arXiv subjects

Mohammadreza Mohajer

Publications and source records attributed to Mohammadreza Mohajer.

4 recordsLinked to original sources

arXiv2026

Crystalline Realizations of 1-Motives: Integral Structure and Classification via Linear Frobenius

We study morphisms and integral weight splittings in the contravariant Dieudonné realization of Deligne $1$-motives over a finite field. Combining the classical Tate theorem with a finite-level divisibility criterion, we prove integral full faithfulness after $p$-adic completion of morphism groups. We identify the geometric extension classes killed by the Barsotti--Tate realization and determine the exact common $p$-power denominator of the weight projectors from the extension maps of a $1$-motive and its Cartier dual. The same invariant measures the integral splitting defect. Frobenius resultants and reduced resultants give uniform bounds, whose sharpness is illustrated by explicit elliptic examples. Semilinear descent expresses crystalline morphisms through linear Frobenius intertwiners; eigenvalue ratios then determine morphism ranks over finite extensions and the minimal common field of definition. We characterize the rational essential image using cyclotomic factors and Honda--Tate multiplicities, and obtain derived realization equivalences for motives generated by curves after scalar extension and idempotent completion. Applications include quaternionic descent and divisor correspondences between smooth projective curves.

math.AG↗

arXiv2026

Annular Frobenius Classification of p-adic Stieltjes--Schwarzian Equations

We give explicit annular normal forms and classify Frobenius structures for $p$-adic Schwarzian equations defined by developing differentials. For a unit of the Robba ring, reduction of order expresses the associated module as a quadratic rank-one twist of a unipotent extension determined by the residue. Frobenius existence is characterized by a square-class condition. At odd primes, there are exactly four differential-module types, and a uniform dominant-monomial hypothesis gives a Frobenius formula in analytic families across the zero-residue locus. For rational nonintegral annular powers, we obtain a diagonal normal form and a congruence criterion at odd primes; both results hold at every prime when the unit factor is an explicit square. Applied to the reducible modular equations $θ^2y-\ell^2E_4y/144=0$, with $\gcd(\ell,6)=1$, this gives exactly two annular types at every prime and determines the least $s\geq1$ for which Frobenius under $q\mapsto q^{p^s}$ exists. The subfamily $\ell=12n+1$ has a single annular type. Heine--Stieltjes residue cancellation and nondegenerate Jacobi configurations provide explicit examples. We also prove an obstruction to constant projective Frobenius equivariance of Robba-ring developing maps.

math.NT↗

arXiv2025

P-adic Period Conjectures for 1-motives: Integration and Linear Relations

We develop a $p$-adic theory of periods for 1-motives, extending the classical theory of complex periods into the non-archimedean setting. For 1-motives with good reduction over $p$-adic local fields, we construct a $p$-adic integration pairing that generalizes the Colmez--Fontaine--Messing theory for abelian varieties. This pairing is bilinear, perfect, Galois-equivariant, and compatible with the Hodge filtration, taking values in a quotient of the de Rham period ring. Building on this construction, we introduce a stratified formalism for $p$-adic periods, defining period spaces at various depths that capture increasingly refined relations among periods, and formulating conjectures that mirror the Grothendieck period conjecture in this new context. The classical period conjecture for 1-motives over $\bar{Q}$, previously resolved via the Huber--Wustholz analytic subgroup theorem, is recovered at depth 1 in our framework. For 1-motives over number fields with good reduction at $p$, we identify canonical $Q$-structures on the $p$-adic realizations arising from rational points of their associated formal $p$-divisible groups. Relative to these structures, we establish the conjectures at depths 1 and 2. A key tool is the development of a $p$-adic analytic subgroup theorem tailored to 1-motives, providing an analogue of Wustholz's classical result. Our work not only yields a $p$-adic counterpart to the Kontsevich--Zagier conjecture for 1-motives but also opens new pathways for the study of linear relations among $p$-adic periods and their transcendence properties.

math.NT↗

arXiv2024

Linear relations of p-adic periods of 1-motives (thesis)

In this thesis, we aim to develop p-adic analogs of known results for classical periods, focusing specifically on 1-motives. We establish an integration theory for 1-motives with good reductions, which generalizes the Colmez-Fontaine-Messing p-adic integration for abelian varieties with good reductions. We also compare the integration pairing with other pairings such as those induced by crystalline theory. Additionally, we introduce a formalism for periods and formulate p-adic period conjectures related to p-adic periods arising from this integration pairing. Broadly, our p-adic period conjecture operates at different depths, with each depth revealing distinct relations among the p-adic periods. Notably, the classical period conjecture (Kontsevich-Zanier conjecture over $\bar{\mathbb{Q}}$) for 1-periods fits within our framework, and, according to the classical subgroup theorem of Huber-Wüstholz for 1-motives, the conjecture for classical periods of 1-motives holds true at depth 1. Finally, we identify three $\mathbb{Q}$-structures arising from $\bar{\mathbb{Q}}$-rational points of the formal p-divisible group associated with a 1-motive $M$ with a good reduction at $p$, and we prove p-adic period conjectures at depths 2 and 1, relative to periods induced by the p-adic integration of $M$ and these $\mathbb{Q}$-structures. Our proof involves a p-adic version of the subgroup theorem that we obtain for 1-motives with good reductions.

math.NT↗