The results concern Ore ring extensions, including differential and skew polynomial extensions, with particular emphasis on the prime radical β . It is shown that for an algebra R over a field with a locally algebraic endomorphism σ , β (R) ⊆β (R[x; σ ]) . An analogous result is obtained for algebras of the form R[x; δ ] with a locally algebraic derivation δ , provided that the prime radical β (R) is δ -stable.
We establish an extension of Viennot’s geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type C analogue of Schensted’s theorem on longest decreasing subsequences. This pairs with our results from [3] on Type C webs to give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a 2k-fold tensor product of the vector representation of 𝔰𝔭_2n equals the number of (n+1) -avoiding matchings of 2k points.
Let p be an odd prime number. Let f be a weight-two normalized Hecke eigen-cuspform that is non-ordinary at p. Let K be an imaginary quadratic field in which p splits. We study the Artin formalism for the two-variable signed p-adic L-functions attached to f over K. In particular, we give evidence of a prediction made by Castella–Ciperiani–Skinner–Sprung and provide a different proof of (Sprung in Adv Math 449:109741, 2024, Lemma 4.34).
The order of a constant cycle curve C ⊂ X on a complex K3 surface, defined by Huybrechts, is a positive integer that measures the obstruction to decomposing the diagonal class Δ _C in the Chow group CH^2(X × C) . In this paper, we compute the order of elliptic constant cycle curves that naturally arise on Kummer surfaces, by passing to the transcendental intermediate Jacobian J_tr^3(X × C) . As a consequence, every n ∈ℕ can be realized as the order of a constant cycle curve on a complex K3 surface.
At the first step of studying order estimates for the q-analogue of the Riemann zeta function, we estimate bounds for it on vertical lines for a fixed parameter q.
We introduce a new invariant of fields that refines their real spectrum and is related to their absolute Galois group: the Artin-Schreier quandle. For formally real number fields, it is freely generated in its variety by a Cantor space of indeterminates. For Laurent series fields, we compute it in terms of the Artin-Schreier quandle of the coefficient field. This result and other examples show that, in general, there are relations.
Using trace formulas for Hecke operators, Eichler first provided a positive solution about basis problems of elliptic cusp forms by quadratic forms. J-L. Waldspurger established that elliptic cusp forms of arbitrary level are spanned by theta series by means of different and interesting ideas and methods. This result is given by Zagier’s analytic theorems, the Siegel main theorem of quadratic forms and the theory of Hecke operators. We intend to generalize Waldspurger’s results and determine theta series which span the space of Hilbert new forms over arbitrary totally real algebraic number fields following Waldspurger’s methods.
It is shown that any localisation of triangulated categories induces (up to an equivalence) a localisation of abelian categories when one passes to their abelianisations. From this one obtains for any enlargement of Grothendieck universes an example of an abelian category and a Serre subcategory within the smaller universe such that the corresponding quotient does only exist within the bigger universe. The second part of this note provides an analogue for the abelian hull of an arbitrary category.
Let 𝒯 be an algebraic triangulated category and 𝒞 an extension-closed subcategory with Hom(𝒞, Σ^<0𝒞)=0. Then 𝒞 has an exact structure induced from exact triangles in 𝒯. Keller and Vossieck say that there exists a triangle functor D^b(𝒞) →𝒯 extending the inclusion 𝒞⊆𝒯. We provide the missing details for a complete proof.
Let $p$ and $q$ be two distinct fixed prime numbers and $(n_i)_{i\geq 0}$ the sequence of consecutive integers of the form $p^a\cdot q^b$ with $a,b\ge 0$. Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size $n_{i+1}-n_i$, with each bound containing an unspecified exponent and implicit constant. We will explicitly bound these four quantities. Earlier Langevin (1976) gave weaker estimates for (only) the exponents. Given a real number $\alpha>1$, there exists a smallest number $m$ such that for every $n\ge m$, there exists an integer $n_i$ in $[n,n\alpha)$. Our effective version of Tijdeman's result immediately implies an upper bound for $m$, which using the Koksma-Erd\H{o}s-Turan inequality we will improve on. We present a fast algorithm to determine $m$ when $\max\{p,q\}$ is not too large and demonstrate it with numerical material. In an appendix we explain, given $n_i$, how to efficiently determine both $n_{i-1}$ and $n_{i+1}$, something closely related to work of B\'erczes, Dujella and Hajdu.
We give a description of the intermediate Jacobian fibration attached to a general complex cubic fourfold $X$ containing a plane as a Lagrangian subfibration of a moduli space of torsion sheaves on the K3 surface associated to $X$ up to a cover. To do so, we propose a general construction of Lagrangian fibrations in Prym varieties as subfibrations of Beauville-Mukai systems over some loci of nodal curves in linear systems on K3 surfaces.
We compare the Iwasawa invariants of fine Selmer groups of $p$-adic Galois representations over admissible $p$-adic Lie extensions of a number field $K$ to the Iwasawa invariants of ideal class groups along these Lie extensions. More precisely, let $K$ be a number field, let $V$ be a $p$-adic representation of the absolute Galois group $G_K$ of $K$, and choose a $G_K$-invariant lattice ${T \subseteq V}$. We study the fine Selmer groups of ${A = V/T}$ over suitable $p$-adic Lie extensions $K_\infty/K$, comparing their corank and $\mu$-invariant to the corank and the $\mu$-invariant of the Iwasawa module of ideal class groups in $K_\infty/K$. In the second part of the article, we compare the Iwasawa $\mu$- and $l_0$-invariants of the fine Selmer groups of CM modular forms on the one hand and the Iwasawa invariants of ideal class groups on the other hand over trivialising multiple $\mathbb{Z}_p$-extensions of $K$.
We compute a formula for the discriminant of tautological bundles on symmetric powers of a complex smooth projective curve. It follows that the Bogomolov inequality does not give a new restriction to stability of these tautological bundles. It only rules out tautological bundles which are already known to have the structure sheaf as a destabilising subbundle.
Let 0 < a ≤ 1/2 and define the quadrilateral zeta function by 2Q(s,a):= ζ (s,a) + ζ (s,1-a) + Li_s (e^2π ia) + Li_s(e^2π i(1-a)) , where ζ (s,a) is the Hurwitz zeta function and Li_s (e^2π ia) is the periodic zeta function. In the present paper, we show that there exists a unique real number a_0 ∈ (0,1/2) such that all real zeros of Q(s, a) are simple and are located only at the negative even integers just like ζ (s) if and only if a_0 < a ≤ 1/2 . Moreover, we prove that Q(s, a) has infinitely many complex zeros in the region of absolute convergence and the critical strip when a ∈ℚ∩ (0,1/2) ∖{1/6, 1/4, 1/3} . The Lerch formula, Hadamard product formula, Riemann-von Mangoldt formula for Q(s, a) are also shown.
Let $X$ be a K3 surface, let $C$ be a smooth curve of genus $g$ on $X$, and let $A$ be a line bundle of degree $d$ on $C$. Then a line bundle $M$ on $X$ with $M\otimes\mathcal{O}_C=A$ is called a lift of $A$ . In this paper, we prove that if the dimension of the linear system $|A|$ is $r\geq2$, $g>2d-4+r(r-1)$, $d\geq 2r+4$, and $A$ computes the Clifford index of $C$, then there exists a base point free lift $M$ of $A$ such that the general member of $|M|$ is a smooth curve of genus $r$. In particular, if $|A|$ is a base point free net which defines a double covering $\pi:C\longrightarrow C_0$ of a smooth curve $C_0\subset\mathbb{P}^2$ of degree $k\geq 4$ branched at distinct $6k$ points on $C_0$, then, by using the aforementioned result, we can also show that there exists a 2:1 morphism $\tilde{\pi}:X\longrightarrow \mathbb{P}^2$ such that $\tilde{\pi}|_C=\pi$.
We state and prove a formula for the adjoint of the nullwert map from spaces of Jacobi cusp forms of lattice index to spaces of modular forms. Furthermore, we prove a nonvanishing result for the image of the adjoint of the nullwert map.
In this paper we study the theta lifting of a weight 2 Bianchi modular form ℱ of level Γ _0(𝔫) with 𝔫 square-free to a weight 2 holomorphic Siegel modular form. Motivated by Prasanna’s work for the Shintani lifting, we define the local Schwartz function at finite places using a quadratic Hecke character χ of square-free conductor 𝔣 coprime to level 𝔫 . Then, at certain 2 by 2 g matrices β related to 𝔣 , we can express the Fourier coefficient of this theta lifting as a multiple of L(ℱ,χ ,1) by a non-zero constant. If the twisted L-value is known to be non-vanishing, we can deduce the non-vanishing of our theta lifting.
In this paper, we consider biconservative and biharmonic isometric immersions into the 4-dimensional Lorentzian space form 𝕃^4(δ ) with constant sectional curvature δ . We obtain some local classifications of biconservative CMC surfaces in 𝕃^4(δ ) . Further, we get complete classification of biharmonic CMC surfaces in the de Sitter 4-space. We also proved that there is no biharmonic CMC surface in the anti-de Sitter 4-space. Further, we get the classification of biconservative, quasi-minimal surfaces in Minkowski-4 space.
In this paper, when 1<p<2, we establish the C-loc(1,alpha)-regularity of weak solutions to thedegenerate subellipticp-Laplacian equation Delta(H,p)u(x) = (6)& sum;X-i=1(i)& lowast;(|del(H)u|(p-2)X(i)u)=0 on SU(3) endowed with the horizontal vector fields X-1,...,X-6. The result can be extendedto a class of compact connected semi-simple Lie group
Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup of $K^\times$. Given a prime $\mathfrak p$ of $K$ we denote by ${\rm ind}_{\mathfrak p}(G)$ the index of the subgroup $(G\bmod\mathfrak p)$ of the multiplicative group of the residue field at $\mathfrak p$. Under the Generalized Riemann Hypothesis we determine the natural density of primes of $K$ for which this index is in a prescribed set $S$ and has prescribed Frobenius in a finite Galois extension $F$ of $K$. We study in detail the natural density in case $S$ is an arithmetic progression, in particular its positivity.
