Exceptional Zero Research Articles

Joint equidistribution of CM points

We prove the mixing conjecture of Michel and Venkatesh for toral packets with negative fundamental discriminants and split at two fixed primes; assuming all splitting fields have no exceptional Landau-Siegel zero. As a consequence we establish for arbitrary products of indefinite Shimura curves the equidistribution of Galois orbits of generic sequences of CM points all whose components have the same fundamental discriminant; assuming the CM fields are split at two fixed primes and have no exceptional zero. The joinings theorem of Einsiedler and Lindenstrauss applies to the toral orbits arising in these results. Yet it falls short of demonstrating equidistribution due to the possibility of intermediate algebraic measures supported on Hecke correspondences and their translates. The main novel contribution is a method to exclude intermediate measures for toral periods. The crux is a geometric expansion of the cross-correlation between the periodic measure on a torus orbit and a Hecke correspondence, expressing it as a short shifted convolution sum. The latter is bounded from above generalizing the method of Shiu and Nair to polynomials in two variables on smooth domains.

Exceptional Jacobi polynomials

In this paper we present a systematic way to describe exceptional Jacobi polynomials via two partitions. We give the construction of these polynomials and restate the known aspects of these polynomials in terms of their partitions. The aim is to show that the use of partitions is an elegant way to label these polynomials. Moreover, we prove asymptotic results according to the regular and exceptional zeros of these polynomials.

Density of multivariate homogeneous polynomials on star like domains

The famous Weierstrass theorem asserts that every continuous function on a compact set in Rd can be uniformly approximated by algebraic polynomials. A related interesting problem consists in studying the same question for the important subclass of homogeneous polynomials containing only monomials of the same degree. The corresponding conjecture claims that every continuous function on the boundary of convex 0-symmetric bodies can be uniformly approximated by pairs of homogeneous polynomials. The main objective of the present paper is to review the recent progress on this conjecture and provide a new unified treatment of the same problem on non convex star like domains. It will be shown that the boundary of every 0-symmetric non convex star like domain contains an exceptional zero set so that a continuous function can be uniformly approximated on the boundary of the domain by a sum of two homogeneous polynomials if and only if the function vanishes on this zero set. Thus the Weierstrass type approximation problem for homogeneous polynomials on non convex star like domains amounts to the study of these exceptional zero sets. We will also present an extension of a theorem of Varjú which describes the exceptional zero sets for intersections of star like domains. These results combined with certain transformations of the underlying region will lead to the discovery of some new classes of convex and non convex domains for which the Weierstrass type approximation result holds for homogeneous polynomials.

κ -Poincaré invariant quantum field theories with Kubo-Martin-Schwinger weight

A natural star product for 4-d $\kappa$-Minkowski space is used to investigate various classes of $\kappa$-Poincar\'e invariant scalar field theories with quartic interactions whose commutative limit coincides with the usual $\phi^4$ theory. $\kappa$-Poincar\'e invariance forces the integral involved in the actions to be a twisted trace, thus defining a KMS weight for the noncommutative (C*-)algebra modeling the $\kappa$-Minkowski space. The associated modular group and Tomita modular operator are characterized. In all the field theories, the twist generates different planar one-loop contributions to the 2-point function which are at most UV linearly diverging. Some of these theories are free of UV/IR mixing. In the others, UV/IR mixing shows up in non-planar contributions to the 2-point function as a polynomial singularity at exceptional zero external momenta while staying finite at non-zero external momenta. These results are discussed together with the possibility for the KMS weight relative to the quantum space algebra to trigger the appearance of KMS state on the algebra of observables.

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.

Exceptional Laguerre Polynomials

AbstractThe aim of this paper is to present the construction of exceptional Laguerre polynomials in a systematic way and to provide new asymptotic results on the location of the zeros. To describe the exceptional Laguerre polynomials, we associate them with two partitions. We find that the use of partitions is an elegant way to express these polynomials and we restate some of their known properties in terms of partitions. We discuss the asymptotic behavior of the regular zeros and the exceptional zeros of exceptional Laguerre polynomials as the degree tends to infinity.

Explicit bounds for primes in arithmetic progressions

We derive explicit upper bounds for various counting functions for primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\mathop{\mathrm{gcd}}\nolimits (a,q)=1$ and $3\leq q\leq10^{5}$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p\equiv a\ (\operatorname{mod}q)$ with $p\leq x$, we show that \[\vert \theta(x;q,a)-{x}/{\varphi(q)}\vert <\frac{1}{160}\frac{x}{\log x}\] for all $x\geq8\cdot10^{9}$, with significantly sharper constants obtained for individual moduli $q$. We establish inequalities of the same shape for the other standard prime-counting functions $\pi(x;q,a)$ and $\psi(x;q,a)$, as well as inequalities for the $n$th prime congruent to $a\ (\operatorname{mod}q)$ when $q\le1200$. For moduli $q>10^{5}$, we find even stronger explicit inequalities, but only for much larger values of $x$. Along the way, we also derive an improved explicit lower bound for $L(1,\chi)$ for quadratic characters $\chi$, and an improved explicit upper bound for exceptional zeros.

Exceptional zeros of polynomials satisfying a three-term recurrence

Let Pm(z) be a sequence of polynomials satisfying the recurrence relation Pm(z)+B(z)Pm−1(z)+A(z)Pm−2(z)=0. Tran showed that with a suitable initial condition, the zeros of Pm(z) lie on a fixed explicit curve C on the complex plane. We study a more general initial condition in which we can count the finite number of zeros lying off C.

An application of the Hardy–Littlewood conjecture

We assume a weak Hardy–Littlewood conjecture and derive an upper bound for a real exceptional zero associated with a prime modulus.

$p$-adic L-functions of automorphic forms and exceptional zeros

We construct p -adic L-functions for automorphic representations of \mathrm{GL} _2 of a number field F , and show that the corresponding p -adic L-function of a modular elliptic curve E over F has an extra zero at the central point for each prime above p at which E has split multiplicative reduction, a part of the exceptional zero conjecture.

On the rank one abelian Gross–Stark conjecture

Let F be a totally real number field, p a rational prime, and \chi a finite order totally odd abelian character of Gal (\overline{F}/F) such that \chi(\mathfrak{p})=1 for some \mathfrak{p}|p . Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the p -adic L -function associated to \chi at its exceptional zero and the \mathfrak{p} -adic logarithm of a p -unit in the \chi component of F_\chi^\times . In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture in the rank one setting assuming two conditions: that Leopoldt's conjecture holds for F and p , and that if there is only one prime of F lying above p , a certain relation holds between the \mathscr{L} -invariants of \chi and \chi^{-1} . The main result of this paper removes both of these conditions, thus giving an unconditional proof of the rank one conjecture.

Factorization of p-adic Rankin L-series

We prove that the p-adic L-series of the tensor square of a p-ordinary modular form factors as the product of the symmetric square p-adic L-series of the form and a Kubota–Leopoldt p-adic L-series. This establishes a generalization of a conjecture of Citro. Greenberg’s exceptional zero conjecture for the adjoint follows as a corollary of our theorem. Our method of proof follows that of Gross, who proved a factorization result for the Katz p-adic L-series associated to the restriction of a Dirichlet character. Whereas Gross’s method is based on comparing circular units with elliptic units, our method is based on comparing these same circular units with a new family of units (called Beilinson–Flach units) that we construct. The Beilinson–Flach units are constructed using Bloch’s intersection theory of higher Chow groups applied to products of modular curves. We relate these units to special values of classical and p-adic L-functions using work of Beilinson (as generalized by Lei–Loeffler–Zerbes) in the archimedean case and Bertolini–Darmon–Rotger (as generalized by Kings–Loeffler–Zerbes) in the p-adic case. Central to our method are two compatibility theorems regarding Bloch’s intersection pairing and the classical and p-adic Beilinson regulators defined on higher Chow groups.

Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions.

The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval R le r le gamma R with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function Phi _{n,nu }(r)=Y_{nu }^{prime }(lambda _{n,nu }) J_{nu }(lambda _{n,nu } r/R)-J_{nu }^{prime }(lambda _{n,nu }) Y_{nu }(lambda _{n,nu } r/R) or linear combinations of the spherical Bessel functions psi _{m,nu }(r)=y_{nu }^{prime }(lambda _{m,nu }) j_{nu }(lambda _{m,nu } r/R)-j_{nu }^{prime }(lambda _{m,nu }) y_{nu }(lambda _{m,nu } r/R). The orthogonality relations with analytical expressions for the normalization constant are given. Explicit expressions for the Lommel integrals in terms of Lommel functions are derived. The cross product zeros Y_{nu }^{prime }(lambda _{n,nu }) J_{nu }^{prime }(gamma lambda _{n,nu })-J_{nu }^{prime }(lambda _{n,nu }) Y_{nu }^{prime }(gamma lambda _{n,nu }) = 0 and y_{nu }^{prime }(lambda _{m,nu }) j_{nu }^{prime }(gamma lambda _{m,nu })-j_{nu }^{prime }(lambda _{m,nu }) y_{nu }^{prime }(gamma lambda _{m,nu }) = 0 are considered in the complex plane for real as well as complex values of the index nu and approximations for the exceptional zero lambda _{1,nu } are obtained. A numerical scheme based on the discretization of the two-dimensional and three-dimensional Laplace operator with Neumann boundary conditions is presented. Explicit representations of the radial part of the Laplace operator in form of a tridiagonal matrix allow the simple computation of the cross product zeros.

Zeros of exceptional Hermite polynomials

We study the zeros of exceptional Hermite polynomials associated with an even partition λ. We prove several conjectures regarding the asymptotic behaviour of both the regular (real) and the exceptional (complex) zeros. The real zeros are distributed as the zeros of usual Hermite polynomials and, after contracting by a factor 2n, we prove that they follow the semi-circle law. The non-real zeros tend to the zeros of the generalized Hermite polynomial Hλ, provided that these zeros are simple. It was conjectured by Veselov that the zeros of generalized Hermite polynomials are always simple, except possibly for the zero at the origin, but this conjecture remains open.

Exceptional zero formulae and a conjecture of Perrin-Riou

Let $$A/\mathbf{Q}$$ be an elliptic curve with split multiplicative reduction at a prime p. We prove (an analogue of) a conjecture of Perrin-Riou, relating p-adic Beilinson–Kato elements to Heegner points in $$A(\mathbf{Q})$$ , and a large part of the rank-one case of the Mazur–Tate–Teitelbaum exceptional zero conjecture for the cyclotomic p-adic L-function of A. More generally, let f be the weight-two newform associated with A, let $$f_{\infty }$$ be the Hida family of f, and let $$L_{p}(f_{\infty },k,s)$$ be the Mazur–Kitagawa two-variable p-adic L-function attached to $$f_{\infty }$$ . We prove a p-adic Gross–Zagier formula, expressing the quadratic term of the Taylor expansion of $$L_{p}(f_{\infty },k,s)$$ at $$(k,s)=(2,1)$$ as a non-zero rational multiple of the extended height-weight of a Heegner point in $$A(\mathbf{Q})$$ .

On Nekovář's Heights, Exceptional Zeros and a Conjecture of Mazur–Tate–Teitelbaum

On Nekovář's Heights, Exceptional Zeros and a Conjecture of Mazur–Tate–Teitelbaum

On Teitelbaum type [formula omitted]-invariants of Hilbert modular forms attached to definite quaternions

We generalize Teitelbaum's work on the definition of the L-invariant to Hilbert modular forms that arise from definite quaternion algebras over totally real fields by the Jacquet–Langlands correspondence. Conjecturally this coincides with the Fontaine–Mazur type L-invariant, defined by applying Fontaine's theory to the Galois representation associated to Hilbert modular forms. An exceptional zero conjecture for the p-adic L-function of Hilbert modular forms is also proposed.

ℒ-invariants and logarithm derivatives of eigenvalues of Frobenius

Let K be a p-adic local field. We study a special kind of p-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms. In particular, we verify that a formula of Colmez can be generalized to our case. We also include a degenerated version of Colmez’s formula.

Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes

Let f be a cuspidal newform with complex multiplication (CM) and let p be an odd prime at which f is non-ordinary. We construct admissible p-adic L-functions for the symmetric powers of f, thus verifying general conjectures of Dabrowski and Panchishkin in this special case. We also construct their "mixed" plus and minus counterparts and prove an analogue of Pollack's decomposition of the admissible p-adic L-functions into mixed plus and minus p-adic L-functions. On the arithmetic side, we define corresponding mixed plus and minus Selmer groups. We unite the arithmetic with the analytic by first formulating the Main Conjecture of Iwasawa Theory relating the plus and minus Selmer groups with the plus and minus p-adic L-functions, and then proving the exceptional zero conjecture for the admissible p-adic L-functions. The latter result takes advantage of recent work of Benois, while the former uses recent work of the second author, as well as the main conjecture of Mazur-Wiles.

Some Properties on Complex Functional Difference Equations

We obtain some results on the transcendental meromorphic solutions of complex functional difference equations of the form∑λ∈Iαλ(z)(∏j=0nf(z+cj)λj)=R(z,f∘p)=((a0(z)+a1(z)(f∘p)+ ⋯ +as(z) (f∘p)s)/(b0(z)+b1(z)(f∘p)+ ⋯ +bt(z)(f∘p)t)), whereIis a finite set of multi-indexesλ=(λ0,λ1,…,λn),c0=0,cj∈ℂ∖{0} (j=1,2,…,n)are distinct complex constants,p(z)is a polynomial, andαλ(z) (λ∈I),ai(z) (i=0,1,…,s), andbj(z) (j=0,1,…,t)are small meromorphic functions relative tof(z). We further investigate the above functional difference equation which has special type if its solution has Borel exceptional zero and pole.