arXiv subjects

Matthew Papanikolas

Publications and source records attributed to Matthew Papanikolas.

4 recordsLinked to original sources

arXiv2025

Hecke $L$-series for Sinha modules

We investigate Goss $L$-functions associated to Anderson $t$-modules defined by Sinha having complex multiplication by Carlitz cyclotomic fields. We show that these $t$-modules are defined over the cyclotomic field and that their $L$-functions are products of Hecke $L$-series for Anderson's Hecke character defined via Coleman functions. Applying identities of Fang and Taelman, we prove that special values of these $L$-functions are expressible in terms of products of values of Thakur's geometric $\Gamma$-function.

math.NT↗

arXiv2022

A note on log-algebraicity on elliptic curves

We analyze log-algebraic power series identities for formal groups of elliptic curves over $\mathbb{Q}$ which arise from modular parametrizations. We further investigate applications to special values of elliptic curve $L$-functions.

math.NT↗

arXiv2020

Equidistribution of Gross points over rational function fields

In this paper we prove a sparse equidistribution theorem for Gross points over the rational function field $\mathbb{F}_q(t)$. We apply this result to study the reduction map from CM Drinfeld modules to supersingular Drinfeld modules. Our proofs rely crucially on a period formula due to M. Papikian and F.-T. Wei/J. Yu, and a Lindelöf-type bound for central values of Rankin-Selberg $L$-functions associated to twists of automorphic forms of Drinfeld-type by ideal class group characters.

math.NT↗

arXiv2005

On the torsion of Jacobians of principal modular curves of level 3^n

We demonstrate that the 3-power torsion points of the Jacobians of the principal modular curves X(3^n) are fixed by the kernel of the canonical outer Galois representation of the pro-3 fundamental group of the projective line minus three points. The proof proceeds by demonstrating the curves in question satisfy a two-part criterion given by Anderson and Ihara. Two proofs of the second part of the criterion are provided; the first relies on a theorem of Shimura, while the second uses the moduli interpretation.

math.NT↗