Variation of the algebraic λ-invariant over a solvable extension

Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$ . Given a modular Galois representation $\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$ which is p-ordinary and also p-distinguished, we shall write $\mathcal{H}(\overline{\rho})$ for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant \begin{equation} \lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of } \text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big) \end{equation} at all $f\in\mathcal{H}(\overline{\rho})$ , under the assumption $\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$ for at least one form f0. We can then easily deduce that $\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$ is constant along branches of $\mathcal{H}(\overline{\rho})$ , generalising a theorem of Emerton, Pollack and Weston for $\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$ . For example, if $\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$ has the structure of a p-adic Lie extension then our formulae include the cases where: either (i) $\mathcal{D}_{\infty}/K$ is a g-fold false Tate tower, or (ii) $\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$ has dimension ≤ 3 and is a pro-p-group.

SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS

We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over $\mathbb{Q}$ of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use $R=T$ theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from $\text{GSp}(4)$ to $U(4)$. In that case we obtain a corollary for abelian surfaces.

A p-ADIC HERMITIAN MAASS LIFT

For K, an imaginary quadratic field with discriminant −DK, and associated quadratic Galois character χK, Kojima, Gritsenko and Krieg studied a Hermitian Maass lift of elliptic modular cusp forms of level DK and nebentypus χK via Hermitian Jacobi forms to Hermitian modular forms of level one for the unitary group U(2, 2) split over K. We generalize this (under certain conditions on K and p) to the case of p-oldforms of level pDK and character χK. To do this, we define an appropriate Hermitian Maass space for general level and prove that it is isomorphic to the space of special Hermitian Jacobi forms. We then show how to adapt this construction to lift a Hida family of modular forms to a p-adic analytic family of automorphic forms in the Maass space of level p.

ON THE EXCEPTIONAL SPECIALIZATIONS OF BIG HEEGNER POINTS

We extend the $p$-adic Gross–Zagier formula of Bertolini et al. [Generalized Heegner cycles and $p$-adic Rankin $L$-series, Duke Math. J.162(6) (2013), 1033–1148] to the semistable non-crystalline setting, and combine it with our previous work [Castella, On the $p$-adic variation of Heegner points, Preprint, 2014, arXiv:1410.6591] to obtain a derivative formula for the specializations of Howard’s big Heegner points [Howard, Variation of Heegner points in Hida families, Invent. Math.167(1) (2007), 91–128] at exceptional primes in the Hida family.

Algebraic functional equation for Hida family

We prove a functional equation for the characteristic ideal of the "big" Selmer group 𝒳(𝒯ℱ/F cyc ) associated to an ordinary Hida family of elliptic modular forms over the cyclotomic ℤp extension of a general number field F, under the assumption that there is at least one arithmetic specialization whose Selmer group is torsion over its Iwasawa algebra. For a general number field, the two-variable cyclotomic Iwasawa main conjecture for ordinary Hida family is not proved and this can be thought of as an evidence to the validity of the Iwasawa main conjecture. The central idea of the proof is to prove a variant of the result of Perrin-Riou [Groupes de Selmer et accouplements; cas particulier des courbes elliptiques, Doc. Math.2003 (2003) 725–760, Extra Volume: Kazuya Kato's fiftieth birthday] by constructing a generalized pairing on the individual Selmer groups corresponding to the arithmetic points and make use of the appropriate specialization techniques of Ochiai [Euler system for Galois deformations, Ann. Inst. Fourier (Grenoble)55(1) (2005) 113–146].

On Greenberg’s L-invariant of the symmetric sixth power of an ordinary cusp form

We derive a formula for Greenberg’s L-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight ≥4, under some technical assumptions. This requires a ‘sufficiently rich’ Galois deformation of the symmetric cube, which we obtain from the symmetric cube lift to GSp(4)/Q of Ramakrishnan–Shahidi and the Hida theory of this group developed by Tilouine–Urban. The L-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg’s L-invariant of all symmetric powers by using appropriate functorial transfers and Hida theory on higher rank groups.

The exceptional zero conjecture for Hilbert modular forms

Using ap-adic analogue of the convolution method of Rankin–Selberg and Shimura, we construct the two-variablep-adicL-function of a Hida family of Hilbert modular eigenforms of parallel weight. It is shown that the conditions of Greenberg–Stevens [R. Greenberg and G. Stevens,p-adic L-functions and p-adic periods of modular forms, Invent. Math.111(1993), 407–447] are satisfied, from which we deduce special cases of the Mazur–Tate–Teitelbaum conjecture in the Hilbert modular setting.

Stark–Heegner points and the Shimura correspondence

Let $g = \sum c(D)q^D$ and $f=\sum a_n q^n$ be modular forms of half-integral weight k+1/2 and integral weight 2k respectively that are associated to each other under the Shimura–Kohnen correspondence. For suitable fundamental discriminants D, a theorem of Waldspurger relates the coefficient c(D) to the central critical value L(f,D,k) of the Hecke L-series of f twisted by the quadratic Dirichlet character of conductor D. This paper establishes a similar kind of relationship for central critical derivatives in the special case k=1, where f is of weight 2. The role of c(D) in our main theorem is played by the first derivative in the weight direction of the Dth Fourier coefficient of a p-adic family of half-integral weight modular forms. This family arises naturally, and is related under the Shimura correspondence to the Hida family interpolating f in weight 2. The proof of our main theorem rests on a variant of the Gross–Kohnen–Zagier formula for Stark–Heegner points attached to real quadratic fields, which may be of some independent interest. We also formulate a more general conjectural formula of Gross–Kohnen–Zagier type for Stark–Heegner points, and present numerical evidence for it in settings that seem inaccessible to our methods of proof based on p-adic deformations of modular forms.