Exceptional Zero Conjecture Research Articles

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.

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.

$p$-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.

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.

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 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.

On special zeros of p-adic L-functions of Hilbert modular forms

Let E be a modular elliptic curve over a totally real number field F. We prove the weak exceptional zero conjecture which links a (higher) derivative of the p-adic L-function attached to E to certain p-adic periods attached to the corresponding Hilbert modular form at the places above p where E has split multiplicative reduction. Under some mild restrictions on p and the conductor of E we deduce the exceptional zero conjecture in the strong form (i.e. where the automorphic p-adic periods are replaced by the \(\mathcal {L}\)-invariants of E defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the p-adic L-function of E in terms of local data.

The Exceptional Zero Conjecture for Symmetric Powers of CM Modular Forms: the Ordinary Case

We prove the exceptional zero conjecture for the symmetric powers of CM cuspidal eigenforms at ordinary primes. In other words, we determine the trivial zeroes of the associated p-adic L-functions, compute the L-invariants, and show that they agree with Greenberg's L-invariants.

CM cycles on Shimura curves, and p-adic L-functions

AbstractLet f be a modular form of weight k≥2 and level N, let K be a quadratic imaginary field and assume that there is a prime p exactly dividing N. Under certain arithmetic conditions on the level N and the field K, one can attach to this data a p-adic L-function Lp (f,K,s) , as done by Bertolini–Darmon–Iovita–Spieß in [Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting, Amer. J. Math. 124 (2002), 411–449]. In the case of p being inert in K, this analytic function of a p-adic variable s vanishes in the critical range s=1,…,k−1 , and one may be interested in the values of its derivative in this range. We construct, for k≥4 , a Chow motive endowed with a distinguished collection of algebraic cycles which encode these values, via the p-adic Abel–Jacobi map. Our main result generalizes the result obtained by Iovita and Spieß in [Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms, Invent. Math. 154 (2003), 333–384], which gives a similar formula for the central value s=k/2 . Even in this case our construction is different from the one found by Iovita and Spieß.

Proof of an exceptional zero conjecture for elliptic curves over function fields

Based on the analogy between number fields and function fields of one variable over finite fields, we formulate and prove an analogue of the exceptional zero conjecture of Mazur, Tate and Teitelbaum for elliptic curves defined over function fields. The proof uses modular parametrization by Drinfeld modular curves and the theory of non-archimedean integration. As an application we prove a refinement of the Birch-Swinnerton-Dyer conjecture if the analytic rank of the elliptic curve is zero.

Teitelbaum's exceptional zero conjecture in the function field case

The exceptional zero conjecture relates the first derivative of the $p$-adic $L$-function of a rational elliptic curve with split multiplicative reduction at $p$ to its complex $L$-function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field $\FQ(T)$ with split multiplicative reduction at two places $\fp$ and $\infty$, avoiding the construction of a $\fp$-adic $L$-function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals over arbitrary function fields. A function field version of Darmon's period conjecture is also obtained.

An Elementary Proof of a Weak Exceptional Zero Conjecture

AbstractIn this paper we extend Darmon's theory of “integration on ℋp × ℋ” to cusp forms f of higher even weight. This enables us to prove a “weak exceptional zero conjecture”: that when the p-adic L-function of f has an exceptional zero at the central point, the ℒ-invariant arising is independent of a twist by certain Dirichlet characters.

Teitelbaum's exceptional zero conjecture in the anticyclotomic setting

Teitelbaum formulated a conjecture relating first derivatives of the Mazur-Swinnerton-Dyer p -adic L -functions attached to modular forms of even weight k ≥ 2 to certain [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-invariants arising from Shimura curve parametrizations. This article formulates an analogue of Teitelbaum's conjecture in which the cyclotomic [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] extension of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is replaced by the anticyclotomic [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]-extension of an imaginary quadratic field. This analogue is then proved by using the Cerednik-Drinfeld theory of p -adic uniformisation of Shimura curves.