arXiv subjects
Publications and source records attributed to Andreas Schweizer.
We prove the rank one case of Skolem's Conjecture on the exponential local-global principle for algebraic functions and discuss its analog for meromorphic functions.
We show that if a complex entire function $f$ and its derivative $f'$ share their simple zeroes and their simple $a$-points for some nonzero constant $a$, then $f\equiv f'$. We also discuss how far these conditions can be relaxed or generalized. Finally, we determine all entire functions $f$ such that for 3 distinct complex numbers $a_1,a_2,a_3$ every simple $a_j$-point of $f$ is an $a_j$-point of $f'$.
Let $K$ be an algebraic function field with constant field ${\mathbb F}_q$. Fix a place $\infty$ of $K$ of degree $δ$ and let $A$ be the ring of elements of $K$ that are integral outside $\infty$. We give an explicit description of the elliptic points for the action of the Drinfeld modular group $G=GL_2(A)$ on the Drinfeld's upper half-plane $Ω$ and on the Drinfeld modular curve $G\!\setminus\!Ω$. It is known that under the {\it building map} elliptic points are mapped onto vertices of the {\it Bruhat-Tits tree} of $G$. We show how such vertices can be determined by a simple condition on their stabilizers. Finally for the special case $δ=1$ we obtain from this a surprising free product decomposition for $PGL_2(A)$.
Let $f$ be a meromorphic function. We suggest a generalization of $f$ and its derivative $f'$ sharing a nonzero value $a$ IM that does not impose any a priori restrictions on the ramification of $f$. Then we discuss some results around the question whether the famous theorem on entire functions $f$ that share two values IM with $f'$ still holds for this weaker notion.
Let $K$ be an algebraic function field of one variable with constant field $k$ and let $C$ be the Dedekind domain consisting of all those elements of $K$ which are integral outside a fixed place $\infty$ of $K$. When $k$ is finite the group $GL_2(C)$ plays a central role in the theory of Drinfeld modular curves analagous to that played by $SL_2(Z)$ in the classical theory of modular forms. When $k$ is finite (resp. infinite) we refer to a group $GL_2(C)$ as an arithmetic (resp. non-arithmetic) Drinfeld modular group. Associated with $GL_2(C)$ is its Bruhat-Tits tree, $T$. The structure of the group is derived from that of the quotient graph $GL_2(C)\backslash T$. Using an elementary approach which refers explicitly to matrices we determine the structure of all the vertex stabilizers of $T$. This extends results of Serre, Takahashi and the authors. We also determine all possible valencies of the vertices of $GL_2(C)\backslash T$ for the important special case where $\infty$ has degree 1.
Let $X$ be a compact Riemann surface of genus $g\geq 2$. A cyclic subgroup of prime order $p$ of $Aut(X)$ is called properly $(p,h)$-gonal if it has a fixed point and the quotient surface has genus $h$. We show that if $p>6h+6$, then a properly $(p,h)$-gonal subgroup of $Aut(X)$ is unique. We also discuss some related results.
We extend some algebraic properties of the classical modular group SL_2(Z) to equivalent groups in the theory of Drinfeld modules, in particular properties which are important in the theory of modular curves. We study cusp amplitudes and the level of a (congruence) subgroup of SL_2(D) for any Dedekind domain D, as ideals of D. In particular, we extend a remarkable result of Larcher. We introduce finer notions of quasi-amplitude and quasi-level, which are not required to be ideals and encode more information about the subgroup. Our results also provide several new necessary conditions for a subgroup of SL_2(D) to be a congruence subgroup.
We explicitly determine the elliptic K3 surfaces with a maximal singular fibre. If the characteristic of the ground field is different from 2, for each of the two possible maximal fibre types, $I_{19}$ and $I^*_{14}$, the surface is unique. In characteristic 2 the maximal fibre types are $I_{18}$ and $I^*_{13}$, and there exist two (resp. one) one-parameter families of such surfaces.
We continue work of Gekeler and others on elliptic curves over ${\mathbb F}_q(T)$ with conductor $\infty\cdot{\mathfrak n}$ where ${\mathfrak n}\in{\mathbb F}_q[T]$ has degree 3. Because of the Frobenius isogeny there are infinitely many curves in each isogeny class, and we discuss in particular which of these curves is the strong Weil curve with respect to the uniformization by the Drinfeld modular curve $X_0({\mathfrak n})$. As a corollary we obtain that the strong Weil curve $E/{\mathbb F}_q(T)$ always gives a rational elliptic surface over $\bar{{\mathbb F}_q}$.
Let K be a function field with constant field k and let "infinity" be a fixed place of K. Let C be the Dedekind domain consisting of all those elements of K which are integral outside "infinity". The group G=GL_2(C) is important for a number of reasons. For example, when k is finite, it plays a central role in the theory of Drinfeld modular curves. Many properties follow from the action of G on its associated Bruhat-Tits tree, T. Classical Bass-Serre theory shows how a presentation for G can be derived from the structure of the quotient graph (or fundamental domain) G\T. The shape of this quotient graph (for any G) is described in a fundamental result of Serre. However there are very few known examples for which a detailed description of G\T is known. (One such is the rational case, C=k[t], i.e. when K has genus zero and "infinity" has degree one.) In this paper we give a precise description of G\T for the case where the genus of K is zero, K has no places of degree one and "infinity" has degree two. Among the known examples a new feature here is the appearance of vertex stabilizer subgroups (of G) which are of quaternionic type.
We present some results on two meromorphic functions from S to the Riemann sphere sharing a number of values where S is a Riemann surface of one of the following types: compact, compact minus finitely many points, the unit disk, a torus, the complex plane.
We show that the Davenport-Stothers inequality from characteristic 0 fails in any characteristic p>3. The proof uses elliptic surfaces over the projective line and inseparable base change. We then present adjusted inequalities. These follow from results of Pesenti-Szpiro. For characteristic 2 and 3, we achieve a similar result in terms of the maximal singular fibres of elliptic surfaces over the projective line. Our ideas are also related to supersingular surfaces (in Shioda's sense).