Uniform local models of quasi-unipotent $p$-adic differential equations

Poornima Belvotagi

Home/Archive/arXiv

arXiv · 2610.36130

Uniform local models of quasi-unipotent $p$-adic differential equations

Poornima Belvotagi

View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct uniform analytic models of quasi-unipotent differential modules over Robba rings. For modules of rank at most $r$ and highest ramification break at most $b$, the inner radius of the model annulus depends only on $r,b$ and the residue characteristic $p$. At each radius, the model admits a basis, possibly depending on the radius, in which the connection matrix is bounded by a constant depending only on $r,b,p$. Both bounds are independent of the coefficient field, and these analytic models require no coefficient extension. A prescribed absolute Frobenius structure extends to such a bounded model with the same radius and connection bound. After a finite coefficient extension, we also realize quasi-constant modules, which carry the finite local monodromy, as differential direct summands of sums of pushforwards of rank-one modules, with uniformly bounded cover degrees and highest breaks of the inducing modules.

Explore related subjects

math.NT math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Poornima Belvotagi. 2026-10-26. Uniform local models of quasi-unipotent $p$-adic differential equations. https://arxiv.org/abs/2610.36130

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Effective Chebotarev-Sato-Tate distribution for Abelian varieties with applications

We establish a uniform effective Chebotarev-Sato-Tate estimate for abelian varieties, conditional on the Mumford-Tate conjecture and the standard analytic conjectures for L-functions attached to the irreducible representations of the Chebotarev-Sato-Tate group. The proof uses spectral majorants and minorants of indicator functions. For regions whose boundaries have finite upper (d-1)-dimensional Minkowski content, the resulting exponent improves that in Bucur et al. (Theorem 1.1, 2024) whenever the Sato-Tate group has rank greater than one. We establish uniform estimates for regions and finite Galois extensions that vary with the counting parameter, and for prime-dependent regions satisfying an explicit variation condition. As an application, we obtain a conditional upper bound for the number of primes at which a weight-two newform with a quadratic coefficient field has a rational Fourier coefficient. We also formulate a discrepancy conjecture, motivated by a probabilistic model, that implies the asymptotic predicted by Murty (Conjecture 3.4, 1999).

math.NT↗

arXiv2026

Archimedean zeta integrals on $ {\rm SO}_{2n} \times {\rm GL}_n$ and $ {\rm Sp}_{2n} \times {\rm GL}_n $

We evaluate the archimedean Rankin-Selberg zeta integrals associated with spherical principal series representations of ${\rm SO}_{2n}(\mathbf R) \times {\rm GL}_n(\mathbf R) $ and $ {\rm Sp}_{2n}(\mathbf R) \times {\rm GL}_n(\mathbf R) $, where $ {\rm SO}_{2n} $ is split and $n\ge2$. We obtain explicit formulas expressing these integrals as quotients of products of gamma functions. Our approach uses Mellin-Barnes representations of spherical Whittaker functions, together with new integral identities for their Mellin transforms and transformations of Barnes integrals. In the even orthogonal case, we establish a recurrence in the rank and evaluate the integral by induction. The symplectic case then follows from an explicit identity relating the two families of zeta integrals.

math.NT↗

arXiv2026

Resolvent Components of Completely Normal Elements and the Morgan-Mullen Conjecture

We study completely normal elements of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, that is, elements that are normal over every intermediate field, through partial Lagrange resolvents. In these coordinates, complete normality splits into independent components, in a way closely related to Hachenberger's decomposition theory. Our main tool is a structure theorem for the components; together with a finite computation, it shows that every component has weight (the sum of the probabilities that its conditions fail) at most $3/4$, uniformly in $n$. Combined with the classical character-sum method, this gives an explicit sufficient criterion for the existence of a primitive completely normal element. We verify the criterion for all $n>q$ except $70$ pairs $(q,n)$, which we settle with explicit elements found by computer. Since the case $n\le q$ is due to Garefalakis and Kapetanakis, this confirms the conjecture of Morgan and Mullen (1996) that every $\mathbb{F}_{q^n}$ contains a primitive element that is completely normal over $\mathbb{F}_q$. The same estimates give two further results. First, Hachenberger's conjectured lower bound $(q-1)^{n'}q^{(p^a-1)n'}$ for the number of completely normal elements, where $p$ is the characteristic and $n=p^an'$ with $p\nmid n'$, holds for all pairs $(q,n)$; it was known for regular pairs. Second, primitivity and complete normality are asymptotically independent: as $q^n\to\infty$, the number of primitive completely normal elements is $(1+o(1))\,φ(q^n-1)/(q^n-1)$ times the number of completely normal elements (for fixed $n$ this was known). Our arguments for $n>q$ and for both further results, including every computer-assisted step, are formalised in Lean 4 with Mathlib; the formalisation takes the case $n\le q$ and an explicit bound of Rosser and Schoenfeld as axioms and relies on Lean's compiler and interpreter for the large computations.

math.NT↗