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Fabien Trihan

Publications and source records attributed to Fabien Trihan.

15 recordsLinked to original sources
arXiv2026

Specialisations of the Burungale-Castella-Skinner main conjecture to $\mathbb Z_p$-lines

Let $p>3$ be a prime, $E/\mathbb Q$ be an elliptic curve and $K$ an imaginary quadratic field satisfying the hypotheses of Burungale-Castella-Skinner, and let $L/K$ be the unique $\mathbb{Z}_p^2$-extension. In this note, by combining their integral two-variable main conjecture with the specialisation formula of the first-named author, we obtain a characteristic-ideal identity over every $\mathbb{Z}_p$-line in $L/K$, involving an explicit local factor. With the characteristic ideal of a non-torsion module defined to be zero, this identity also applies when the two-variable Perrin-Riou element specialises to zero. We call such lines exceptional and prove that only finitely many occur. We also prove that the cyclotomic line is non-exceptional with trivial local factor, and that the anticyclotomic line is exceptional. Finally, we bound the number of exceptional lines by the cyclotomic augmentation order and prove that if the $p$-primary Tate-Shafarevich group over $K$ is finite and the cyclotomic $p$-adic height pairing is non-degenerate, then this number is at most ${\rm rank}(E(K))$. In particular, when $E(K)$ has rank one, the anticyclotomic line is the unique exceptional line.

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arXiv2026

Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields

Let $A$ be an ordinary elliptic curve over a global function field $K$ of characteristic $p$, assumed semistable at every place, and let $L/K$ be a $\mathbb{Z}_p^d$-extension ramified only at finitely many places where $A$ has ordinary reduction. Building on the framework of [Tan26] (arXiv:2603.10576), we prove the Iwasawa Main Conjecture for $A$ over $L$, subject to a technical $\mu$-invariant hypothesis that is already detected after specialization to the unramified $\mathbb{Z}_p$-extension. The principal new input is a `$\chi$-formula' that compares appropriate $\chi$-isotypic characteristic ideals of Selmer modules with the corresponding specializations of the $p$-adic $L$-function. Finally, to show that our $\mu$-hypothesis is non-vacuous, we prove, for $p>3$, that the hypothesis holds on a Zariski open dense locus in the moduli of semistable elliptic curves.

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arXiv2024

On a Birch and Swinnerton-Dyer type conjecture for the Hasse-Weil-Artin $L$-functions in characteristic $p>0$

Given an abelian variety $A$ over a global function field $K$ of characteristic $p>0$ and an irreducible complex continuous representation $\psi$ of the absolute Galois group of $K$, we obtain a BSD-type formula for the leading term of Hasse--Weil--Artin $L$-function for $(A,\psi)$ at $s=1$ under certain technical hypotheses. The formula we obtain can be applied quite generally; for example, it can be applied to the $p$-part of the leading term even when $\psi$ is weakly wildly ramified at some place under additional hypotheses. Our result is the function field analogue of the work of D. Burns and D. Macias Castillo, built upon the work on the equivariant refinement of the BSD conjecture by D. Burns, M. Kakde and the first-named author. To handle the $p$-part of the leading term, we need the Riemann--Roch theorem for equivariant vector bundles on a curve over a finite field generalising the work of S. Nakajima, B. K\"ock, and H. Fischbacher-Weitz and B. K\"ock, which is of independent interest.

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arXiv2023

The $\mu$-invariant change for abelian varieties over finite $p$-extensions of global fields

We extend the work of Lai, Longhi, Suzuki, the first two authors and study the change of $\mu$-invariants, with respect to a finite Galois p-extension $K'/K$, of an ordinary abelian variety $A$ over a $\mathbb{Z}_p^d$-extension of global fields $L/K$ that ramifies at a finite number of places at which $A$ has ordinary reductions. In characteristic $p>0$, we obtain an explicit bound for the size $\delta_v$ of the local Galois cohomology of the Mordell-Weil group of $A$ with respect to a $p$-extension ramified at a supersingular place $v$. Next, in all characteristics, we describe the asymptotic growth of $\delta_v$ along a multiple $\mathbb{Z}_p$-extension $L/K$ and provide a lower bound for the change of $\mu$-invariants of $A$ from the tower $L/K$ to the tower $LK'/K'$. Finally, we present numerical evidence supporting these results.

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arXiv2020

On the $μ$-invariants of abelian varieties over function fields of positive characteristic

Let $A$ be an abelian variety over a global function field $K$ of characteristic $p$. We study the $μ$-invariant appearing in the Iwasawa theory of $A$ over the unramified $\mathbb{Z}_p$-extension of $K$. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate-Shafarevich group of $A$ and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate-Shafarevich group (which is now the $μ$-invariant) in terms of other quantities including the Faltings height of $A$ and Frobenius slopes of the numerator of the Hasse-Weil $L$-function of $A / K$ assuming the conjectural Birch-Swinnerton-Dyer formula. Our next result is to prove this $μ$-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the "$μ=0$" locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset.

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arXiv2019

Tamagawa number formula with coefficients over varieties in positive characteristic

We express the order of the pole and the leading coefficient of the L-function of a (large class of) -adic coefficients (any prime) over a quasi-projective variety over a finite field of characteristic p. This is a generalization of the result of Milne-Ramachandran with coefficients. The new key ingredient is the use of F-gauges and their equivalence in the derived category with Raynaud modules proved by Ekedahl.

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arXiv2019

On the non commutative Iwasawa main conjecture for abelian varieties over function fields

We establish the Iwasawa main conjecture for semi-stable abelian varieties over a function field of characteristic $p$ under certain restrictive assumptions. Namely we consider $p$-torsion free $p$-adic Lie extensions of the base field which contain the constant $\mathbb Z_p$-extension and are everywhere unramified. Under the classical $μ=0$ hypothesis we give a proof which mainly relies on the interpretation of the Selmer complex in terms of $p$-adic cohomology [TV] together with the trace formulas of [EL1].

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arXiv2014

Pontryagin duality for Iwasawa modules and abelian varieties

We prove a functional equation for two projective systems of finite abelian $p$-groups, $\{\fa_n\}$ and $\{\fb_n\}$, endowed with an action of $\ZZ_p^d$ such that $\fa_n$ can be identified with the Pontryagin dual of $\fb_n$ for all $n$. Let $K$ be a global field. Let $L$ be a $\ZZ_p^d$-extension of $K$ ($d\geq 1$), unramified outside a finite set of places. Let $A$ be an abelian variety over $K$. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of $A$.

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arXiv2013

On the Iwasawa Main conjecture of abelian varieties over function fields

We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over $Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set of places, and semistable abelian varieties over the arithmetic $Z_p$-extension of a function field. One of the tools we use in our proof is a pseudo-isomorphism relating the duals of the Selmer groups of $A$ and its dual abelian variety $A^t$. This holds as well over number fields and is a consequence of a quite general algebraic functional equation.

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arXiv2007

Sur l'holonomie de D-modules arithmétiques associés à des F-isocristaux surconvergents sur des courbes lisses

We show that the arithmetic D-module associated to an overconvergent F-isocrystal over a smooth curve is holonomic. We first prove that unipotent F-isocrystals are holonomic D-module by using the fact that such F-isocrystals come from logarithmic F-isocrystals. We deduce the general case from the semi-stable theorem for F-isocrystals over curves of Matsuda-Trihan which relies on the p-adic monodromy theorem independently proved by André, Kedlaya and Mebkhout. The main result has already been proved by D. Caro.

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arXiv2007

On the Selmer groups of abelian varieties over function fields of characteristic p>0

In this paper, we study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We propose also a conjecture which might be considered as a counterpart of the principal conjecture in the case over a number field. \par This is a preprint which is distributed since 2005 which is still in the process of submision. Following a recent modification of some technical mistakes in the previous version of the paper as well as an amelioration of the presentation of the paper, we decide wider distribution via the archive.

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