On some rational generating series occuring in arithmetic geometry

The authors defined in their previous paper ''$h^1\ne h_1$ for Anderson t-motives'' the notion of an affine equation associated to a t-motive. We study in the present paper general affine equations and their systems. We define L-functions of an affine equation. We show that its local factor is an algebraic function; we state a problem to prove an analog of the Lefschetz trace formula for the global L-function. We define t-holonomic sequences which are solutions of any affine equations. We state some conjectures and open questions related to t-holonomic sequences, for example formulas of their asymptotic growth. We show that some of the properties of t-holonomic sequences are analogous to the properties of the classical holonomic sequences, for example the closure properties. Further, we show that systems of affine equations can be reduced to one equation by a process analogous to the elimination of variables in a system of linear equations; this gives us a notion of non-commutative determinants. Finally, we consider some explicit examples of affine equations that come from Drinfeld modules, their duals and some other t-motives.

math.NT↗

arXiv2026

The prime geodesic theorem for the modular orbifold

We establish the prime geodesic theorem for the modular orbifold with exponent $\frac{9}{13} = 0.692307 \cdots$ in the error term. By periodicity and Heath-Brown's quadratic large sieve, we estimate the sixth moment of the error term in an approximation to the Zagier $L$-series at $s = 1$. The smooth explicit formula of Soundararajan and Young then leads to the stated exponent without recourse to subconvexity for quadratic Dirichlet $L$-functions.

math.NT↗

arXiv2026

Dwork Motives, Monodromy and Potential Automorphy

In this paper we study certain families of motives, which arise as direct summands of the cohomology of the Dwork family. We computationally find examples of interesting families with the following three properties. Firstly, their geometric monodromy group is Zariski dense in $\operatorname{SL}_n$. Secondly, they realise many different unipotent operators as the monodromy operator at $t = \infty$. Thirdly, all their Hodge numbers are $\leq 1$. This has consequences for Galois representations. Namely, if a nilpotent operator $N$ appears as the monodromy at $t = \infty$ in one of our families, we can construct potentially automorphic representations with $\ell$-adic monodromy given by $N$ at a fixed prime $p$. As another application, we obtain a new proof of some cases of the recent local-global compatibility theorem of Matsumoto.

math.NT↗