arXiv subjects

Andreas Mihatsch

Publications and source records attributed to Andreas Mihatsch.

12 recordsLinked to original sources

arXiv2025

Unitary Shimura varieties at ramified primes and arithmetic transfer

We consider unitary Shimura varieties at places where the totally real field ramifies over $\mbQ$. Our first result constructs comparison isomorphisms between absolute and relative local models in this context, which relies on a reformulation of the Eisenstein condition of Rapoport--Zink and Rapoport--Smithling--Zhang. Related to that, we also provide a moduli description for the integral models of RSZ unitary Shimura varieties in new cases. Our second result lifts the comparison of local models to categories of $p$-divisible groups and, as a corollary, to various kinds of Rapoport--Zink spaces. Our third result is a proof of the arithmetic transfer conjecture of the third author in full generality. Using our statements about Rapoport--Zink spaces, we extend the previous proof from the unramified case to that of all $p$-adic local fields ($p$ odd).

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arXiv2025

Gaussian test functions and Jacquet-Rallis transfer

We construct Gaussian test functions for the general linear side of the Jacquet-Rallis relative trace formula comparison. These are functions which are defined in terms of their orbital integrals and transfer to the compact unitary group. Our construction relies on the formalism of Kudla-Millson and simple geometric properties of symmetric spaces. In particular, it also provides an explicit formula in terms of the Howe operator.

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arXiv2024

$δ$-Forms on Lubin--Tate Space

We extend Gubler--Künnemann's theory of $δ$-forms from algebraic varieties to good Berkovich spaces. This is based on the observation that skeletons in such spaces satisfy a tropical balance condition. Our main result is that complete intersection formal models of cycles give rise to Green $δ$-forms for their generic fibers. We moreover show that, in certain situations, intersection numbers on formal models are given by the $\star$-product of their Green currents. In this way, we generalize some results for divisor intersection to higher codimension situations. We illustrate the mentioned results in the context of an intersection problem for Lubin--Tate spaces.

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arXiv2024

Arithmetic transfer for inner forms of $GL_{2n}$

We formulate Guo--Jacquet type fundamental lemma conjectures and arithmetic transfer conjectures for inner forms of $GL_{2n}$. Our main results confirm these conjectures for division algebras of invariant $1/4$ and $3/4$.

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arXiv2024

A linear AFL for quaternion algebras

We prove new fundamental lemma and arithmetic fundamental lemma identities for general linear groups over quaternion division algebras. In particular, we verify the transfer conjeture and the arithmetic transfer conjecture from arXiv:2307.11716 in cases of Hasse invariant 1/2.

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arXiv2024

On the Linear AFL: The Non-Basic Case

The linear Arithmetic Fundamental Lemma (AFL) conjecture compares intersection numbers on Lubin--Tate deformation spaces with derivatives of orbital integrals. It has been introduced for elliptic orbits in arXiv:1803.07553 and arXiv:2010.07365. In these cases, the relevant intersection problem is formulated for the basic isogeny class. In the present article, we extend the theory to all orbits and all isogeny classes. Our main result is a reduction of the non-basic cases of the AFL to the basic ones, which is achieved by exploiting the connected-étale sequence. Our theory will be relevant in the global setting, where also locally non-elliptic orbits may contribute in a non-trivial way.

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arXiv2022

On Tropical Intersection Theory

We develop a tropical intersection formalism of forms and currents that extends classical tropical intersection theory in two ways. First, it allows to work with arbitrary polytopes, also non-rational ones. Second, it allows for smooth differential forms as coefficients. The intersection product in our formalism can be defined through the diagonal intersection method of Allermann--Rau or the fan displacement rule. We prove with a limiting argument that both definitions agree.

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arXiv2022

On the Arithmetic Fundamental Lemma conjecture over a general $p$-adic field

We prove the arithmetic fundamental lemma conjecture over a general $p$-adic field with odd residue cardinality $q\geq \dim V$. Our strategy is similar to the one used by the second author during his proof of the AFL over $\mathbb{Q}_p$ (arXiv:1909.02697), but only requires the modularity of divisor generating series on the Shimura variety (as opposed to its integral model). The resulting increase in flexibility allows us to work over an arbitrary base field. To carry out the strategy, we also generalize results of Howard (arXiv:1303.0545) on CM-cycle intersection and of Ehlen--Sankaran (arXiv:1607.06545) on Green function comparison from $\mathbb{Q}$ to general totally real base fields.

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arXiv2021

Local Constancy of Intersection Numbers

We prove that, in certain situations, intersection numbers on formal schemes that come in profinite families vary locally constantly in the parameter. To this end, we define the product $S\times M$ of a profinite set $S$ with a locally noetherian formal scheme $M$ and study intersections thereon. Our application is to the Arithmetic Fundamental Lemma of W. Zhang where the result helps to remove a restriction in its recent proof, cf. arXiv:1909.02697.

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arXiv2019

Relative unitary RZ-spaces and the Arithmetic Fundamental Lemma

We verify new cases of the Arithmetic Fundamental Lemma (AFL) of Wei Zhang. This relies on a recursive algorithm which allows, under certain conditions, to reduce the AFL identity in question to an AFL identity in lower dimension. The main ingredient for this reduction is a comparison isomorphism between different moduli problems of PEL-type for p-divisible groups. The construction of this comparison isomorphism is based on the theory of relative displays and frames, as developed by Tobias Ahsendorf, Eike Lau and Thomas Zink.

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arXiv2015

An Arithmetic Transfer Identity

We prove a variant of the Arithmetic Fundamental Lemma conjecture of Wei Zhang for n=2. More precisely, we consider the deformation lengths of certain quasi-homomorphisms of quasi-canonical lifts in the sense of Gross. We prove the existence of a test function on a symmetric space related to GL_2 whose orbital integrals over GL_1 equal the deformation lengths in question.

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arXiv2015

On the Arithmetic Fundamental Lemma through Lie algebras

We prove that the Arithmetic Fundamental Lemma conjecture of Wei Zhang is equivalent to a similar conjecture, but for Lie algebras, in the case of non-degenerate intersection. We use this result to give a simplified proof of the AFL for $n=3$. The idea for the reduction to the Lie algebra is due to Wei Zhang.

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