We give infinitely many knots with the same HOMFLYPT and Q polynomials, which are distinguished by the Kauffman polynomial. They are symmetric unions of the trefoil knot.

Fix primes 𝑝 and ℓ with ℓ ≠ 𝑝. If (𝐴, 𝜆) is a 𝑔-dimensional principally polarized abelian variety, an (ℓ)^𝑔-isogeny of (𝐴, 𝜆) has kernel a maximal isotropic subgroup of the ℓ-torsion of 𝐴; the image has a natural principal polarization. In this paper we study the isogeny graphs of (ℓ)^𝑔-isogenies of principally polarized superspecial abelian varieties in characteristic 𝑝. We define three isogeny graphs associated to such (ℓ)^𝑔-isogenies—the big isogeny graph 𝐺𝑟_𝑔(ℓ, 𝑝), the little isogeny graph 𝑔𝑟_𝑔(ℓ, 𝑝), and the enhanced isogeny graph \widetilde{𝑔𝑟}_𝑔(ℓ, 𝑝). We apply strong approximation for the quaternionic unitary group to prove both that 𝑔𝑟_𝑔(ℓ, 𝑝) and 𝐺𝑟_𝑔(ℓ, 𝑝) are connected and that they are not bipartite. The connectedness of the enhanced isogeny graph \widetilde{𝑔𝑟}_𝑔(ℓ, 𝑝) then follows. The quaternionic unitary group has previously been applied to moduli of abelian varieties in characteristic 𝑝 (sometimes invoking strong approximation) by Chai, Ekedahl/Oort, and Chai/Oort. The adjacency matrices of the three isogeny graphs are given in terms of the Brandt matrices defined by Hashimoto, Ibukiyama, Ihara, and Shimizu. We study some basic properties of these Brandt matrices and recast the theory using the notion of Brandt graphs. We show that the isogeny graphs 𝐺𝑟_𝑔(ℓ, 𝑝) and 𝑔𝑟_𝑔(ℓ, 𝑝) are in fact our Brandt graphs. We give the ℓ-adic uniformization of 𝑔𝑟_𝑔(ℓ, 𝑝) and \widetilde{𝑔𝑟}_𝑔(ℓ, 𝑝). The (ℓ + 1)-regular isogeny graph 𝐺𝑟₁(ℓ, 𝑝) for supersingular elliptic curves is well known to be Ramanujan. We calculate the Brandt matrices for a range of 𝑔 > 1, ℓ, and 𝑝. These calculations give four examples with 𝑔 > 1 where the regular graph 𝐺𝑟_𝑔(ℓ, 𝑝) has two vertices and is Ramanujan, and all other examples we computed with 𝑔 > 1 and two or more vertices were not Ramanujan. In particular, the (ℓ)^𝑔-isogeny graph is not in general Ramanujan for 𝑔 > 1.

Let 𝐹 be a number field and 𝐷 a quaternion algebra over 𝐹.Take a cuspidal automorphic representation 𝜋 of 𝐷_𝔸^× with trivial central character and a cusp form 𝜙 in 𝜋. Using the prehomogeneous zeta function, we find an explicit mean value of the toric periods of 𝜙 with respect to quadratic algebras over 𝐹. The result can also be written as a mean value formula for the central values of automorphic 𝐿-functions twisted by quadratic characters.

The subject of this paper is a kind of dynamical systems called quantum walks. We study one-dimensional homogeneous analytic quantum walks 𝑈. We explain how to identify the space of all the uniform intertwining operators between these walks. We can also determine whether 𝑈 can be realized by a (not necessarily homogeneous) continuous-time uniform quantum walk on ℤ. Several examples of quantum walks, which can not be realized by continuous-time uniform quantum walks, are presented. The 4-state Grover walk is one of them. Before stating the main theorems, we clarify the definition of one-dimensional quantum walks. For the first half of this paper, we study basic properties of one-dimensional quantum walks, which are not necessarily homogeneous. An equivalence relation between quantum walks called similarity is also introduced. This allows us to manipulate quantum walks in a flexible manner.

We prove the Zariski density of certain set of CM points lying in self-products of elliptic modular curves, considered by Hida. A new ingredient in our argument is a finiteness result on irreducible components for the inverse images in infinite coverings of self-products of modular curves.

We consider the incompressible and stationary Stokes equations on an infinite two-dimensional wedge with non-scaling invariant Navier-slip boundary conditions. We prove well-posedness and higher regularity of the Stokes problem in a certain class of weighted Sobolev spaces.

The novelty of this work is the occurrence of two different scalings in the boundary condition, which is not treated so far for the Stokes system in unbounded wedge-type domains. These difficulties are overcome by first constructing a variational solution in a second-order weighted Sobolev space and subsequently proving higher regularity up to the tip of the wedge by employing an iterative scheme. We believe that this method can be used for other problems with variational structure and multiple scales.

Let 𝑘 be an algebraically closed uncountable field of characteristic zero. Let 𝑅 be a complete local hypersurface over 𝑘. Denote by CM(𝑅) the category of maximal Cohen–Macaulay 𝑅-modules and by D^sg(𝑅) the singularity category of 𝑅. Denote by CM₀(𝑅) the full category of CM(𝑅) consisting of modules that are locally free on the punctured spectrum of 𝑅, and by D^sg₀(𝑅) the full subcategory of D^sg(𝑅) consisting of objects that are locally zero on the punctured spectrum of 𝑅. In this paper, under the assumption that 𝑅 has finite or countable CM-representation type, we completely classify the extension-closed subcategories of CM₀(𝑅) in dimension at most two, and the extension-closed subcategories of D^sg₀(𝑅) in arbitrary dimension.

Let Π_𝑏 be a bounded 𝑛 parameter paraproduct with symbol 𝑏. We demonstrate that this operator is in the Schatten class 𝕊^𝑝, 0 < 𝑝 < ∞, if the symbol is in the 𝑛 parameter Besov space 𝔹^𝑝. Our result covers both the dyadic and continuous version of the paraproducts in the multiparameter setting.

In this paper, we consider the zeta function of graphs and the twisted Alexander polynomials of knots. Since they have been developed separately, we explain the relationship between them through the notion of knot graphs. We also provide a new volume presentation of hyperbolic knots using matrix-weighted graphs and Bell polynomials.

We state several questions, and prove some partial results, about the Chow ring 𝐴^*(𝑋) of complete intersections in projective space. For one thing, we prove that if 𝑋 is a general Calabi–Yau hypersurface, the intersection product 𝐴²(𝑋) ⋅ 𝐴^𝑖(𝑋) is one-dimensional, for any 𝑖 > 0. We also show that quintic threefolds have a multiplicative Chow–Künneth (MCK) decomposition. We wonder whether all Calabi–Yau hypersurfaces might have an MCK decomposition, and prove this is the case conditional to a conjecture of Voisin.