Analytic Rank Research Articles

On Shafarevich-Tate groups and analytic ranks in families of modular forms, I. Hida families

Let $f$ be a newform of weight $2$, square-free level and trivial character, let $A_f$ be the abelian variety attached to $f$ and for every good ordinary prime $p$ for $f$ let $\boldsymbol f^{(p)}$ be the $p$-adic Hida family through $f$. We prove that, for all but finitely many primes $p$ as above, if $A_f$ is an elliptic curve such that $A_f(\mathbb Q)$ has rank $1$ and the $p$-primary part of the Shafarevich-Tate group of $A_f$ over $\mathbb Q$ is finite then all specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have finite ($p$-primary) Shafarevich-Tate group and $1$-dimensional image of the relevant $p$-adic \'etale Abel-Jacobi map. Analogous results are obtained also in the rank $0$ case. As a second contribution, with no restriction on the dimension of $A_f$ but assuming the non-degeneracy of certain height pairings \`a la Gillet-Soul\'e between Heegner cycles, we show that if $f$ has analytic rank $1$ then, for all but finitely many $p$, all specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have analytic rank $1$. This result provides some evidence in rank $1$ and weight larger than $2$ for a conjecture of Greenberg predicting that the analytic ranks of even weight modular forms in a Hida family should be as small as allowed by the functional equation, with at most finitely many exceptions.

Nonvanishing of L-function of some Hecke characters on cyclotomic fields

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the p-th Fermat curve where 2 is a quadratic residue modulo p, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.

Predicting root numbers with neural networks

In this paper, we report on two machine learning experiments in search of statistical relationships between Dirichlet coefficients and root numbers or analytic ranks of certain low-degree [Formula: see text]-functions. The first experiment is to construct interpretable models based on murmurations, a recently discovered correlation between Dirichlet coefficients and root numbers. We show experimentally that these models achieve high accuracy by learning a combination of Mestre–Nagao-type heuristics and murmurations, noting that the relative importance of these features varies with degree. The second experiment is to search for a low-complexity statistic of Dirichlet coefficients that can be used to predict root numbers in polynomial time. We give experimental evidence and provide heuristics that suggest this cannot be done with standard machine learning techniques.

A higher Gross–Zagier formula and the structure of Selmer groups

We describe a Kolyvagin system-theoretic refinement of Gross–Zagier formula by comparing Heegner point Kolyvagin systems with Kurihara numbers when the root number of a rational elliptic curve E E over an imaginary quadratic field K K is − 1 -1 . When the root number of E E over K K is 1, we first establish the structure theorem of the p ∞ p^\infty -Selmer group of E E over K K . The description is given by the values of certain families of quaternionic automorphic forms, which is a part of bipartite Euler systems. By comparing bipartite Euler systems with Kurihara numbers, we also obtain an analogous refinement of Waldspurger formula. No low analytic rank assumption is imposed in both refinements. We also prove the equivalence between the non-triviality of various “Kolyvagin systems” and the corresponding main conjecture localized at the augmentation ideal. As consequences, we obtain new applications of (weaker versions of) the Heegner point main conjecture and the anticyclotomic main conjecture to the structure of p ∞ p^\infty -Selmer groups of elliptic curves of arbitrary rank. In particular, the Heegner point main conjecture localized at the augmentation ideal implies the strong rank one p p -converse to the theorem of Gross–Zagier and Kolyvagin.

On Shafarevich-Tate groups and analytic ranks in families of modular forms, II. Coleman families

On Shafarevich-Tate groups and analytic ranks in families of modular forms, II. Coleman families

Analytic ranks of automorphic L$L$‐functions and Landau–Siegel zeros

AbstractWe relate the study of Landau–Siegel zeros to the ranks of Jacobians of modular curves for large primes . By a conjecture of Brumer–Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level have analytic rank . We show that either Landau–Siegel zeros do not exist, or that, for wide ranges of , almost all such newforms have analytic rank . In particular, in wide ranges, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes in a wide range, we show that the rank of is asymptotically equal to the rank predicted by the Brumer–Murty conjecture.

L-functions of elliptic curves modulo integers

In 1985, Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes (see [9]). If E/K is an elliptic curve with nonconstant j-invariant defined over a function field K of characteristic p≥5, we know that its L-function L(T,E/K) is a polynomial in Z[T] (see [5, p. 11]). Inspired by Schoof, we study the reduction of L(T,E/K) modulo integers. We obtain three main results. Firstly, if E/K has non-trivial K-rational N-torsion for some positive integer N coprime with p, we extend a formula for L(T,E/K)modN due to Hall (see [4, p. 133, Theorem 4]) to all quadratic twists Ef/K with f∈K×∖K×2. Secondly, without any condition on the 2-torsion subgroup of E(K), we give a formula for the quotient modulo 2 of L-functions of any two quadratic twists of E/K. Thirdly, we use these results to compute the global root numbers of an infinite family of quadratic twists of an elliptic curve and, under some assumption, find in most cases the exact analytic rank of these twists. We also illustrate that in favourable situations, our second main result allows one to compute much more efficiently L(T,Ef/K)mod2 than an algorithm of Baig and Hall (see [1]). Finally, we use our formulas to compute some degree 2 L-functions directly.

On the distribution of analytic ranks of elliptic curves

On the distribution of analytic ranks of elliptic curves

Partition and analytic rank are equivalent over large fields

Partition and analytic rank are equivalent over large fields

THE DIAGONAL CYCLE EULER SYSTEM FOR

Abstract We construct an anticyclotomic Euler system for the Rankin–Selberg convolutions of two modular forms, using p-adic families of generalised Gross–Kudla–Schoen diagonal cycles. As applications of this construction, we prove new results on the Bloch–Kato conjecture in analytic ranks zero and one, and a divisibility towards an Iwasawa main conjecture.

The Second Moment Theory of Families of 𝐿-Functions–The Case of Twisted Hecke 𝐿-Functions

For a fairly general family of L L -functions, we survey the known consequences of the existence of asymptotic formulas with power-saving error term for the (twisted) first and second moments of the central values in the family. We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime q q , and prove that it satisfies such formulas. We derive arithmetic consequences: a positive proportion of central values L ( f ⊗ χ , 1 / 2 ) L(f\otimes \chi ,1/2) are non-zero, and indeed bounded from below; there exist many characters χ \chi for which the central L L -value is very large; the probability of a large analytic rank decays exponentially fast. We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols.

Analytic ranks of elliptic curves over number fields

Let E E be an elliptic curve over Q \mathbb {Q} . Then, we show that the average analytic rank of E E over cyclic extensions of degree l l over Q \mathbb {Q} with l l a prime not equal to 2 2 , is at most 2 + r Q ( E ) 2+r_\mathbb {Q}(E) , where r Q ( E ) r_\mathbb {Q}(E) is the analytic rank of the elliptic curve E E over Q \mathbb {Q} . This bound is independent of the degree l l . Using a recent result of Bhargava, Taniguchi and Thorne [Improved error estimates for the Davenport–Heilbronn theorems, arxiv.org/abs/2107.12819, 2021], we obtain a non-trivial upper bound on the average analytic rank of E E over S 3 S_3 -fields.

On L-derivatives and biextensions of Calabi–Yau motives

Abstract We prove that certain differential operators of the form $ DLD $ with $ L $ hypergeometric and $ D=z\frac{\partial }{dz} $ are of Picard–Fuchs type. We give closed hypergeometric expressions for minors of the biextension period matrices that arise from certain rank 4 weight 3 Calabi–Yau motives presumed to be of analytic rank 1. We compare their values numerically to the first derivative of the $ L $ -functions of the respective motives at $ s=2 $ .

The average analytic rank of elliptic curves with prescribed torsion

We show that average analytic rank of elliptic curves with prescribed torsion $G$ is bounded for every torsion group $G$ under GRH for elliptic curve $L$-functions.

A divisibility related to the Birch and Swinnerton-Dyer conjecture

Let E/Q be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero that the order of the torsion subgroup of E/Q divides the product of the order of the Shafarevich–Tate group of E/Q, the (global) Tamagawa number of E/Q, and the Tamagawa number of E/Q at infinity. This consequence of the Birch and Swinnerton-Dyer conjecture was noticed by Agashe and Stein in 2005. In this paper, we prove this divisibility statement unconditionally in many cases, including the case where the curve E/Q is semi-stable.

The universal p-adic Gross–Zagier formula

Let \(\mathrm {G}\) be the group \( (\mathrm {GL}_{2}\times \mathrm{GU}(1))/\mathrm {GL}_{1}\) over a totally real field F, and let \(\mathscr {X}\) be a Hida family for \(\mathrm {G}\). Revisiting a construction of Howard and Fouquet, we construct an explicit section \(\mathscr {P}\) of a sheaf of Selmer groups over \(\mathscr {X}\). We show, answering a question of Howard, that \(\mathscr {P}\) is a universal Heegner class, in the sense that it interpolates geometrically defined Heegner classes at all the relevant classical points of \(\mathscr {X}\). We also propose a ‘Bertolini–Darmon’ conjecture for the leading term of \(\mathscr {P}\) at classical points. We then prove that the p-adic height of \(\mathscr {P}\) is given by the cyclotomic derivative of a p-adic L-function. This formula over \(\mathscr {X}\) (which is an identity of functionals on some universal ordinary automorphic representations) specialises at classical points to all the Gross–Zagier formulas for \(\mathrm {G}\) that may be expected from representation-theoretic considerations. Combined with a result of Fouquet, the formula implies the p-adic analogue of the Beilinson–Bloch–Kato conjecture in analytic rank one, for the selfdual motives attached to Hilbert modular forms and their twists by CM Hecke characters. It also implies one half of the first example of a non-abelian Iwasawa main conjecture for derivatives, in \(2[F:\mathbf {Q}]\) variables. Other applications include two different generic non-vanishing results for Heegner classes and p-adic heights.

On the Bloch–Kato conjecture for the symmetric cube

We prove one inclusion in the Iwasawa main conjecture, and the Bloch–Kato conjecture in analytic rank 0, for the symmetric cube of a level 1 modular form.

One-level density of the family of twists of an elliptic curve over function fields

We fix an elliptic curve E/Fq(t) and consider the family {E⊗χD} of E twisted by quadratic Dirichlet characters. The one-level density of their L-functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside (−1,1). As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least 12.5% of the family have rank zero, and at least 37.5% have rank one. The Katz and Sarnak philosophy predicts that those percentages should both be 50% and that the average analytic rank should be 1/2. We finish by computing the one-level density of E twisted by Dirichlet characters of order ℓ≠2 coprime to q. We obtain a restriction of (−1/2,1/2) on the support with a unitary symmetry.

Average rank of quadratic twists with a rational point of almost minimal height

Given a family of quadratic twists of a fixed elliptic curve defined over Q $\mathbb {Q}$ , we investigate the average rank in the subfamily of twists having a non-torsion rational point of almost minimal height. We show in particular that the average analytic rank is greater than one.

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

AbstractWe discuss the$\ell $-adic case of Mazur’s ‘Program B’ over$\mathbb {Q}$: the problem of classifying the possible images of$\ell $-adic Galois representations attached to elliptic curvesEover$\mathbb {Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes$\ell =2$and$\ell \ge 13$are addressed by prior work, so we focus on the remaining primes$\ell = 3, 5, 7, 11$. For each of these$\ell $, we compute the directed graph of arithmetically maximal$\ell $-power level modular curves$X_H$, compute explicit equations for all but three of them and classify the rational points on all of them except$X_{\mathrm {ns}}^{+}(N)$, for$N = 27, 25, 49, 121$and two-level$49$curves of genus$9$whose Jacobians have analytic rank$9$.Aside from the$\ell $-adic images that are known to arise for infinitely many${\overline {\mathbb {Q}}}$-isomorphism classes of elliptic curves$E/\mathbb {Q}$, we find only 22 exceptional images that arise for any prime$\ell $and any$E/\mathbb {Q}$without complex multiplication; these exceptional images are realised by 20 non-CM rationalj-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on$X_{\mathrm {ns}}^+(\ell )$with$\ell \ge 19$, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the$\ell $-adic images of Galois for any elliptic curve over$\mathbb {Q}$.In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on$\Gamma _1(N)$are of$\operatorname {GL}_2$-type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of$X_H$.