Birch And Swinnerton-Dyer Conjecture Research Articles

Iwasawa theory and BSD conjecture

In this survey article, we introduce the backgrounds for BSD (Birch and Swinnerton-Dyer) conjecture—one of the seven millennium problems, and Iwasawa theory,which is a main tool of the study of BSD conjecture. Then we discuss the methods and results to study them, focusing on recent progress.

Asymptotic Sign Coherence Conjecture

The sign coherence phenomenon is an important feature of c-vectors in cluster algebras with principal coefficients. In this note, we consider a more general version of c-vectors defined for arbitrary cluster algebras of geometric type and formulate a conjecture describing their asymptotic behavior. This conjecture, which is called the asymptotic sign coherence conjecture, states that for any infinite sequence of matrix mutations that satisfies certain natural conditions, the corresponding c-vectors eventually become sign coherent. We prove this conjecture for rank 2 cluster algebras of infinite type and for a particular sequence of mutations in a cluster algebra associated with the Markov quiver.

Interpolation for Curves in Projective Space with Bounded Error

AbstractGiven $n$ general points $p_1, p_2, \ldots , p_n \in{\mathbb{P}}^r$ it is natural to ask whether there is a curve of given degree $d$ and genus $g$ passing through them; by counting dimensions a natural conjecture is that such a curve exists if and only if $$\begin{equation*}n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor.\end{equation*}$$The case of curves with nonspecial hyperplane section was recently studied in [2], where the above conjecture was shown to hold with exactly three exceptions.In this paper, we prove a “bounded-error analog” for special linear series on general curves; more precisely we show that existence of such a curve subject to the stronger inequality $$\begin{equation*}n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor - 3.\end{equation*}$$Note that the $-3$ cannot be replaced with $-2$ without introducing exceptions (as a canonical curve in ${\mathbb{P}}^3$ can only pass through nine general points, while a naive dimension count predicts twelve).We also use the same technique to prove that the twist of the normal bundle $N_C(-1)$ satisfies interpolation for curves whose degree is sufficiently large relative to their genus, and deduce from this that the number of general points contained in the hyperplane section of a general curve is at least $$\begin{equation*}\min\left(d, \frac{(r - 1)^2 d - (r - 2)^2 g - (2r^2 - 5r + 12)}{(r - 2)^2}\right).\end{equation*}$$As explained in [7], these results play a key role in the author’s proof of the maximal rank conjecture [9].

NEW BOUNDS ON THE TORAL RANK WITH APPLICATIONS TO COHOMOLOGICALLY SYMPLECTIC SPACES

We use Boij–Soderberg theory to give two lower bounds for the dimension of the cohomology of a finite CW-complex in terms of the toral rank and certain Betti numbers of the space. One of our bounds turns out to be particularly effective for c-symplectic spaces, proving the toral rank conjecture for c-symplectic spaces of formal dimension ≤ 8.

Symmetric quotients of knot groups and a filtration of the Gordian graph

AbstractWe define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The nth subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridional Rank Conjecture for a family of knots with unknotting number one yet arbitrarily high bridge number.

On the main conjecture of Iwasawa theory for certain non‐cyclotomic Zp‐extensions

Let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, with ring of integers $\mathcal{O}$ and Hilbert class field $H$. Suppose $p\nmid [H:K]$ is a prime number which splits in $K$, say $p\mathcal{O}=\mathfrak{p}\mathfrak{p}^*$. Let $H_\infty=HK_\infty$ where $K_\infty$ is the unique $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. Write $M(H_\infty)$ for the maximal abelian $p$-extension of $H_\infty$ unramified outside the primes above $\mathfrak{p}$, and set $X(H_\infty)=\mathrm{Gal}(M(H_\infty)/H_\infty)$. In this paper, we establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_\infty)$. As a consequence, we have that if $X(H_\infty)=0$, the relevant $L$-values are $\mathfrak{p}$-adic units. In addition, the main conjecture for $X(H_\infty)$ has implications toward (a) the BSD Conjecture for a class of CM elliptic curves; (b) weak $\mathfrak{p}$-adic Leopoldt conjecture.

Family of curves with large unitary summand in the Hodge bundle

In this note, we construct non-isotrivial families of curves of genus $$g\ge 2$$ , where the rank of the unitary summand contained in the Hodge bundle can be as large as $$(2g+1)/3$$ , and hence disprove Xiao’s conjecture for the unitary rank.

Bounds on special values of L-functions of elliptic curves in an Artin\u2013Schreier family

We study a certain Artin–Schreier family of elliptic curves over the function field . We prove an asymptotic estimate on the special values of their L-function in terms of the degree of their conductor; we show that the special values are, in a sense, ‘asymptotically as large as possible’. We also provide an explicit expression for their L-function. The proof of the main result uses this expression and a detailed study of the distribution of character sums related to Kloosterman sums. Via the BSD conjecture, the main result translates into an analogue of the Brauer–Siegel theorem for these elliptic curves.

Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

Let k be an algebraically closed field and A the polynomial algebra in r variables with coefficients in k. In case the characteristic of k is 2, Carlsson [9] conjectured that for any DG-A-module M of dimension N as a free A-module, if the homology of M is nontrivial and finite dimensional as a k-vector space, then 2r≤N. Here we state a stronger conjecture about varieties of square-zero upper triangular N×N matrices with entries in A. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when N<8 or r<3 without any restriction on the characteristic of k. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture and give new results when N>4 and r=2.

Heegner points on Hijikata-Pizer-Shemanske curves and the Birch and Swinnerton-Dyer conjecture

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rathergeneral type of quaternionic orders. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In particular, under mild technical conditions, we show the existence of non-torsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence.

A non-abelian conjecture of Tate–Shafarevich type for hyperbolic curves

Let X denote a hyperbolic curve over $$\mathbb {Q}$$ and let p denote a prime of good reduction. The third author’s approach to integral points, introduced in Kim (Invent Math 161:629–656, 2005; Publ Res Inst Math Sci 45:89–133, 2009), endows $$X({\mathbb {Z}_p})$$ with a nested sequence of subsets $$X({\mathbb {Z}_p})_n$$ which contain $$X(\mathbb {Z})$$ . These sets have been computed in a range of special cases (Balakrishnan et al., J Am Math Soc 24:281–291, 2011; Dan-Cohen and Wewers, Proc Lond Math Soc 110:133–171, 2015; Dan-Cohen and Wewers, Int Math Res Not IMRN 17:5291–5354, 2016; Kim, J Am Math Soc 23:725–747, 2010); there is good reason to believe them to be practically computable in general. In 2012, the third author announced the conjecture that for n sufficiently large, $$X(\mathbb {Z}) = X({\mathbb {Z}_p})_n$$ . This conjecture may be seen as a sort of compromise between the abelian confines of the BSD conjecture and the profinite world of the Grothendieck section conjecture. After stating the conjecture and explaining its relationship to these other conjectures, we explore a range of special cases in which the new conjecture can be verified.

Distortion for abelian subgroups of Out(Fn)

We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn–Handel. As an application, we prove the rank conjecture for [Formula: see text].

Transformation properties for Dyson’s rank function

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $R(\zeta,q)$, where $R(z,q)$ is the two-variable generating function of Dyson's rank function and $\zeta$ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon and McIntosh, and motivated by Dyson's question, Bringmann, Ono and Rhoades studied transformation properties of $R(\zeta,q)$. In this paper we strengthen and extend the results of Bringmann, Rhoades and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo $5$ from the Lost Notebook has an analogue for all primes greater than $3$. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod $7$.

The Elkies curve has rank 28 subject only to GRH

In 2006, Elkies presented an elliptic curve with 28 independent rational points. We prove that subject to GRH, this curve has Mordell-Weil rank equal to 28 and analytic rank at most 28. We prove similar results for a previously unpublished curve of Elkies having rank 27. We also prove that subject to GRH, certain specific elliptic curves have Mordell-Weil ranks 20, 21, 22, 23, and 24. This complements the work of Jonathan Bober, who proved this claim subject to both the Birch and Swinnerton-Dyer rank conjecture and GRH. This gives some new evidence that the Birch and Swinnerton-Dyer rank conjecture holds for elliptic curves over Q of very high rank. Our results about Mordell-Weil ranks are proven by computing the 2-ranks of class groups of cubic fields associated to these elliptic curves. As a consequence, we also succeed in proving that, subject to GRH, the class group of a particular cubic field has 2-rank equal to 22 and that the class group of a particular totally real cubic field has 2-rank equal to 20.

The Mercat Conjecture for stable rank 2 vector bundles on generic curves

We prove the Mercat Conjecture for rank $2$ vector bundles on generic curves of every genus. For odd genus, we identify the effective divisor in ${\cal M}_g$ where the Mercat Conjecture fails and compute its slope.

Wedge operations and doubling operations of real toric manifolds

This paper deals with two things. First, the cohomology of canonical extensions of real topological toric manifolds is computed when coefficient ring G is a commutative ring in which 2 is unit in G. Second, the author focuses on a specific canonical extensions called doublings and presents their various properties. They include existence of infinitely many real topological toric manifolds admitting complex structures, and a way to construct infinitely many real toric manifolds which have an odd torsion in their cohomology groups. Moreover, some questions about real topological toric manifolds related to Halperin’s toral rank conjecture are presented.

Nuclear norm of higher-order tensors

We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field --- the value of the nuclear norm of a real 3-tensor depends on whether we regard it as a real 3-tensor or a complex 3-tensor with real entries. We show that every tensor has a nuclear norm attaining decomposition and every symmetric tensor has a symmetric nuclear norm attaining decomposition. There is a corresponding notion of nuclear rank that, unlike tensor rank, is upper semicontinuous. We establish an analogue of Banach's theorem for tensor spectral norm and Comon's conjecture for tensor rank --- for a symmetric tensor, its symmetric nuclear norm always equals its nuclear norm. We show that computing tensor nuclear norm is NP-hard in several sense. Deciding weak membership in the nuclear norm unit ball of 3-tensors is NP-hard, as is finding an $\varepsilon$-approximation of nuclear norm for 3-tensors. In addition, the problem of computing spectral or nuclear norm of a 4-tensor is NP-hard, even if we restrict the 4-tensor to be bi-Hermitian, bisymmetric, positive semidefinite, nonnegative valued, or all of the above. We discuss some simple polynomial-time approximation bounds. As an aside, we show that the nuclear $(p,q)$-norm of a matrix is NP-hard in general but can be computed in polynomial-time if $p=1$, $q = 1$, or $p=q=2$, with closed-form expressions for the nuclear $(1,q)$- and $(p,1)$-norms.

Machine learning in the string landscape

We utilize machine learning to study the string landscape. Deep data dives and conjecture generation are proposed as useful frameworks for utilizing machine learning in the landscape, and examples of each are presented. A decision tree accurately predicts the number of weak Fano toric threefolds arising from reflexive polytopes, each of which determines a smooth F-theory compactification, and linear regression generates a previously proven conjecture for the gauge group rank in an ensemble of frac{4}{3}times 2.96times {10}^{755} F-theory compactifications. Logistic regression generates a new conjecture for when E6 arises in the large ensemble of F-theory compactifications, which is then rigorously proven. This result may be relevant for the appearance of visible sectors in the ensemble. Through conjecture generation, machine learning is useful not only for numerics, but also for rigorous results.

Combinatorial and inductive methods for the tropical maximal rank conjecture

We produce new combinatorial methods for approaching the tropical maximal rank conjecture, including inductive procedures for deducing new cases of the conjecture on graphs of increasing genus from any given case. Using explicit calculations in a range of base cases, we prove this conjecture for the canonical divisor, and in a wide range of cases for m=3, extending previous results for m=2.

On $L$-functions of quadratic $\mathbb {Q}$-curves

Let $K$ be a quadratic number field of discriminant $\Delta_K$, let $E$ be a $\mathbb Q$-curve without CM completely defined over $K$ and let $\omega_E$ be an invariant differential on $E$. Let $L(E,s)$ be the $L$-function of $E$. In this setting, it is known that $L(E,s)$ possesses an analytic continuation to $\mathbb C$. The period of $E$ can be written (up to a power of $2$) as the product of the Tamagawa numbers of $E$ with $\Omega_E/\sqrt{|\Delta_K|}$, where $\Omega_E$ is a quantity, independent of $\omega_E$, which encodes the real periods of $E$ when $K$ is real and the covolume of the period lattice of $E$ when $K$ is imaginary. In this paper we compute, under the generalized Manin conjecture, an effective nonzero integer $Q=Q(E,\omega_E)$ such that if $L(E,1)\neq 0$ then $L(E,1)\cdot Q\cdot\sqrt{|\Delta_K|}/\Omega_E$ is an integer. Computing $L(E,1)$ up to sufficiently high precision, our result allows us to prove that $L(E,1)=0$ whenever this is the case and to compute the $L$-ratio $L(E,1)\cdot\sqrt{|\Delta_K|}/\Omega_E$ when $L(E,1)\neq 0$. An important ingredient is an algorithm to compute a newform $f$ of weight $2$ level $\Gamma_1(N)$ such that $L(E,s)=L(f,s)\cdot L({}^{\sigma\!} f,s)$, for ${}^{\sigma\!} f$ the unique Galois conjugate of $f$. As an application of these results, we verify the validity of the weak BSD conjecture for some $\mathbb Q$-curves of rank $2$ and we will compute the $L$-ratio of a curve of rank $0$.