Anticyclotomic P-adic L-functions Research Articles

An explicit comparison of anticyclotomic p-adic L-functions for Hida families

We provide comparison results for anticyclotomic [Formula: see text]-adic [Formula: see text]-functions attached to Hida families of modular forms. The main result is a comparison between the anticyclotomic restriction of the three variable [Formula: see text]-adic [Formula: see text]-function introduced by Skinner and Urban, and the anticyclotomic [Formula: see text]-function constructed by means of [Formula: see text]-adic families of Gross points in the setting of definite quaternion algebras. Several auxiliary results involving anticyclotomic [Formula: see text]-adic [Formula: see text]-functions introduced by Chida-Hsieh and Büyükboduk–Lei are also proved.

Plectic Stark–Heegner points

We propose a conjectural construction of determinants of global points on modular elliptic curves over arbitrary number fields, generalizing both the p-adic construction of Heegner points via Čerednik–Drinfeld uniformization and the definition of classical Stark–Heegner points. In alignment with Nekovář and Scholl's plectic conjectures, we expect the non-triviality of these plectic Stark–Heegner points to control the Mordell–Weil group of higher rank elliptic curves. We provide some indirect evidence for our conjectures by showing that higher order derivatives of anticyclotomic p-adic L-functions compute plectic invariants.

Generalized Heegner cycles on Mumford curves

We study generalised Heegner cycles, originally introduced by Bertolini–Darmon–Prasanna for modular curves in Bertolini et al. (Duke Math J 162(6):1033–1148, 2013), in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic p-adic L-function attached to a Coleman family $$f_\infty $$ and an imaginary quadratic field K, constructed in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014). While in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014) only the restriction to the central critical line of this 2 variable p-adic L-function is considered, our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic p-adic L-function restricted to non necessarily central critical lines as a combination of the image of generalized Heegner cycles under a p-adic Abel–Jacobi map. In studying generalised Heegner cycles in the context of Mumford curves, we also obtain an extension of a result of Masdeu (Compos Math 148(4):1003–1032, 2012) for the (one variable) anticyclotomic p-adic L-function of a modular form f and K at non-central critical integers.

Anticyclotomic p-adic L-functions and Ichino’s formula

We give a new construction of a $p$-adic $L$-function $\mathcal{L}(f,\Xi)$, for $f$ a holomorphic newform and $\Xi$ an anticyclotomic family of Hecke characters of $\mathbb{Q}(\sqrt{-d})$. The construction uses Ichino's triple product formula to express the central values of $L(f,\xi,s)$ in terms of Petersson inner products, and then uses results of Hida to interpolate them. The resulting construction is well-suited for studying what happens when $f$ is replaced by a modular form congruent to it modulo $p$, and has future applications in the case where $f$ is residually reducible.

Anticyclotomic 𝑝-adic 𝐿-functions and the exceptional zero phenomenon

Let A A be a modular elliptic curve over a totally real field F F , and let K / F K/F be a totally imaginary quadratic extension. In the event of exceptional zero phenomenon, we prove a formula for the derivative of the multivariable anticyclotomic p p -adic L L -function attached to ( A , K ) (A,K) , in terms of the Hasse-Weil L L -function and certain p p -adic periods attached to the respective automorphic forms. Our methods are based on a new construction of the anticyclotomic p p -adic L L -function by means of the corresponding automorphic representation.

Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes

Let E be a rational elliptic curve and let p be an odd prime of additive reduction. Let K be an imaginary quadratic field and fix a positive integer c prime to the conductor of E. The main goal of the present article is to define an anticyclotomic p-adic L-function L attached to E/K when E/Qp attains semistable reduction over an abelian extension. We prove that L satisfies the expected interpolation properties; namely, we show that if χ is an anticyclotomic character of conductor cpn, then χ(L) is equal (up to explicit constants) to L(E,χ,1) or L′(E,χ,1).

Exceptional zero formulae for anticyclotomic p-adic L-functions of elliptic curves in the ramified case

Iwasawa theory of modular forms over anticyclotomic Zp-extensions of imaginary quadratic fields K has been studied by several authors, starting from the works of Bertolini–Darmon and Iovita–Spiess, under the crucial assumption that the prime p is unramified in K. We start in this article the systematic study of anticyclotomic p-adic L-functions when p is ramified in K. In particular, when f is a weight 2 modular form attached to an elliptic curve E/Q having multiplicative reduction at p, and p is ramified in K, we show an analogue of the exceptional zeroes phenomenon investigated by Bertolini–Darmon in the setting when p is inert in K. More precisely, we consider situations in which the p-adic L-function Lp(E/K) of E over the anticyclotomic Zp-extension of K does not vanish identically but, by sign reasons, has a zero at certain characters χ of the Hilbert class field of K. In this case we show that the value at χ of the first derivative of Lp(E/K) is equal to the formal group logarithm of the specialization at p of a global point on the elliptic curve (actually, this global point is a twisted sum of Heegner points). This generalizes similar results of Bertolini–Darmon, available when p is inert in K and χ is the trivial character.

On the derivative of anticyclotomic p-adic L-functions for Hilbert modular forms

We construct anticyclotomic [Formula: see text]-adic [Formula: see text]-functions for ordinary Hilbert modular forms and a formula relating the first derivative of this [Formula: see text]-adic [Formula: see text]-function at the central point and Teieltbaum’s [Formula: see text]-invariant defined by Chida, Mok and Park. Our results extend previous works of Bertolini, Darmon, Iovita and Speiss to Hilbert modular forms of parallel weight [Formula: see text].

A p-adic Waldspurger formula

In this article, we study p-adic torus periods for certain p-adic-valued functions on Shimura curves of classical origin. We prove a p-adic Waldspurger formula for these periods as a generalization of recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic p-adic L-function of Rankin–Selberg type. At a character of positive weight, the p-adic L-function interpolates the central critical value of the complex Rankin–Selberg L-function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding p-adic torus period.

On the non-vanishing mod ℓ of central L-values with anticyclotomic twists for Hilbert modular forms

In this article, we generalize some works of Masataka Chida and Ming-Lun Hsieh on anticyclotomic p-adic L-functions attached to modular forms to Hilbert case. We construct a class of anticyclotomic p-adic L-functions for ordinary Hilbert modular forms and derive the interpolation formula at all critical specializations. Moreover, we prove results on the vanishing of μ-invariant of these p-adic L-functions and the non-vanishing of central L-values with anticyclotomic twists.

Special values of anticyclotomic L-functions for modular forms

Abstract In this article, we generalize some works of Bertolini–Darmon and Vatsal on anticyclotomic L-functions attached to modular forms of weight two to higher weight case. We construct a class of anticyclotomic p-adic L-functions for ordinary modular forms and derive the functional equation and the interpolation formula at all critical specializations. Moreover, we prove results on the vanishing of μ-invariant of these p-adic L-functions and the non-vanishing of central L-values with anticyclotomic twists.

Anticyclotomic p-adic L-function of Central Critical Rankin-Selberg L-value

Let M be an imaginary quadratic field, f a Hecke eigenform on GL2(Q) and \pi the unitary base-change to M of the automorphic representation associated to f. Take a unitary arithmetic Hecke character \chi of M inducing the inverse of the central character of f. The celebrated formula of Waldspurger relates the square of an integral L(f) of f and \chi over the idele class group of M to the central critical value L(1/2,{\pi}\otimes{\chi}). In this paper, we present a construction of a new p-adic L-function that interpolates L(f) over arithmetic characters for a cusp form f in the spirit of the landmark result of Katz where he did that for Eisenstein series.

A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing

Let E be an elliptic curve over Q attached to a newform f of weight 2 on Γ 0 (N) , and let K be a real quadratic field in which all the primes dividing N are split. This paper relates the canonical ℝ/ℤ -valued "circle pairing" on E(K) defined by Mazur and Tate [MT1] to a period integral I ′ (f,k) defined in terms of f and k . The resulting conjecture can be viewed as an analogue of the classical Birch and Swinnerton-Dyer conjecture, in which I ′ (f,k) replaces the derivative of the complex L -series L(f,K,s) and the circle pairing replaces the Néron-Tate height. It emerges naturally as an archimedean fragment of the theory of anticyclotomic p-adic L-functions developed in [BD], and has been tested numerically in a variety of situations. The last section formulates a conjectural variant of a formula of Gross, Kohnen, and Zagier [GKZ] for the Mazur-Tate circle pairing.

Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms

In this paper we prove a Gross-Zagier type formula for the anticyclotomic p-adic L-function of an elliptic modular form f of higher weight and of multiplicative type at p. For such f we also decribe explicitely the local Galois representation attached to it at p and compare the Fontaine-Mazur and Teitelbaum L-invariants.