Exceptional Zero Research Articles

Derivatives of Beilinson–Flach classes, Gross–Stark formulas and a $p$-adic Harris–Venkatesh conjecture

We propose an alternative approach to the study of exceptional zeros from the point of view of Euler systems. As a first application, we give a new proof of a conjecture of Darmon, Lauder and Rotger regarding the computation of the \mathcal{L} -invariant of the adjoint of a weight one modular form in terms of units and p -units. While in our previous work with Rotger the essential ingredient was the use of Galois deformations techniques, we discuss a new method exclusively using the properties of Beilinson–Flach classes. One of the key ingredients is the computation of a cyclotomic derivative of a cohomology class in the framework of Perrin-Riou theory, which can be seen as a counterpart to the earlier work of Loeffler, Venjakob, and Zerbes. In our second application, we illustrate how these techniques could lead to a better understanding of this setting by introducing a new motivic p -adic L -function whose special values encode information just about the unit of the adjoint (and not also the p -unit), in the spirit of the conjectures of Harris and Venkatesh. We further discuss conjectural connections with the arithmetic of triple products of Coleman families.

Sifting for small primes from an arithmetic progression

In this work and its sister paper (Friedlander and Iwaniec (2023)), we give a new proof of the famous Linnik theorem bounding the least prime in an arithmetic progression. Using sieve machinery in both papers, we are able to dispense with the log-free zero density bounds and the repulsion property of exceptional zeros, two deep innovations begun by Linnik and relied on in earlier proofs.

On the exceptional zeros of 𝑝-non-ordinary 𝑝-adic 𝐿-functions and a conjecture of Perrin-Riou

Our goal in this article is to prove a form of p p -adic Birch and Swinnerton-Dyer formula for the second derivative of the p p -adic L L -function associated to a newform f f which is non-crystalline semistable at p p at its central critical point, by expressing this quantity in terms of a p p -adic (cyclotomic) regulator defined on an extended trianguline Selmer group. We also prove a two-variable version of this result for height pairings we construct by considering infinitesimal deformations afforded by a Coleman family passing through f f . This, among other things, leads us to a proof of an appropriate version of Perrin-Riou’s conjecture in this set up.

Convergence Rates of Exceptional Zeros of Exceptional Orthogonal Polynomials

We consider the zeros of exceptional orthogonal polynomials (XOP). Exceptional orthogonal polynomials were originally discovered as eigenfunctions of second order differential operators that exist outside the classical Bochner–Brenke classification due to the fact that XOP sequences omit polynomials of certain degrees. This omission causes several properties of the classical orthogonal polynomial sequences to not extend to the XOP sequences. One such property is the restriction of the zeros to the convex hull of the support of the measure of orthogonality. In the XOP case, the zeros that exist outside the classical intervals are called exceptional zeros and they often converge toward easily identifiable limit points as the degree becomes large. We deduce the exact rate of convergence and verify that certain estimates that previously appeared in the literature are sharp.

On the non-critical exceptional zeros of Katz p-adic L-functions for CM fields

The primary goal of this article is to study p-adic Beilinson conjectures in the presence of exceptional zeros for Artin motives over CM fields. In more precise terms, we address a question raised by Hida and Tilouine on the order of vanishing of Katz p-adic L-functions associated to CM fields, by means of a leading term formula we prove in terms of Rubin–Stark elements. In the particular case when the CM field in question is imaginary quadratic, our leading term formula and its consequences we record herein are unconditional.

Primes in the Chebotarev density theorem for all number fields (with an Appendix by Andrew Fiori)

Let L/K be any Galois extension of number fields such that L≠Q, and let C be a conjugacy class in Gal(L/K). We show that there exists a degree-one unramified prime p of K such that σp=C and Np≤dLB with B=310. This improves upon B=12577 obtained by Ahn and Kwon by making the argument of Lagarias, Montgomery, and Odlyzko explicit. Our improvements come from using the following key ideas: new weights, an optimized analysis on the location of the potential exceptional zeros for ζL(s), and a new version of Turán's power sum method which gives a stronger Deuring-Heilbronn phenomenon for ζL(s). We also use Fiori's numerical verification for number fields up to a certain discriminant height. Other results include a lower bound for the number of unramified primes p of K such that σp=C.

On combinatorial properties and the zero distribution of certain Sheffer sequences

We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form G(x,z)=Q(z)xQ(−z)1−x, where Q is a quadratic polynomial with real zeros. By using the properties of Riordan matrices we address combinatorial properties and interpretations of our Sheffer sequence of polynomials and their coefficients. We also show that apart from two exceptional zeros, the zeros of polynomials with large enough degree in such a Sheffer sequence lie on the line x=1/2+it.

Generalized Kato classes and exceptional zero conjectures

Generalized Kato classes and exceptional zero conjectures

ℒ-Invariants, p-Adic Heights, and Factorization of p-Adic L-Functions

Abstract We continue with our study of the noncritical exceptional zeros of Katz’ $p$-adic $L$-functions attached to a CM field $K$, following two threads. In the 1st thread, we redefine our (group-ring-valued) ${\mathcal {L}}$-invariant associated with each ${\mathbb {Z}}_p$-extension $K_\Gamma $ of $K$ in terms of $p$-adic height pairings and interpolate them as $K_\Gamma $ varies to a universal (multivariate) group-ring-valued ${\mathcal {L}}$-invariant. In the 2nd thread, we use our results to study the exceptional zeros of the Rankin–Selberg $p$-adic $L$-functions at noncritical specializations of the self-products of nearly ordinary CM families, via the factorization statements we establish. The factorization theorems are extensions of the results due to Greenberg and Palvannan.

Exceptional zeros and the Goldbach problem

We show that the assumption of a weak form of the Hardy-Littlewood conjecture on the Goldbach problem suffices to disprove the possible existence of exceptional zeros of Dirichlet L-functions.

On exceptional zeros of Garrett–Hida p-adic L-functions

This article proves a case of the p-adic Birch and Swinnerton-Dyer conjecture for Garrett p-adic L-functions of [6], in the exceptional zero setting of extended analytic rank 2.

Heegner Points and Exceptional Zeros of Garrett p-Adic L-Functions

This article proves a case of the p-adic Birch and Swinnerton–Dyer conjecture for Garrett p-adic L-functions of (Bertolini et al. in On p-adic analogues of the Birch and Swinnerton–Dyer conjecture for Garrett L-functions, 2021), in the imaginary dihedral exceptional zero setting of extended analytic rank 4.

Derived Beilinson–Flach elements and the arithmetic of the adjoint of a modular form

Kings, Lei, Loeffler and Zerbes constructed in \[LLZ], \[KLZ1] a three-variable Euler system $\kappa(\textbf{g},\textbf{h})$ of Beilinson–Flach elements associated to a pair of Hida families $(\textbf{g},\textbf{h})$ and exploited it to obtain applications to the arithmetic of elliptic curves, extending the earlier work \[BDR]. The aim of this article is to show that this Euler system also encodes arithmetic information concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when $\textbf{g}$ and $\textbf{h}$ specialize in weight 1 to $p$-stabilizations of eigenforms such that one is dual to the other. We encounter an exceptional zero phenomenon which forces the specialization of $\kappa(\textbf{g},\textbf{h})$ to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of the main conjecture of \[DLR4] on iterated integrals and the main conjecture of \[DR1] for Beilinson–Flach elements in the adjoint setting. The main point of this paper is that the methods of \[DLR1], \[DLR4] and \[CH], where the above conjectures are proved when the weight 1 eigenforms have CM, do not apply to our setting and new ideas are required. In the previous works, a crucial ingredient is a factorization of $p$-adic $L$-functions, which in our scenario is not available due to the lack of critical points. Instead we resort to the principle of improved Euler systems and $p$-adic $L$-functions to reduce our problems to questions which can be resolved using Galois deformation theory. We expect this approach may be adapted to prove other cases of the Elliptic Stark Conjecture and of its generalizations that appear in the literature.

Hida families and p-adic triple product L-functions

We construct the three-variable $p$-adic triple product $L$-functions attached to Hida families of elliptic newforms and prove the explicit interpolation formulae at all critical specializations by establishing explicit Ichino's formulae for the trilinear period integrals of automorphic forms. Our formulae perfectly fit the conjectural shape of $p$-adic $L$-functions predicted by Coates and Perrin-Riou. As an application, we prove the factorization of certain unbalanced $p$-adic triple product $L$-functions into a product of anticyclotomic $p$-adic $L$-functions for modular forms. By this factorization, we obtain a construction of the square root of the anticyclotomic $p$-adic $L$-functions for elliptic curves in the definite case via the diagonal cycle Euler system \`a la Darmon and Rotger and obtain a Greenberg-Stevens style proof of anticyclotomic exceptional zero conjecture for elliptic curves due to Bertolini and Darmon.

The exceptional zero phenomenon for elliptic units

The exceptional zero phenomenon has been widely studied in the realm of p -adic L -functions, where the starting point lies in the foundational works of Ferrero–Greenberg, Katz, Gross, or Mazur–Tate–Teitelbaum, among others. This phenomenon also appears in the study of Euler systems, and in this case one is led to study higher order derivatives of cohomology classes in order to extract the arithmetic information which is usually encoded in the explicit reciprocity laws which make the connection with p -adic L -funtions. In this work, we focus on the elliptic units of an imaginary quadratic field and study this exceptional zero phenomenon, proving an explicit formula relating the logarithm of a derived elliptic unit either to special values of Katz’s two variable p -adic L -function or to its derivatives. Further, we interpret this fact in terms of an \mathcal L -invariant and relate this result to other approaches to the exceptional zero phenomenon, most notably to the work of Bley, and we also compare this setting with other scenarios concerning Heegner points and Beilinson–Flach elements.

On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields

We formulate a multivariable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curves A over number fields K. It generalizes the one-variable conjecture of Mazur, Tate, and Teitelbaum, who studied the case K=Q and the phenomenon of exceptional zeros. We discuss old and new theoretical evidence toward our conjecture and in particular we fully prove it, under mild conditions, in the following situation: K is imaginary quadratic, A=EK is the base change to K of an elliptic curve over the rationals, and the rank of A is either 0 or 1. The proof is naturally divided into a few cases. Some of them are deduced from the purely cyclotomic case of elliptic curves over Q, which we obtain from a refinement of recent work of Venerucci alongside the results of Greenberg, Stevens, Perrin-Riou, and the author. The only genuinely multivariable case (rank 1, two exceptional zeros, three partial derivatives) is newly established here. Its proof generalizes to show that the “almost-anticyclotomic” case of our conjecture is a consequence of conjectures of Bertolini and Darmon on families of Heegner points, and of (partly conjectural) p-adic Gross–Zagier and Waldspurger formulas in families.

An effective Chebotarev density theorem for families of number fields, with an application to $$\ell $$-torsion in class groups

We prove a new effective Chebotarev density theorem for Galois extensions $$L/\mathbb {Q}$$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of L); this theorem holds for the Galois closures of “almost all” number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of L, without assuming GRH. We give many different “appropriate families,” including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidalL-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidalL-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of L-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $$\ell $$-torsion in class groups, for all integers $$\ell \ge 1$$, applicable to infinite families of fields of arbitrarily large degree.

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.

Exceptional zeros and $\mathcal {L}$-invariants of Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime $\mathfrak{p}$ of F above p we prove the existence of an L-invariant $\mathcal{L}_{\mathfrak{p}}$, depending only on $\mathfrak{p}$ and f, such that when the p-adic L-function of f has an exceptional zero at $\mathfrak{p}$, its derivative can be related to the classical L-value multiplied by $\mathcal{L}_{\mathfrak{p}}$. The proof uses cohomological methods of Darmon and Orton, who proved similar results for GL(2) over the rationals. When p is not split and f is the base-change of a classical modular form F, we relate $\mathcal{L}_{\mathfrak{p}}$ to the L-invariant of F, resolving a conjecture of Trifkovi\'{c} in this case.

Functoriality of Automorphic $\mathrm{L}$-Invariants and Applications

We study the behaviour of automorphic \mathrm{L} -invariants associated to cuspidal representations of \mathrm{GL}(2) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard non-vanishing hypothesis on automorphic \mathrm{L} -functions and some technical restrictions on the automorphic representation and the base field we get a simple proof of the equality of automorphic and arithmetic \mathrm{L} -invariants. This together with Spieß' results on p -adic \mathrm{L} -functions yields a new proof of the exceptional zero conjecture for modular elliptic curves – at least, up to sign.