arXiv subjects
Publications and source records attributed to Daniele Casazza.
We prove the factorization conjecture for triple-product $p$-adic $L$-functions formulated in a companion article in the special case when two of the (three) factors have complex multiplication.
Our main objective in this paper (which is expository for the most part) is to study the necessary steps to prove a factorization formula for a certain triple product $p$-adic $L$-function guided by the Artin formalism. The key ingredients are: a) the explicit reciprocity laws governing the relationship of diagonal cycles and generalized Heegner cycles to $p$-adic $L$-functions; b) a careful comparison of Chow--Heegner points and twisted Heegner points in Hida families, via formulae of Gross--Zagier type.
This paper introduces a systematic approach towards the inverse problem for arboreal Galois representations of finite index attached to quadratic polynomials. Let $F$ be a field of characteristic $\neq 2$, $f\in F[x]$ be monic and quadratic and $ρ_f$ be the arboreal Galois representation associated to $f$, taking values in the group $Ω_{\infty}$ of automorphisms of the infinite binary tree. We give a complete description of the maximal closed subgroups of each closed subgroup of index at most two of $Ω_{\infty}$ in terms of linear relations modulo squares among certain universal functions evaluated in elements of the critical orbit of $f$. We use such description in order to derive necessary and sufficient criteria for the image of $ρ_f$ to be a given subgroup of index two of $Ω_\infty$. These depend exclusively on the arithmetic of the critical orbit of $f$. Afterwards, we prove that if $ϕ=x^2+t\in\mathbb Q(t)[x]$, then there exist exactly five distinct subgroups of index two of $Ω_{\infty}$ that can appear as images of $ρ_{ϕ_{t_0}}$ for infinitely many $t_0\in\mathbb Q$, where $ϕ_{t_0}$ is the specialized polynomial. We show that two of them appear infinitely often, and if Vojta's conjecture over $\mathbb Q$ holds true, then so do the remaining ones. Finally, we give an explicit description of the derived series of each subgroup of index two. Using this, we introduce a sequence of combinatorial invariants for subgroups of index two of $Ω_\infty$. With a delicate use of these invariants we are able to establish that such subgroups are pairwise non-isomorphic as topological groups, a result of independent interest. This implies, in particular, that the five aforementioned groups are pairwise distinct topological groups, and therefore yield five genuinely different instances of the infinite inverse Galois problem over $\mathbb Q$.
We build a one-variable $p$-adic $L$-function attached to two Hida families of ordinary $p$-stabilised newforms $\mathbf{f}$, $\mathbf{g}$, interpolating the algebraic part of the central values of the complex $L$-series $L(f \otimes \textrm{Ad}(g), s)$ when $f$ and $g$ range over the classical specialisations of $\mathbf{f}$, $\mathbf{g}$ on a suitable line of the weight space. The construction rests on two major results: an explicit formula for the relevant complex central $L$-values, and the existence of non-trivial $Λ$-adic Shintani liftings and Saito--Kurokawa liftings studied in a previous work by the authors. We also illustrate that, under an appropriate sign assumption, this $p$-adic $L$-function arises as a factor of a triple product $p$-adic $L$-function attached to $\mathbf{f}$, $\mathbf{g}$, and $\mathbf{g}$.
In this note we give a detailed construction of a $Λ$-adic $\mathfrak{d}$-th Shintani lifting. We derive a $Λ$-adic version of Kohnen's formula relating Fourier coefficients of half-integral weight modular forms and special values of twisted L-series. As a by-product, we derive a mild generalization of such classical formulae, and also point out a relation between Fourier coefficients of $Λ$-adic $\d$-th Shintani liftings and Stark--Heegner points.
This article is devoted to the elliptic Stark conjecture formulated by Darmon, Lauder and Rotger [DLR], which proposes a formula for the transcendental part of a $p$-adic avatar of the leading term at $s=1$ of the Hasse-Weil-Artin $L$-series $L(E,\varrho_1\otimes \varrho_2,s)$ of an elliptic curve $E$ twisted by the tensor product $\varrho_1\otimes \varrho_2$ of two odd $2$-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a $2\times 2$ $p$-adic regulator involving the $p$-adic formal group logarithm of suitable Stark points on $E$. This conjecture was proved in [DLR] in the setting where $\varrho_1$ and $\varrho_2$ are induced from characters of the same imaginary quadratic field $K$. In this note we prove a refinement of this result, that was discovered experimentally in Remark 3.4 of [DLR] in a few examples. Namely, we are able to determine the algebraic constant up to which the main theorem of [DLR] holds in a particular setting where the Hida-Rankin $p$-adic $L$-function associated to a pair of Hida families can be exploited to provide an alternative proof of the same result. This constant encodes local and global invariants of both $E$ and $K$.