Abstract We prove that an abelian variety and its dual over a global field have the same Faltings height and, more precisely, have isomorphic Hodge line bundles, including their natural metrized bundle structures. More carefully treating real places, we also show that these abelian varieties have the same real and global periods that appear in the Birch–Swinnerton-Dyer conjecture.

Abstract We prove that the Tate conjecture for divisors is “generically true” for mod ⁡ p \operatorname{mod}p reductions of complex projective varieties with h 2 , 0 = 1 h^{2,0}=1 , under a mild assumption on moduli. By refining this general result, we establish a new case of the BSD conjecture over global function fields, and the Tate conjecture for a class of general type surfaces of geometric genus 1.

Given a smooth totally nonholonomic distribution on a smooth manifold, we construct a singular distribution capturing essential abnormal lifts which is locally generated by vector fields with controlled divergence. Then, as an application, we prove the Sard conjecture for rank 3 distribution in dimension 4 and generic distributions of corank 1 .

Abstract We prove non‐vanishing theorems for the central values of ‐series of quadratic twists of the Gross elliptic curve with complex multiplication by the imaginary quadratic field , where is any prime congruent to 7 modulo 8. This completes the non‐vanishing theorems proven by Coates and the second author in which the primes were taken to be congruent to 7 modulo 16. From this, we obtain the finiteness of the Mordell–Weil group and the Tate–Shafarevich group for these curves. For a prime lying above the prime 2, we also prove a converse theorem in the rank 0 case and the ‐part of the Birch–Swinnerton–Dyer conjecture for the higher‐dimensional abelian varieties obtained by restriction of scalars.

Let E / Q E/\mathbf {Q} be an elliptic curve and let p p be an odd prime of good reduction for E E . Let K K be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which p p splits. The goal of this paper is two-fold: (1) we formulate a p p -adic BSD conjecture for the p p -adic L L -function L p B D P L_\mathfrak {p}^{\mathrm {BDP}} introduced by Bertolini–Darmon–Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148]; and (2) for an algebraic analogue F p ¯ B D P F_{\overline {\mathfrak {p}}}^{\mathrm {BDP}} of L p B D P L_\mathfrak {p}^{\mathrm {BDP}} , we show that the “leading coefficient” part of our conjecture holds, and that the “order of vanishing” part follows from the expected “maximal non-degeneracy” of an anticyclotomic p p -adic height. In particular, when the Iwasawa–Greenberg Main Conjecture ( F p ¯ B D P ) = ( L p B D P ) (F_{\overline {\mathfrak {p}}}^{\mathrm {BDP}})=(L_\mathfrak {p}^{\mathrm {BDP}}) is known, our results determine the leading coefficient of L p B D P L_{\mathfrak {p}}^{\mathrm {BDP}} at T = 0 T=0 up to a p p -adic unit. Moreover, by adapting the approach of Burungale–Castella–Kim [Algebra Number Theory 15 (2021), pp. 1627–1653], we prove the main conjecture for supersingular primes p p under mild hypotheses. In the p p -ordinary case, and under some additional hypotheses, similar results were obtained by Agboola–Castella [J. Théor. Nombres Bordeaux 33 (2021), pp 629–658], but our method is new and completely independent from theirs, and apply to all good primes.

We introduce several methods to study the rank of the sum of squares (SoS) hierarchy for problems over the Boolean hypercube. We apply our techniques to improve upon existing results, thus answering several open questions. We answer the question by Laurent regarding the SoS rank of the empty integral hull (EIH) problem. We prove that the SoS rank is between [Formula: see text] and [Formula: see text]. We refute the Lee-Prakash-de Wolf-Yuen (LPdWY) conjecture for the SoS rank of symmetric quadratic functions in n variables with roots placed in points k – 1 and k that conjectured the lower bound of [Formula: see text]. We prove that the SoS rank for SQFs is at most [Formula: see text]. We answer another question by Laurent for an instance of the min knapsack problem parameterized by P. We prove a nearly tight SoS rank between [Formula: see text] and [Formula: see text]. Finally, we refute the conjecture by Bienstock-Zuckerberg that presumed the SoS rank lower bound of [Formula: see text] for an instance of the set cover problem. We refute the conjecture by providing an [Formula: see text] SoS certificate for this problem. Funding: A. Kurpisz was supported by Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (SNSF) [Grant PZ00P2_174117], A. Potechin was supported in part by the National Science Foundation (NSF) [Grant CCF2008920], and E. Wirth was supported by DFG under Germany’s Excellence Strategy, The Berlin Mathematics Research Center (MATH+) [Grant EXC-2046/1, project ID 390685689, BMS Stipend].

The total rank conjecture in characteristic 2

Abstract We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$ . We apply our methods to all 28 Atkin–Lehner quotients of $X_0(N)$ of genus $2$ , all 97 genus $2$ curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least $2$ . We also give an example where we verify that the order of the Tate–Shafarevich group is $7^2$ and agrees with the order predicted by the BSD Conjecture.

Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu-Farkas strong maximal rank conjecture, in genus 22 and 23. This constitutes a major step forward in Farkas' program to prove that the moduli spaces of curves of genus 22 and 23 are of general type. Our techniques involve a combination of the Eisenbud-Harris theory of limit linear series, and the notion of linked linear series developed by Osserman.

Abstract In 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.

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

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.

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 .

Abstract Assuming 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.

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.

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.

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

Vertex and edge metric dimensions of cacti