Fundamental domains for congruence subgroups of SL2 in positive characteristic

In this work, we construct fundamental domains for congruence subgroups of $SL_2(F_q[t])$ and $PGL_2(F_q[t])$. Our method uses Gekeler's description of the fundamental domains on the Bruhat- Tits tree $X = X_{q+1}$ in terms of cosets of subgroups. We compute the fundamental domains for a number of congruence subgroups explicitly as graphs of groups using the computer algebra system Magma

Lattices with and lattices without spectral gap

International audienceLet $G = G(k)$ be the $k$-rational points of a simple algebraic group G over a local field k and let $\Gamma$ be a lattice in G. We show that the regular representation $\rho_{\Gamma\setminus G}$ of $G$ on $L^{2}(\Gamma\setminus G)$ has a spectral gap, that is, the restriction of $\rho_{\Gamma\setminus G}$ to the orthogonal of the constants in $L^{2}(\Gamma\setminus G)$ has no almost invariant vectors. On the other hand, we give examples of locally compact simple groups $G$ and lattices $\Gamma$ for which $L^{2}(\Gamma\setminus G)$ has no spectral gap. This answers in the negative a question asked by Margulis [Marg91, Chapter III, 1.12]. In fact, $G$ can be taken to be the group of orientation preserving automorphisms of a k-regular tree for $k > 2$

Graph Laplacians, component groups and Drinfeld modular curves

Let $\frak{p}$ be a prime ideal of $\mathbb{F}_q[T]$. Let $J_0(\frak{p})$ be the Jacobian variety of the Drinfeld modular curve $X_0(\frak{p})$. Let $\Phi$ be the component group of $J_0(\frak{p})$ at the place $1/T$. We use graph Laplacians to estimate the order of $\Phi$ as $\mathrm{deg}(\frak{p})$ goes to infinity. This estimate implies that $\Phi$ is not annihilated by the Eisenstein ideal of the Hecke algebra $\mathbb{T}(\frak{p})$ acting on $J_0(\frak{p})$ once the degree of $\frak{p}$ is large enough. We also obtain an asymptotic formula for the size of the discriminant of $\mathbb{T}(\frak{p})$ by relating this discriminant to the order of $\Phi$; in this problem the order of $\Phi$ plays a role similar to the Faltings height of classical modular Jacobians. Finally, we bound the spectrum of the adjacency operator of a finite subgraph of an infinite diagram in terms of the spectrum of the adjacency operator of the diagram itself; this result has applications to the gonality of Drinfeld modular curves

A combinatorial Li-Yau inequality and rational points on curves

We present a method to control gonality of nonarchimedean curves based on graph theory. Let k denote a complete nonarchimedean valued field.We first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally,we apply the results to give a lower bound for the gonality of arbitraryDrinfeld modular curves over finite fields and for general congruence subgroups Γ of Γ (1) that is linear in the index [Γ (1) : Γ ], with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian

Drinfeld modules may not be for isogeny based cryptography

Elliptic curves play a prominent role in cryptography. For instance, the hardness of the elliptic curve discrete logarithm problem is a foundational assumption in public key cryptography. Drinfeld modules are positive characteristic function field analogues of elliptic curves. It is natural to ponder the existence/security of Drinfeld module analogues of elliptic curve cryptosystems. But the Drinfeld module discrete logarithm problem is easy even on a classical computer. Beyond discrete logarithms, elliptic curve isogeny based cryptosystems have have emerged as candidates for post-quantum cryptography, including supersingular isogeny Diffie-Hellman (SIDH) and commutative supersingular isogeny Diffie-Hellman (CSIDH) protocols. We formulate Drinfeld module analogues of these elliptic curve isogeny based cryptosystems and devise classical polynomial time algorithms to break these Drinfeld analogues catastrophically