Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions

Let A be an abelian variety over a number field k and F a finite cyclic extension of k of p-power degree for an odd prime p. Under certain technical hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number conjecture (eTNC) for A, F/k and p as an explicit family of p-adic congru- ences involving values of derivatives of the Hasse-Weil L-functions of twists of A, normalised by completely explicit twisted regulators. This reinterpretation makes the eTNC amenable to numerical verification and furthermore leads to explicit predictions which refine well-known conjectures of Mazur and Tate

On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair (h1(A/F)(1),Z[Gal(F/k)]). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural p-adic congruences between the values at s = 1 of derivatives of the Hasse–Weil L-functions associated to twists of A, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions

Equivariant Birch-Swinnerton-Dyer conjecture for the base change of elliptic curves: An example

Let E be an elliptic curved defined over \Q and let K/\Q be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for h^1(E\times_{\Q} K)(1) viewed as a motive over \Q with coefficients in \Q[G] relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper we prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S_3-extension of \Q.Comment: 21 page

Mock plectic points

A $p$-arithmetic subgroup of $\mathrm{SL}_2(\mathbb{Q})$ like the Ihara group $\Gamma := \mathrm{SL}_2(\mathbb{Z}[1/p])$ acts by M\"obius transformations on the Poincar\'e upper half plane $\mathcal{H}$ and on Drinfeld's $p$-adic upper half plane $\mathcal{H}_p := \mathbb{P}_1(\mathbb{C}_p)\setminus\mathbb{P}_1(\mathbb{Q}_p)$. The diagonal action of $\Gamma$ on the product is discrete, and the quotient $\Gamma\backslash(\mathcal{H}_p\times \mathcal{H})$ can be envisaged as a "mock Hilbert modular surface". According to a striking prediction of Nekov\'a$\check{\text{r}}$ and Scholl, the CM points on genuine Hilbert modular surfaces should give rise to "plectic Heegner points" that encode non-trivial regulators attached, notably, to elliptic curves of rank two over real quadratic fields. This article develops the analogy between Hilbert modular surfaces and their mock counterparts, with the aim of transposing the plectic philosophy to the mock Hilbert setting, where the analogous plectic invariants are expected to lie in the alternating square of the Mordell-Weil group of certain elliptic curves of rank two over $\mathbb{Q}$

Elliptic curves of rank two and generalized Kato classes

Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts for their main limitation, namely that they are torsion in (analytic) rank >1. This partly expository article discusses the generalised Kato classes introduced in Bertolini et al. (J Algebr Geom 24:569–604, 2015) and Darmon and Rotger (J AMS 2016), stressing their analogy with Heegner points but explaining why they are expected to give non-trivial, canonical elements of the idoneous Selmer group in settings where the classical L-function (of Hasse–Weil–Artin type) that governs their behaviour has a double zero at the centre. The generalised Kato class denoted ¿(f,g,h) is associated to a triple (f, g, h) consisting of an eigenform f of weight two and classical p-stabilised eigenforms g and h of weight one, corresponding to odd two-dimensional Artin representations Vg and Vh of Gal(H/Q) with p-adic coefficients for a suitable number field H. This class is germane to the Birch and Swinnerton-Dyer conjecture over H for the modular abelian variety E over Q attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that ¿(f,g,h) lies in the pro-p Selmer group of E over H precisely when L(E,Vgh,1)=0, where L(E,Vgh,s) is the L-function of E twisted by Vgh:=Vg¿Vh. In the setting of interest, parity considerations imply that L(E,Vgh,s) vanishes to even order at s=1, and the Selmer class ¿(f,g,h) is expected to be trivial when ords=1L(E,Vgh,s)>2. The main new contribution of this article is a conjecture expressing ¿(f,g,h) as a canonical point in (E(H)¿Vgh)GQ when ords=1L(E,Vgh,s)=2. This conjecture strengthens and refines the main conjecture of Darmon et al. (Forum Math Pi 3:e8, 2015) and supplies a framework for understanding the results of Darmon et al. (2015), Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016).Peer ReviewedPostprint (published version

On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair (h1(A/F)(1),Z[Gal(F/k)]). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural p-adic congruences between the values at s = 1 of derivatives of the Hasse–Weil L-functions associated to twists of A, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions

Torsion homology growth and cycle complexity of arithmetic manifolds

Let M be an arithmetic hyperbolic 3-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of M, where each basis element is represented by a surface of ‘low’ genus, and give evidence for this. We explain the relationship between this conjecture and the study of torsion homology growth

The Cassels-Tate pairing on polarized abelian varieties

Let (A,\lambda) be a principally polarized abelian variety defined over a global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd \rightarrow \Q/\Z. If A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on \Sha(A)_\nd. These criteria are expressed in terms of an element c \in \Sha(A)_\nd that is canonically associated to the polarization \lambda. In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine effectively whether \#\Sha(A) (if finite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperelliptic curves of even genus g \ge 2 over \Q have a Jacobian with nonsquare \#\Sha (if finite). For example, it appears that this density is about 13% for curves of genus 2. The proof makes use of a general result relating global and local densities; this result can be applied in other situations.Comment: 41 pages, published versio