Murmurations and Sato–Tate conjectures for high rank zetas of elliptic curves

On the failure of the integral Tate conjecture for products with projective hypersurfaces

Abstract Let be a smooth cubic hypersurface, and let be the variety of lines on . We prove the surjectivity of the cylinder maps on the Chow groups of and if contains a one‐cycle of degree . Mongardi and Ottem previously proved the integral Hodge conjecture for curve classes on hyperkähler manifolds. Using the cylinder maps, we provide an alternative proof for the of a smooth complex cubic fourfold , which is a special hyperkähler fourfold. In addition, we confirm the integral Tate conjecture for of a smooth cubic fourfold over a finitely generated field.

We study zero-cycles on rationally connected varieties defined over characteristic zero Laurent fields with algebraically closed residue fields. We show that the degree map induces an isomorphism for rationally connected threefolds defined over such fields. In general, the degree map is an isomorphism if rationally connected varieties defined over algebraically closed fields of characteristic zero satisfy the integral Hodge/Tate conjecture for one-cycles, or if the Tate conjecture is true for divisor classes on surfaces defined over finite fields. To prove these results, we introduce techniques from the minimal model program to study the Gersten type complex defined by Kato/Bloch–Ogus. We also propose a conjecture about the Kato homology of a rationally connected fibration.

Abstract Let $a(n)$ be the nth Dirichlet coefficient of the automorphic L-function or the Rankin–Selberg L-function. We investigate the cancellation of $a(n)$ over sequences linked to the Waring–Goldbach problem, by establishing a non-trivial bound for the additive twisted sums over primes on ${\mathrm {GL}}_m$ . The bound does not depend on the generalized Ramanujan conjecture or the non-existence of Landau–Siegel zeros. Furthermore, we present an application associated with the Sato–Tate conjecture and propose a conjecture about the Goldbach conjecture on average bound.

Let E E be an elliptic curve over Q \mathbb {Q} with Mordell–Weil rank 2 2 and p p be an odd prime of good ordinary reduction. For every imaginary quadratic field K K satisfying the Heegner hypothesis, there is (subject to the Shafarevich–Tate conjecture) a line, i.e., a free Z p \mathbb {Z}_p -submodule of rank 1 1 , in E ( K ) ⊗ Z p E(K)\otimes \mathbb {Z}_p given by universal norms coming from the Mordell–Weil groups of subfields of the anticyclotomic Z p \mathbb {Z}_p -extension of K K ; we call it the shadow line. When the twist of E E by K K has analytic rank 1 1 , the shadow line is conjectured to lie in E ( Q ) ⊗ Z p E(\mathbb {Q})\otimes \mathbb {Z}_p ; we verify this computationally in all our examples. We study the distribution of shadow lines in E ( Q ) ⊗ Z p E(\mathbb {Q})\otimes \mathbb {Z}_p as K K varies, framing conjectures based on the computations we have made.

As a continuation of our earlier paper [Z. Shi and L. Weng, Murmurations and Sato–Tate conjectures for high rank zetas of elliptic curves, preprint, arXiv:2410.04952], we offer a new approach to murmurations and Sato–Tate laws for higher rank zetas of elliptic curves. Our approach here does not depend on the Riemann hypothesis for the so-called [Formula: see text]-invariant [Formula: see text] in rank [Formula: see text] even for the Sato–Tate law, rather, on a much refined structure, a similar version of which was already observed earlier by Zagier and the senior author of this paper in [L. Weng and D. Zagier, Higher rank zeta functions for elliptic curves, Proc. Natl. Acad. Sci USA 117(9) (2020) 4546–4558] when the rank [Formula: see text] Riemann hypothesis was established. Namely, instead of the rank [Formula: see text] Riemann hypothesis bounds [Formula: see text] on which our first paper is based, we use the asymptotic bounds [Formula: see text]. Accordingly, rank [Formula: see text] Sato–Tate law can be established and rank [Formula: see text] murmurations can be formulated equally well, similar to the corresponding structures in the abelian framework for Artin zetas of elliptic curves.

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.

We give a geometric criterion to check the validity of the integral Tate conjecture for one-cycles on a smooth projective variety that is separably rationally connected in codimension one, and to check that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on a separably rationally connected variety defined over a global function field. We prove that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on all geometrically rational surfaces defined over a global function field, and to the Hasse principle for rational points on del Pezzo surfaces of degree four defined over a global function field of odd characteristic. Along the way, we also prove some results about the space of one-cycles on a smooth projective variety that is separably rationally connected in codimension one, which leads to the equality of the coniveau filtration and the strong coniveau filtration on degree 3 3 homology of such varieties.

Let S be an integral variety over a finitely generated field k , with generic point \eta , and A\rightarrow S an abelian scheme. The Hilbert irreducibility theorem and the Tate conjectures imply that the following local-global principle always holds if k is infinite. Given an abelian variety \mathfrak{A} over k , for every closed point s\in S , \mathfrak{A} is a geometric isogeny factor of A_{s} if and only if \mathfrak{A}\times_{k}k(\eta) is a geometric isogeny factor of A_{\eta} . If k is finite, the problem is more subtle. We construct an obstruction – the ghost of A\rightarrow S – which completely controls the failure of the above local-global principle and is a motive built from the weight zero part of the representation of the geometric monodromy on the \ell -adic Tate module of A_{\eta} ( \ell\not=p ). In particular, this enables us to show that the above local-global principle fails for certain abelian schemes built by Katz and Bültel.

Abstract We prove the Ramanujan and Sato–Tate conjectures for Bianchi modular forms of weight at least $2$ . More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\operatorname {\mathrm {GL}}_2(\mathbf {A}_F)$ of parallel weight, where F is any CM field. We deduce these theorems from a new potential automorphy theorem for the symmetric powers of $2$ -dimensional compatible systems of Galois representations of parallel weight.

We study Gushel-Mukai (GM) varieties of dimension 4 or 6 in characteristic p. Our main result is the Tate conjecture for all such varieties over finitely generated fields of characteristic p ≥ 5. In the case of GM sixfolds, we follow the method used by Madapusi Pera in his proof of the Tate conjecture for K3 surfaces. As input for this, we prove a number of basic results about GM sixfolds, such as the fact that there are no nonzero global vector fields. For GM fourfolds, we prove the Tate conjecture by reducing it to the case of GM sixfolds by making use of the notion of generalised partners plus the fact that generalised partners in characteristic 0 have isomorphic Chow motives in middle degree. Several steps in the proofs rely on results in characteristic 0 that are proven in our paper [19] .

We study algebraic cycles on complex Gushel-Mukai (GM) varieties. We prove the generalised Hodge conjecture, the (motivated) Mumford-Tate conjecture, and the generalised Tate conjecture for all GM varieties. We compute all integral Chow groups of GM varieties, except for the only two infinite-dimensional cases (1-cycles on GM fourfolds and 2-cycles on GM sixfolds). We prove that if two GM varieties are generalised partners or generalised duals, their rational Chow motives in middle degree are isomorphic.Comment: 23 pages, final version, in special volume in honour of C. Voisin

Patterns of primes in joint Sato–Tate distributions

From the generalized Riemann hypothesis for motivic L -functions, we derive an effective version of the Sato–Tate conjecture for an abelian variety A defined over a number field k with connected Sato–Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato–Tate measure that only depends on certain invariants of A . We discuss three applications of this conditional result. First, for an abelian variety defined over k , we consider a variant of Linnik’s problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius traces attain the integral part of the Hasse–Weil bound. Third, for a pair of abelian varieties A and A' defined over k with no common factors up to k -isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces of A and A' have opposite sign.

We prove that the $p^\infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a ‘flat Tate conjecture’ for divisors. We also study other geometric Galois-invariant $p^\infty$-torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely $p$-divisible. We explain how the existence of these $p$-divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron–Severi groups in characteristic $p$.

Let A and B be abelian varieties defined over the function field k(S) of a smooth algebraic variety S/k. We establish criteria, in terms of restriction maps to subvarieties of S, for existence of various important classes of k(S)-homomorphisms from A to B, e.g., for existence of k(S)-isogenies. Our main tools consist of Hilbertianity methods, Tate conjecture as proven by Tate, Zarhin and Faltings, and of the minuscule weights conjecture of Zarhin in the case when the base field is finite.

Abstract The classical Mordell–Weil theorem implies that an abelian variety A over a number field K has only finitely many K-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension $K^{\mathrm {cyc}}=K{\mathbb Q}^{\mathrm {ab}}$ by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety A over the infinite algebraic extension $K_B$ obtained by adjoining the coordinates of all torsion points of an abelian variety B. Assuming the Mumford–Tate conjecture, and up to a finite extension of the base field K, we give a necessary and sufficient condition for the finiteness of $A(K_B)_{\mathrm tors}$ in terms of Mumford–Tate groups. We give a complete answer when both abelian varieties have dimension at most 3, or when both have complex multiplication.

On the observability of Galois representations and the Tate conjecture

Using the formalism of Newton hyperplane arrangements, we resolve the open questions regarding angle rank left over from work of the first two authors with Roe and Vincent. As a consequence we end up generalizing theorems of Lenstra–Zarhin and Tankeev proving several new cases of the Tate conjecture for abelian varieties over finite fields. We also obtain an effective version of a recent theorem of Zarhin bounding the heights of coefficients in multiplicative relations among Frobenius eigenvalues.