecture and a p-adic Stark conjecture which can be seen as an explicit strengthening of conjectures by Perrin-Riou and Benois in the context of Artin motives. We show that these conjectures imply the p-part of the Tamagawa number conjecture for Artin motives at s=0 and we obtain unconditional results on the torsionness of Selmer groups. We also relate our new conjectures with various main conjectures and variants of p-adic Stark conjectures that appear in the literature. In the case of monomial representations, we prove that our conjectures are essentially equivalent to some newly introduced Iwasawa-theoretic conjectures for Rubin-Stark elements. We derive from this a p-adic Beilinson-Stark formula for finite-order characters of an imaginary quadratic field in which p is inert. Along the way, we prove that the Gross-Kuz'min conjecture unconditionally holds for abelian extensions of imaginary quadratic fields. [ABSTRACT FROM AUTHOR]

Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted L-functions. These ratios conjectures are very powerful and can be used to determine many statistics of L-functions, including moments or statistics about the distribution of zeros. We consider the family of real Dirichlet characters, and use multiple Dirichlet series to prove the ratios conjectures with one shift in the numerator and denominator in some range of the shifts. This range can be improved by extending the family to include non-primitive characters. All of the results are conditional under the Generalized Riemann hypothesis. This extended range is good enough to enable us to compute an asymptotic formula for the sum of shifted logarithmic derivatives near the critical line. As an application, we compute the one-level density for test functions whose Fourier transform is supported in \left (-2,2\right), including lower-order terms. [ABSTRACT FROM AUTHOR]

Let p be a prime and G a finite group. We propose a strong bound for the number of p'-degree irreducible characters of G in terms of the commutator factor group of a Sylow p-subgroup of G. The bound arises from a recent conjecture of Navarro and Tiep [Forum Math. Pi 9 (2021), pp. 1–28] on fields of character values and a phenomenon called the continuity of p-rationality level of p'-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture (see G. Navarro [Ann. of Math. (2) 160 (2004), pp. 1129–1140]). We achieve both the bound and the continuity property for p=2. [ABSTRACT FROM AUTHOR]