Standard Conjectures Research Articles

The Hodge standard conjecture for self-products of K3 surfaces

The Hodge standard conjecture for self-products of K3 surfaces

Perfect powers in elliptic divisibility sequences

AbstractLet be an elliptic curve given by an integral Weierstrass equation. Let be a point of infinite order, and let be the elliptic divisibility sequence generated by . This paper is concerned with a question posed in 2007 by Everest, Reynolds and Stevens: does contain only finitely many perfect powers? We answer this question positively under the following three additional assumptions: is non‐integral; , where is the discriminant of ; , where denotes the connected component of identity. Our method makes use of Galois representations attached to elliptic curves defined over totally real fields, and their modularity. We can deduce the same theorem without assumptions (ii) and (iii), provided that we assume some standard conjectures from the Langlands programme.

The Lefschetz standard conjectures for IHSMs of generalized Kummer deformation type in certain degrees

The Lefschetz standard conjectures for IHSMs of generalized Kummer deformation type in certain degrees

Chow–Lefschetz motives

We develop Milne’s theory of Lefschetz motives for general adequate equivalence relations and over a not necessarily algebraically closed base field. The corresponding categories turn out to enjoy all properties predicted by standard and less standard conjectures, in a stronger way: algebraic and numerical equivalences agree in this context. We also compute the Tannakian group associated to a Weil cohomology in a different and more conceptual way than Milne’s case-by-case approach.

Root numbers of a family of elliptic curves and two applications

For each t∈Q∖{−1,0,1}, define an elliptic curve over Q by Et:y2=x(x+1)(x+t2).Using a formula for the root number W(Et) as a function of t and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves E/Q whose Mordell–Weil group contains Z×Z/2Z×Z/4Z, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.

A dynamical analogue of a question of Fermat

Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter quadratic polynomials with orbits containing three consecutive squares. In addition, we show that there exists at least one polynomial of the form $x^2+c$ with a rational point whose orbit under this map contains four consecutive squares. This can be viewed as a dynamical analogue of a question of Fermat on rational squares in arithmetic progression. Finally, assuming a standard conjecture on exact periods of periodic points of quadratic polynomials over the rational field, we give necessary and sufficient conditions under which the orbit of a periodic point contains only rational squares.

Gaussian boson sampling with click-counting detectors

Gaussian boson sampling constitutes a prime candidate for an experimental demonstration of quantum advantage within reach with current technological capabilities. The original proposal employs photon-number-resolving detectors, however, these are not widely available. Nevertheless, inexpensive threshold detectors can be combined into a single click-counting detector to achieve approximate photon-number resolution. We investigate the problem of sampling from a general multimode Gaussian state using click-counting detectors and show that the probability of obtaining a given outcome is related to a matrix function which is dubbed as the Kensingtonian. We show how the Kensingtonian relates to the Torontonian and the Hafnian, thus bridging the gap between known Gaussian boson sampling variants. We then prove that, under standard complexity-theoretical conjectures, the model cannot be simulated efficiently. Published by the American Physical Society 2024

On the standard conjecture for a fourfold with $1$-parameter fibration by Abelian varieties

It is proved that the Grothendieck standard conjecture $B(X)$ of Lefschetz type holds for a smooth complex projective 4-dimensional variety $X$ provided that there exists a morphism of $X$ onto a smooth projective curve whose generic scheme fibre is an Abelian variety with bad semi-stable reduction at some place of the curve.

Standard conjecture D for local stacky matrix factorizations

Standard conjecture D for local stacky matrix factorizations

On the standard conjecture for a fourfold with $1$-parameter fibration by Abelian varieties

Доказано, что стандартная гипотеза Гротендика $B(X)$ типа Лефшеца верна для любого гладкого комплексного проективного 4-мерного многообразия $X$, допускающего морфизм на гладкую проективную кривую, общим схемным слоем которого является абелево многообразие с плохой полустабильной редукцией в некоторой точке кривой. Библиография: 41 наименование.

2‐Selmer parity for hyperelliptic curves in quadratic extensions

AbstractWe study the 2‐parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a generalisation of a formula of Kramer and Tunnell relating local invariants of the curve, which may be of independent interest. A new feature of this generalisation is the appearance of terms which govern whether or not the Cassels–Tate pairing on the Jacobian is alternating, which first appeared in work of Poonen–Stoll. We establish the local formula in many instances and show that in remaining cases, it follows from standard global conjectures.

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 .

The numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension

We prove the numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension, and also in some related cases.

Asymptotic Fermat for signature (4,2,p) over number fields

Let K be a number field. Using the modular method, we prove asymptotic results on solutions of the Diophantine equation x4−y2=zp over K, assuming some deep but standard conjectures of the Langlands programme when K has at least one complex embedding. On the other hand, we give unconditional results in the case of totally real extensions having odd narrow class number and a unique prime above 2. When modularity of elliptic curves over K is known, for example when K is real quadratic or the r-layer of the cyclotomic Z2-extension of Q, effective asymptotic results hold.

Arithmetic statistics and diophantine stability for elliptic curves

We study the growth and stability of the Mordell–Weil group and Tate–Shafarevich group of an elliptic curve defined over the rationals, in various cyclic Galois extensions of prime power order. Mazur and Rubin introduced the notion of diophantine stability for the Mordell–Weil group an elliptic curve $$E_{/{{\mathbb {Q}}}}$$ at a given prime p. Inspired by their definition of stability for the Mordell–Weil group, we introduce an analogous notion of stability for the Tate–Shafarevich group, called -stability. From the perspective of Iwasawa theory, it benefits us to introduce a stronger notion of diophantine stability for the Mordell–Weil group. It is shown that any non-CM elliptic curve of rank 0 defined over the rationals is strongly diophantine stable and -stable at $$100\%$$ of primes p. Next, we show that standard conjectures on rank distribution give lower bounds for the proportion of rational elliptic curves E that are strongly diophantine stable at a fixed prime $$p\ge 11$$ . Related questions are studied for rank jumps and growth of ranks Tate–Shafarevich groups on average in prime power cyclic extensions.

On Siegel’s problem for $E$-functions

Siegel defined in 1929 two classes of power series, the E -functions and G -functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. He asked whether any E -function can be represented as a polynomial with algebraic coefficients in a finite number of E -functions of the form {}_pF_q(\lambda z^{q-p+1}) , q\ge p\ge 1 , with rational parameters. The case of E -functions of differential order less than or equal to 2 was settled in the affirmative by Gorelov in 2004, but Siegel's question is open for higher order. We prove here that if Siegel's question has a positive answer, then the ring \mathbf{G} of values taken by analytic continuations of G -functions at algebraic points must be a subring of the relatively “small” ring \mathbf{H} generated by algebraic numbers, 1/\pi and the values of the derivatives of the Gamma function at rational points. Because that inclusion seems unlikely (and contradicts standard conjectures), this points towards a negative answer to Siegel's question in general. As intermediate steps, we first prove that any element of \mathbf{G} is a coefficient of the asymptotic expansion of a suitable E -function, which completes previous results of ours. We then prove (in two steps) that the coefficients of the asymptotic expansion of a hypergeometric E -function with rational parameters are in \mathbf{H} . Finally, we prove a similar result for G -functions.

The connected wedge theorem and its consequences

In the AdS/CFT correspondence, bulk causal structure has consequences for boundary entanglement. In quantum information science, causal structures can be replaced by distributed entanglement for the purposes of information processing. In this work, we deepen the understanding of both of these statements, and their relationship, with a number of new results. Centrally, we present and prove a new theorem, the n-to-n connected wedge theorem, which considers n input and n output locations at the boundary of an asymptotically AdS2+1 spacetime described by AdS/CFT. When a sufficiently strong set of causal connections exists among these points in the bulk, a set of n associated regions in the boundary will have extensive-in-N mutual information across any bipartition of the regions. The proof holds in three bulk dimensions for classical spacetimes satisfying the null curvature condition and for semiclassical spacetimes satisfying standard conjectures. The n-to-n connected wedge theorem gives a precise example of how causal connections in a bulk state can emerge from large-N entanglement features of its boundary dual. It also has consequences for quantum information theory: it reveals one pattern of entanglement which is sufficient for information processing in a particular class of causal networks. We argue this pattern is also necessary, and give an AdS/CFT inspired protocol for information processing in this setting.Our theorem generalizes the 2-to-2 connected wedge theorem proven in [3]. We also correct some errors in the proof presented there, in particular a false claim that existing proof techniques work above three bulk dimensions.

Embedding Divisor and Semi-Prime Testability in f-Vectors of Polytopes

We obtain computational hardness results for f-vectors of polytopes by exhibiting reductions of the problems DIVISOR and SEMI-PRIME TESTABILITY to problems on f-vectors of polytopes. Further, we show that the corresponding problems for f-vectors of simplicial polytopes are polytime solvable. The situation where we prove this computational difference (conditioned on standard conjectures on the density of primes and on \(P \ne NP \)) is when the dimension d tends to infinity and the number of facets is linear in d.

Quantum generalizations of the polynomial hierarchy with applications to QMA(2)

The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, \(\rm{QCPH}\), uses classical proofs, and the second, \(\rm{QPH}\), uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, \(\rm{Q\Sigma_3}\), into NEXP using the ellipsoid method for efficiently solving semidefinite programs. These results yield two implications for \(\rm{QMA(2)}\), the variant of Quantum Merlin-Arthur (\(\rm{QMA}\)) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if \(\rm{QCPH = QPH}\) (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs ``equivalent''), then QMA(2) is in the counting hierarchy (specifically, in \({\rm P}^{{\rm pp}^{{\rm pp}}}\)). Second, because \(\rm{QMA(2)}\subseteq \rm{Q\Sigma_3}\), \(\rm{QMA(2)}\) is strictly contained in NEXP unless \(\rm{QMA(2)}=\rm{Q\Sigma_3}\) (i.e., alternating quantifiers do not help in the presence of ``unentanglement'').

A new bound for the Brown–Erdős–Sós problem

Let f(n,v,e) denote the maximum number of edges in a 3-uniform hypergraph not containing e edges spanned by at most v vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges e≥3, what is the smallest integer d=d(e) such that f(n,e+d,e)=o(n2)? This question has its origins in work of Brown, Erdős and Sós from the early 70's and the standard conjecture is that d(e)=3 for every e≥3. The state of the art result regarding this problem was obtained in 2004 by Sárközy and Selkow, who showed that f(n,e+2+⌊log2⁡e⌋,e)=o(n2). The only improvement over this result was a recent breakthrough of Solymosi and Solymosi, who improved the bound for d(10) from 5 to 4. We obtain the first asymptotic improvement over the Sárközy–Selkow bound, showing thatf(n,e+O(log⁡e/log⁡log⁡e),e)=o(n2).