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The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\\mathbb F_p(t)$, when $p$ is prime and $r\\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\\mathbb F_q(t^1/d)$.

For any α < 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class CtCx α that have nonempty, compact support in time on R × T3 and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for α < 1/3 due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent α = 1/3 marks the threshold for conservation of energy for weak solutions in the class Lt ∞Cx α. The previous best results were solutions in the class CtCx α for α < 1/5, due to [Isett], and in the class Lt 1 Cx α for α < 1/3 due to [Buckmaster, De Lellis, Székelyhidi], both based on the method of convex integration developed for the incompressible Euler equations by [De Lellis, Székelyhidi]. The present proof combines the method of convex integration and a new “Gluing Approximation” technique. The convex integration part of the proof relies on the “Mikado flows” introduced by [Daneri, Székelyhidi] and the framework of estimates developed in the author's previous work.

The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the biharmonic equation, i.e., Δ2x=0. A well-known conjecture proposed by the author in 1991 on biharmonic submanifolds states that every biharmonic submanifold of a Euclidean space is minimal, well known today as Chen’s biharmonic conjecture. On the other hand, independently, G.-Y. Jiang investigated biharmonic maps between Riemannian manifolds as the critical points of the bi-energy functional. In 2002, R. Caddeo, S. Montaldo, and C. Oniciuc pointed out that both definitions of biharmonicity of the author and G.-Y. Jiang coincide for the class of Euclidean submanifolds. Since then, the study of biharmonic submanifolds and biharmonic maps has attracted many researchers, and many interesting results have been achieved. A comprehensive survey of important results on this conjecture and on many related topics was presented by Y.-L. Ou and B.-Y. Chen in their 2020 book. The main purpose of this paper is to provide a detailed survey of recent developments in those subjects after the publication of Ou and Chen’s book.

Let R=C[x1,x2,…,xn]/(f1,…,fm)R= {\\Bbb C}[x_1,x_2,\\ldots , x_n]/(f_1,\\ldots , f_m) be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, differential geometry, and rational homotopy theory is the non-existence of negative weight derivations on RR. Alexsandrov conjectured that there are no negative weight derivations when RR is a complete intersection algebra, and Yau conjectured there are no negative weight derivations on RR when RR is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on RR when the degrees of f1,…,fmf_1,\\ldots ,f_m are bounded below by a constant CC depending only on the weights of x1,…,xnx_1,\\ldots ,x_n. Moreover this bound CC is improved in several special cases.

We formulate a conjecture which generalizes Darmon’s ‘refined class number formula’. We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the ‘except $2$-part’ of Darmon’s conjecture, which was first proved by Mazur and Rubin.

We investigate the Gross–Prasad conjecture and its refinement for the Bessel periods in the case of $(\\mathrm {SO}(5), \\mathrm {SO}(2))$. In particular, by combining several theta correspondences, we prove the Ichino–Ikeda-type formula for any tempered irreducible cuspidal automorphic representation. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two, which are Hecke eigenforms, to central special values of $L$-functions. The formula is regarded as a natural generalization of the Böcherer conjecture to the non-trivial toroidal character case.

In this paper, we show that every (2 n-1 + 1)-vertex induced subgraph of the n-dimensional cube graph has maximum degree at least n . This is the best possible result, and it improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.

This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than 4 · 10¹⁸. Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to 8.37 · 10²⁶. The empirical data collected during this extensive verification effort, namely, counts and first occurrences of so-called minimal Goldbach partitions with a given smallest prime and of gaps between consecutive primes with a given even gap, are used to test several conjectured formulas related to prime numbers. In particular, the counts of minimal Goldbach partitions and of prime gaps are in excellent accord with the predictions made using the prime k-tuple conjecture of Hardy and Littlewood (with an error that appears to be $\\mathrm{O}(\\sqrt{\\mathrm{t}\\text{\\hspace{0.17em}}\\mathrm{log}\\text{\\hspace{0.17em}}\\mathrm{log}\\text{\\hspace{0.17em}}\\mathrm{t})}$, where t is the true value of the quantity being estimated). Prime gap moments also show excellent agreement with a generalization of a conjecture made in 1982 by Heath-Brown.

In 2014, Pila and Tsimerman gave a proof of the Ax–Schanuel conjecture for the $j$ -function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax–Schanuel conjecture. In this article, we show that the hyperbolic Ax–Schanuel conjecture can be used to reduce the Zilber–Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila–Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila–Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber–Pink conjecture for curves in abelian varieties.

In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by Voisin. The book focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by Voisin looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.