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Florian Breuer

Publications and source records attributed to Florian Breuer.

At least 19 recordsLinked to original sources
arXiv2026

Divisibility of the coefficients of modular polynomials

Let $N>1$ and let $Φ_N(X,Y)\in\mathbb{Z}[X,Y]$ be the modular polynomial which vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of degree $N$. In this note we study the divisibility of the coefficients of $Φ_N(X+J, Y+J)$ for certain algebraic numbers $J$, in particular $J=0$ and other singular moduli. It turns out that these coefficients are highly divisible by small primes at which $J$ has supersingular reduction.

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arXiv2025

Partitions and Hecke images

We obtain a new family of relations satisfied by the partition function. In contrast with most partition relations, these involve non-trivial roots of unity. We present two proofs, one using the fact that the discriminant modular form is a multiplicative Hecke eigenform, and one direct proof using q-series.

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arXiv2025

Quadratic units and cubic fields

We investigate Eisenstein discriminants, which are squarefree integers $d \equiv 5 \pmod{8}$ such that the fundamental unit $\varepsilon_d$ of the real quadratic field $K=\mathbb{Q}(\sqrt{d})$ satisfies $\varepsilon_d \equiv 1 \pmod{2\mathcal{O}_K}$. These discriminants are related to a classical question of Eisenstein and have connections to the class groups of orders in quadratic fields as well as to real cubic fields. We present numerical computations of Eisenstein discriminants up to $10^{11}$, suggesting that their counting function up to $x$ is approximated by $π_{\mathcal{E}}(x) \approx \frac{1}{3π^2}x - 0.024x^{5/6}$. This supports a conjecture of Stevenhagen while revealing a surprising secondary term, which is similar to (but subtly different from) the secondary term in the counting function of real cubic fields. We include technical details of our computation method, which uses a modified infrastructure approach implemented on GPUs.

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arXiv2024

Drinfeld modular polynomials of level $T$

We investigate Drinfeld modular polynomials parametrizing $T$-isogenies between Drinfeld $\mathbb{F}_q[T]$-modules of rank $r\geq 2$. By providing an explicit classification of such isogenies, we derive explicit bounds on the $T$-degrees of the coefficients of the associated modular polynomials. In particular, we obtain exact expressions for the height (i.e. degree of the largest coefficient) of these modular polynomials. Numerical computations show that the bounds on the smaller coefficients are often sharp, too.

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arXiv2024

Heights of Drinfeld modular polynomials and Hecke images

We obtain explicit upper and lower bounds on the size of the coefficients of the Drinfeld modular polynomials $Φ_N$ for any monic $N\in\mathbb{F}_q[t]$. These polynomials vanish at pairs of $j$-invariants of Drinfeld $\mathbb{F}_q[t]$-modules of rank 2 linked by cyclic isogenies of degree $N$. The main term in both bounds is asymptotically optimal as $\mathrm{deg}(N)$ tends to infinity. We also obtain precise estimates on the Weil height and Taguchi height of Hecke images of Drinfeld modules of rank 2.

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arXiv2024

SeCritMass: Threshold Secret Petitions

We introduce the notion of an $n$-threshold secret petition, in which users add encrypted signatures to a petition, and the signatures are decrypted if and only if at least $n$ signatures have been gathered. This solves the coordination problem in which users wish to sign a petition or commit to a cause, but do not want to be identified as having signed it before enough others have signed it too. We present an implementation of such a petition based on the ElGamal cryptosystem. Applications include reporting misconduct in situations were complainants hesitate to come forward alone, such as in allegations of sexual harassment or police brutality.

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arXiv2023

Explicit bounds on the coefficients of the modular polynomials and the size of $X_0(N)$

We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials $Φ_N$ for the elliptic $j$-function. These bounds make explicit the best previously known asymptotic bounds. We then give an explicit version of Silverman's Hecke points estimates. Finally, we give an asymptotic comparison between the Faltings height of the modular curve $X_0(N)$ and the height of the modular polynomial $Φ_N$.

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arXiv2020

Heights and isogenies of Drinfeld modules

We provide explicit bounds on the difference of heights of isogenous Drinfeld modules. We derive a finiteness result in isogeny classes. In the rank 2 case, we also obtain an explicit upper bound on the size of the coefficients of modular polynomials attached to Drinfeld modules.

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arXiv2019

The generic monodromy of Drinfeld modular varieties in special characteristic

By combining theorems of Drinfeld and Strauch, we show that the monodromy representation on the special fibre of a Drinfeld modular variety, with level not divisible by the characteristic, is surjective. We illustrate this result in the special case of Drinfeld $\mathbb{F}_q[t]$-modules in level $t$, and apply this to show that the Kronecker factors of a Drinfeld modular polynomial in rank $r$ are irreducible.

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arXiv2019

Lower bounds for periods of Ducci sequences

A Ducci sequence is a sequence of integer $n$-tuples obtained by iterating the map \[ D : (a_1, a_2, \ldots, a_n) \mapsto \big(|a_1-a_2|,|a_2-a_3|,\ldots,|a_n-a_1|\big). \] Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given $n$. We prove lower bounds for $P(n)$ by counting certain partitions.

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arXiv2018

Drinfeld modular forms of arbitrary rank, Part I: Analytic Theory

This is the first of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank r. In the present part, we develop the analytic theory. Most of the work goes into defining and studying the u-expansion of a weak Drinfeld modular form, whose coefficients are weak Drinfeld modular forms of rank r-1. Based on that we give a precise definition of when a weak Drinfeld modular form is holomorphic at infinity and thus a Drinfeld modular form in the proper sense.

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arXiv2018

Drinfeld modular forms of arbitrary rank, Part II: Comparison with Algebraic Theory

This is the second of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present part, we compare the analytic theory with the algebraic one that was begun in a paper of the third author. For any arithmetic congruence subgroup and any integral weight we establish an isomorphism between the space of analytic modular forms with the space of algebraic modular forms defined in terms of the Satake compactification. From this we deduce the important result that this space is finite dimensional.

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arXiv2018

Drinfeld modular forms of arbitrary rank, Part III: Examples

This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define Eisenstein series, as well as the action of Hecke operators upon them, coefficient forms and discriminant forms. In the special case A=F_q[t] we show that all modular forms for GL_r(Γ(t)) are generated by certain weight one Eisenstein series, and all modular forms for GL_r(A) and SL_r(A) are generated by certain coefficient forms and discriminant forms. We also compute the dimensions of the spaces of such modular forms.

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arXiv2016

A note on Gekeler's h-function

We give a brief introduction to Drinfeld modular forms, concentrating on the many equivalent constructions of the form h of weight q+1 and type 1, to which we contribute some new characterizations involving Moore determinants, and an application to the Weil pairing on Drinfeld modules. We also define Drinfeld modular functions of non-zero type and provide a moduli interpretation of these.

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arXiv2015

Drinfeld modular polynomials in higher rank II: Kronecker congruences

This is a sequel to the paper [F. Breuer, H.-G. Rück, Drinfeld modular polynomials in higher rank, J. Number Theory 129 (2009), 59-83.], in which we introduced Drinfeld modular polynomials of higher rank, using an analytic construction. These polynomials relate the isomorphism invariants of Drinfeld F_q[T]-modules of rank r\geq 2 linked by isogenies of a specified type. In the current paper, we give an algebraic construction of greater generality, and prove a generalization of the Kronecker congruences relations, which describe what happens when modular polynomials associated to P-isogenies are reduced modulo a prime P. We also correct an error in [loc. cit.].

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arXiv2015

Galois groups associated to generic Drinfeld modules and a conjecture of Abhyankar

Let $ϕ$ be a rank $r$ Drinfeld $\BF_q[T]$-module determined by $ϕ_T(X) = TX+g_1X^q+...+g_{r-1}X^{q^{r-1}}+X^{q^r}$, where $g_1,...,g_{r-1}$ are algebraically independent over $\BF_q(T)$. Let $N\in\BF_q[T]$ be a polynomial, and $k/\BF_q$ an algebraic extension. We show that the Galois group of $ϕ_N(X)$ over $k(T,g_1,...,g_{r-1})$ is isomorphic to $\GL_r(\BF_q[T]/N\BF_q[T])$, settling a conjecture of Abhyankar.

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