Toward a generalization of Lehmer's problem to adelic curves

We develop a theory of vector valued automorphic forms associated to the Weil representation $ω_f$ and corresponding to vector valued modular forms transforming with the ``finite'' Weil representation $ρ_L$. For each prime $p$ we determine the structure of a vector valued spherical Hecke algebra depending on $ω_f$, which acts on the space of automorphic forms.

math.NT↗

arXiv2026

Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields

Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in $\mathbb{Z}_p$-towers of imaginary quadratic fields $K$. For a odd prime $p$, the lines $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ are identified with $\mathbb{Z}_p$-extensions $ K_{a,b}/K $. Under certain conditions on $ K $ that involve explicit elliptic curves, we identify a line $(a_0,b_0) \in \mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ such that for all $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ with $(a, b)\not\equiv (a_0, b_0)\pmod{p}$, Hilbert's tenth problem has a negative answer in all finite layers of $ K_{a,b} $. Using results of Bhargava et al., we prove unconditionally that a positive proportion of imaginary quadratic fields meet our criterion when $p=3$. For $p=11,13,31,37$, the analogous conclusions obtained from the rank-zero twist families of Kriz--Li are conditional on the vanishing of the $p$-primary Tate--Shafarevich groups for a positive relative proportion of those twists.

math.NT↗

arXiv2026

The standard $L$-function attached to a vector valued modular form

We define two $L$-functions associated to a common vector valued eigenform $f$ transforming with the ``finite'' Weil representation. The first one can be seen as a standard zeta function defined by the eigenvalues of $f$. The second one can be interpreted as standard $L$-function defined as an Euler product where each $p$-factor is a rational function in terms of two unramified characters of the $p$-adic field $\Q_p$. We show that both $L$-functions are related and prove further that they both can be continued meromorphically to the whole complex $s$-plane.

math.NT↗