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Florian Breuer

Publications and source records attributed to Florian Breuer.

28 records · Page 2Linked to original sources
arXiv2015

Explicit Drinfeld moduli schemes and Abhyankar's generalized iteration conjecture

Let $k$ be a field containing $\mathbb{F}_q$. Let $ψ$ be a rank $r$ Drinfeld $\mathbb{F}_q[t]$-module determined by $ψ_t(X) = tX+a_1X^q+\cdots+a_{r-1}X^{q^{r-1}}+X^{q^r}$, where $t,a_1,\ldots,a_{r-1}$ are algebraically independent over $k$. Let $n\in\mathbb{F}_q[T]$ be a monic polynomial. We show that the Galois group of $ψ_n(X)$ over $k(t,a_1,\ldots,a_{r-1})$ is isomorphic to $\mathrm{GL}_r(\mathbb{F}_q[t]/n\mathbb{F}_q[t])$, settling a conjecture of Abhyankar. Along the way we obtain an explicit construction of Drinfeld moduli schemes of level $tn$.

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arXiv2009

Special subvarieties of Drinfeld modular varieties

We explore an analogue of the André-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety $X$ of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if $X$ is a "special" subvariety (i.e. $X$ is defined by requiring additional endomorphisms). We prove this conjecture in two cases. Firstly when $X$ contains a Zariski-dense set of CM points with a certain behaviour above a fixed prime (which is the case if these CM points lie in one Hecke orbit), and secondly when $X$ is a curve containing infinitely many CM points without any additional assumptions.

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arXiv2008

Torsion bounds for elliptic curves and Drinfeld modules

We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and Pink-Ruetsche, respectively. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.

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arXiv2006

Heegner points and the rank of elliptic curves over large extensions of global fields

Let k be a global field, $\bar{k}$ a separable closure of k, and $G_k$ the absolute Galois group $\Gal(\bar{k}/k)$ of $\bar{k}$ over k. For every g in $G_k$, let $\bar{k}^g$ be the fixed subfield of $\bar{k}$ under g. Let E/k be an elliptic curve over k. We show that for each g in $G_k$, the Mordell-Weil group $E(\bar{k}^g)$ has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametrized by a Drinfeld modular curve, and secondly when k is a totally real number field and E/k is parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points on E defined over ring class fields.

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arXiv2004

The Andre-Oort conjecture for products of Drinfeld modular curves

Let $Z=X_1\times...\times X_n$ be a product of Drinfeld modular curves. We characterize those algebraic subvarieties $X \subset Z$ containing a Zariski-dense set of CM points, i.e. points corresponding to $n$-tuples of Drinfeld modules with complex multiplication (and suitable level structure). This is a characteristic $p$ analogue of a special case of the André-Oort conjecture. We follow closely the approach used by Bas Edixhoven in characteristic zero, see math.NT/0302138. Note that in this paper we assume that the characteristic $p$ is odd, and we only treat the case of Drinfeld $F_q[T]$-modules.

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arXiv2004

CM points on products of Drinfeld modular curves

Let X be a product of Drinfeld modular curves over a general base ring A of odd characteristic. We classify those subvarieties of X which contain a Zariski-dense set of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function fields.

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arXiv2004

Monodromy Groups associated to Non-Isotrivial Drinfeld Modules in Generic Characteristic

Let $ϕ$ be a non-isotrivial family of Drinfeld A-modules of rank r in generic characteristic with a suitable level structure over a connected smooth algebraic variety X. Suppose that the endomorphism ring of $ϕ$ is equal to A. Then we show that the closure of the analytic fundamental group of X in $SL_r(\mathbb{A}_F^f)$ is open, where $\mathbb{A}_F^f$ denotes the ring of finite adeles of the quotient field F of A. From this we deduce two further results: (1) If X is defined over a finitely generated field extension of F, the image of the arithmetic étale fundamental group of X on the adelic Tate module of $ϕ$ is open in $GL_r(\mathbb{A}_F^f)$. (2) Let $ψ$ be a Drinfeld A-module of rank r defined over a finitely generated field extension of F, and suppose that $ψ$ cannot be defined over a finite extension of F. Suppose again that the endomorphism ring of $ψ$ is A. Then the image of the Galois representation on the adelic Tate module of $ψ$ is open in $GL_r(\mathbb{A}_F^f)$. Finally, we extend the above results to the case of arbitrary endomorphism rings.

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arXiv2004

Images of isogeny classes on modular elliptic curves

Let K be a number field and E/K a modular elliptic curve, with modular parametrization $X_0(N) \to E$ defined over K. The purpose of this note is to study the images in E of classes of isogenous points in X_0(N).

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arXiv2003

Higher Heegner points on elliptic curves over function fields

Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E, defined over a $Z_p^{\infty}$-tower of finite extensions of k, and show that these Heegner points generate a group of infinite rank. This is a function field analogue of a result of C.Cornut and V.Vatsal

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