arXiv subjects

Romain Branchereau

Publications and source records attributed to Romain Branchereau.

8 recordsLinked to original sources

arXiv2026

Theta correspondence and the Borisov-Gunnells relations

We consider a geometric theta correspondence from the first homology of a modular curve, to modular forms of weight $2$. Using Stevens' description of the homology, we find that this map sends modular symbols to product of weight one Eisenstein series, modular caps to weight $2$ Eisenstein series, and hyperbolic cycles to diagonal restrictions of Hilbert-Eisenstein series. We use it to revisit work of Borisov and Gunnells, and explain its connection to a theorem of Li. In particular, we give a geometric proof of certain relations between Eisenstein series.

math.NT↗

arXiv2026

Kudla-Millson lift of toric cycles and restriction of Hilbert modular forms

Let $V$ be quadratic space of even dimension and of signature $(p, q)$ with $p \geq q > 0$. We show that the Kudla-Millson lift of toric cycles - attached to algebraic tori - is a cusp form that is the diagonal restriction of a Hilbert modular form of parallel weight one. We deduce a formula relating the dimension of the span of such diagonal restrictions and the dimension of the span of toric and special cycles.

math.NT↗

arXiv2026

Eisenstein classes and generating series of modular symbols in $\mathrm{SL}_N$

We define a theta lift between the homology in degree $N-1$ of a locally symmetric space associated to $\mathrm{SL}_N(\mathbb{R})$ and the space of modular forms of weight $N$, similar to the Kudla-Millson lift in the orthogonal setting. We show that the Fourier coefficients of this lift are Poincaré duals of modular symbols associated to maximal parabolic subgroups. The constant term is a canonical cohomology classes obtained by transgressing the Euler class of a torus bundle. When $N=2$, we show that the lift surjects on the space of weight 2 modular forms spanned by an Eisenstein series and the eigenforms with non-vanishing L-function.

math.NT↗

arXiv2025

A regularized theta lift on the symmetric space of $SL_N$

We define a regularized lift from harmonic weak Maass forms of weight $2-N$ to differential forms of degree $N-1$ on the symmetric space $\SL_N(\R)/\SO(N)$, that are smooth outside of certain modular symbols. We show that this lift is adjoint to the derivative of a theta lift. We compute periods of the regularized lift over tori and relate them to Fourier coefficients of Hilbert-Eisenstein series.

math.NT↗

arXiv2024

An upper bound on the denominator of Eisenstein classes in Bianchi manifolds

A general conjecture of Harder relates the denominator of the Eisenstein cohomology of certain locally symmetric spaces to special values of $L$-functions. In this paper we consider the locally symmetric space $\operatorname{SL}_2(\mathcal{O}) \backslash \mathbb{H}_3$ where $\mathcal{O}$ is the ring of integers of an imaginary quadratic field $K$ and $\mathbb{H}_3$ is the hyperbolic $3$-space. Tobias Berger proves a lower bound on the denominator of the Eisenstein cohomology in certain cases. The goal of this paper is to show how results of Ito and Sczech can be used to prove an upper bound on the denominator in terms of a special value of a Hecke $L$-function. When the class number of $K$ is one, we combine this result with Berger's result to obtain the exact denominator.

math.NT↗

arXiv2023

Maximal operators on hyperbolic triangles

We characterize the boundedness properties on the spaces $L^p(\mathbb{H}^2)$ of the maximal operator $M_\mathcal{B}$ where $\mathcal{B}$ is an arbitrary family of hyperbolic triangles stable by isometries.

math.CA↗

arXiv2022

Diagonal restriction of Eisenstein series and Kudla-Millson theta lift

We consider the Kudla-Millson theta series associated to a quadratic space of signature $(N,N)$. By combining a `see-saw' argument with the Siegel-Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree $N$ is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon-Pozzi-Vonk to totally real fields.

math.NT↗

arXiv2022

The Kudla-Millson form via the Mathai-Quillen formalism

In \cite{km2}, Kudla and Millson constructed a $q$-form $φ_{KM}$ on an orthogonal symmetric space using Howe's differential operators. It is a crucial ingredient in their theory of theta lifting. This form can be seen as a Thom form of a real oriented vector bundle. In \cite{mq} Mathai and Quillen constructed a {\em canonical} Thom form and we show how to recover the Kudla-Millson form via their construction. A similar result was obtained by \cite{garcia} for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to an arbitrary signature.

math.NT↗