Birch And Swinnerton-Dyer Conjecture Research Articles

Total Cuntz semigroup, extension, and Elliott Conjecture with real rank zero

AbstractIn this paper, we exhibit two unital, separable, nuclear ‐algebras of stable rank one and real rank zero with the same ordered scaled total K‐theory, but they are not isomorphic with each other, which forms a counterexample to the Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of the total K‐theory. For the general setting, with a new invariant, the total Cuntz semigroup [2], we classify a large class of ‐algebras obtained from extensions. The total Cuntz semigroup, which distinguishes the algebras of our counterexample, could possibly classify all the ‐algebras of stable rank one and real rank zero.

Coxeter Quotients of Knot Groups through 16 Crossings

We find explicit maximal rank Coxeter quotients for the knot groups of 595 , 515 out of the 1 , 701 , 936 knots through 16 crossings. We thus calculate the bridge numbers and verify Cappell and Shaneson’s Meridional Rank Conjecture for these knots. In addition, we provide a computational tool for establishing the conjecture for those knots beyond 16 crossings whose meridional ranks can be detected via finite Coxeter quotients.

On [formula omitted]-neighborly reorientations of oriented matroids

We study the existence and the number of k-neighborly reorientations of an oriented matroid. This leads to k-variants of McMullen’s problem and Roudneff’s conjecture, the case k=1 being the original statements. Adding to results of Larman and García-Colín, we provide new bounds on k-McMullen’s problem and prove the conjecture for several ranks and k by computer. Further, we show that k-Roudneff’s conjecture for fixed rank and k reduces to a finite case analysis. As a consequence we prove the conjecture for odd rank r and k=r−12 as well as for rank 6 and k=2 with the aid of the computer.

Meridional rank and bridge number of knotted 2-spheres

Abstract The meridional rank conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper, we investigate the analogous conjecture for knotted spheres in $S^4$ . Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order in Coxeter groups and alternating groups, which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres.

Dynamics of quadratic polynomials and rational points on a curve of genus 4

Let f t ( z ) = z 2 + t f_t(z)=z^2+t . For any z ∈ Q z\in \mathbb {Q} , let S z S_z be the collection of t ∈ Q t\in \mathbb {Q} such that z z is preperiodic for f t f_t . In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove a uniform result regarding the size of S z S_z over z ∈ Q z\in \mathbb {Q} . In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve C C of genus 4 4 defined over Q \mathbb {Q} . We use Chabauty’s method, which requires us to determine the Mordell-Weil rank of the Jacobian J J of C C . We give two proofs that the rank is 1 1 : an analytic proof, which is conditional on the BSD rank conjecture for J J and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree 12 12 and degree 24 24 , respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for J J .

On the parity conjecture for abelian surfaces

AbstractAssuming finiteness of the Tate–Shafarevich group, we prove that the Birch–Swinnerton–Dyer conjecture correctly predicts the parity of the rank of semistable principally polarised abelian surfaces. If the surface in question is the Jacobian of a curve, we require that the curve has good ordinary reduction at 2‐adic places.

On the 2-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of ℚ(−7)

For a modular elliptic curve [Formula: see text] and its quadratic twists [Formula: see text], we give equivalent conditions such that the [Formula: see text]-Selmer group [Formula: see text] is minimal, namely, it is of order [Formula: see text]. One of these conditions is described by the [Formula: see text]-value [Formula: see text]. The other conditions are described by quadratic and biquadratic residue symbol, so explicit and computable (and one can compute the density of [Formula: see text]). Also we prove the full Birch–Swinnerton-Dyer conjecture when the equivalent conditions are satisfied. This generalizes a result by J. Coates, Y. Li, Y. Tian and S. Zhai.

The bridge number of arborescent links with many twigs

We prove the meridional rank conjecture for arborescent links associated to plane trees with the following property: all branching points carry a straight branch to at least three leaves. The proof involves an upper bound on the bridge number in terms of the maximal number of link components of the underlying tree, valid for all arborescent links.

Determination of the modular Jacobian varieties J_1(M,MN) with the Mordell–Weil rank zero

In this paper, we determine all modular Jacobian varieties J_1(M,MN) over the cyclotomic fields {{mathbb {Q}}}(zeta _M) with the Mordell–Weil rank zero assuming the Birch–Swinnerton-Dyer conjecture, following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.

The Toral Rank Conjecture and variants of equivariant formality

An action of a compact Lie group is called equivariantly formal, if the Leray–Serre spectral sequence of its Borel fibration degenerates at the E2-term. This term is as prominent as it is restrictive.In this article, also motivated by the lack of junction between the notion of equivariant formality and the concept of formality of spaces (surging from rational homotopy theory) we suggest two new variations of equivariant formality: “MOD-formal actions” and “actions of formal core”.We investigate and characterize these new terms in many different ways involving various tools from rational homotopy theory, Hirsch–Brown models, A∞-algebras, etc., and, in particular, we provide different applications ranging from actions on symplectic manifolds and rationally elliptic spaces to manifolds of non-negative sectional curvature.A major motivation for the new definitions was that an almost free action of a torus Tn↷X possessing any of the two new properties satisfies the toral rank conjecture, i.e. dim⁡H⁎(X;Q)≥2n. This generalizes and proves the toral rank conjecture for actions with formal orbit spaces.

Vertex and edge metric dimensions of cacti

In a graph G , a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S . The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G . In Sedlar and Škrekovski (0000) we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of these configurations is present in the graph. In this paper we extend the result to cactus graphs i.e. graphs in which all cycles are pairwise edge disjoint. We do so by defining a unicyclic subgraph of G for every cycle of G and applying the already introduced approach for unicyclic graphs which involves the configurations. The obtained results enable us to prove the cycle rank conjecture for cacti. They also yield a simple upper bound on metric dimensions of cactus graphs and we conclude the paper by conjecturing that the same upper bound holds in general.

Constructing reducible Brill–Noether curves II

This paper is the second of a two-part series by the author devoted to the following fundamental problem in the theory of algebraic curves in projective space: Which reducible curves arise as limits of smooth curves of general moduli? Special cases of this question studied by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57 ), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137 ), and others have been critical in the resolution of many problems in the theory of algebraic curves over the past half century. In this paper, we give sharp bounds on this problem for space curves, when the nodes are general points and the components are general in moduli. We also systematically study a variant in projective spaces of arbitrary dimension when the nodes are general in a hyperplane. The results given here significantly extend those cases established in the previous paper in this series (Eric Larson, Constructing reducible Brill-Noether curves, To appear in documentamathematica, arxiv:1603.02301), as well as those cases established by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57 ), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137 ) , and others. As explained in (Eric Larson, Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980), the reducible curves constructed in this paper also play a critical role in the author’s proof of the maximal rank conjecture in a subsequent paper (Eric Larson Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980).

Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

Let X be one of the 28 Atkin–Lehner quotients of a curve X 0 (N) such that X has genus 2 and its Jacobian variety J is absolutely simple. We show that the Shafarevich–Tate group Ш(J/ℚ) is trivial. This verifies the strong BSD conjecture for J.

Local symmetry rank bound for positive intermediate Ricci curvatures

We use a local argument to prove if an $r$-dimensional torus acts isometrically and effectively on a connected $n$-dimensional manifold which has positive $k^\mathrm{th}$-intermediate Ricci curvature at some point, then $r \leq \lfloor \frac{n+k}{2} \rfloor$. This symmetry rank bound generalizes those established by Grove and Searle for positive sectional curvature and Wilking for quasipositive curvature. As a consequence, we show that the symmetry rank bound in the Maximal Symmetry Rank Conjecture for manifolds of non-negative sectional curvature holds for those which also have positive intermediate Ricci curvature at some point. In the process of proving our symmetry rank bound, we also obtain an optimal dimensional restriction on isometric immersions of manifolds with non-positive intermediate Ricci curvature into manifolds with positive intermediate Ricci curvature, generalizing a result by Otsuki.

VALUES OF ZETA FUNCTIONS OF ARITHMETIC SURFACES AT

AbstractWe show that the conjecture of [27] for the special value at$s=1$of the zeta function of an arithmetic surface is equivalent to the Birch–Swinnerton–Dyer conjecture for the Jacobian of the generic fibre.

Hitchin fibrations, abelian surfaces, and the P=W conjecture

We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 2 curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role.

Lower-Order Biases in the Second Moment of Dirichlet Coefficients in Families of L-Functions

Let be a nontrivial one-parameter family of elliptic curves over , with . Consider the k th moments of the Dirichlet coefficients . Rosen and Silverman proved Nagao’s conjecture relating the first moment to the family’s rank over , and Michel proved if j(T) is not constant then the second moment equals . Cohomological arguments show the lower order terms are of sizes and 1. In every case, we can analyze in closed form, the largest lower order term in the second moment expansion that does not average to zero is on average negative, though numerics suggest this may fail for families of moderate rank. We prove this Bias Conjecture for several large classes of families, including families with rank, complex multiplication, and constant j(T)-invariant. We also study the analogous Bias Conjecture for families of Dirichlet characters, holomorphic forms on GL, and their symmetric powers and Rankin-Selberg convolutions. We identify all lower order terms in large classes of families, shedding light on the arithmetic objects controlling these terms. The negative bias in these lower order terms has implications toward the excess rank conjecture and the behavior of zeros near the central point.

Tiling number problem and $\boldsymbol{\theta}$-congruent number problem

The tiling number problem is closely related to the arithmetic of elliptic curves. In this paper, we construct a family of tiling numbers with multiple prime factors by using Birch lemma and the induction method developed by one of the authors(Ye Tian) to prove the non-torsioness of Heegner points. In addition, we prove the $2$-part BSD conjecture of the related elliptic curves. We deal with the quadratic twist family of elliptic curves without complex multiplication and the distribution of $2$-Selmer groups of this family is not predicted by the general conjecture or the known results.

Torus actions, maximality, and non-negative curvature

Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

The rank two p-curvature conjecture on generic curves

We prove the p-curvature conjecture for rank two vector bundles with connection on generic curves. Our ingredients include new deformation techniques for vector bundles with vanishing p-curvatures on families of varieties, as well as topological criteria for finiteness of surface group representations in terms of simple loops and pairs of pants.