The paper contains an expanded version of the talk delivered by the first author during the conference ALANT3 in Będlewo in June 2014. We survey recent results on independence of systems of Galois representations attached to ℓ-adic cohomology of schemes. Some other topics ranging from the Mumford-Tate conjecture and the Geyer-Jarden conjecture to applications of geometric class field theory are also considered. In addition, we have highlighted a variety of open questions which can lead to interesting...

Let $J_0\left(𝔫\right)$ be the Jacobian variety of the Drinfeld modular curve $X_0\left(𝔫\right)$ over ${𝔽}_q\left(t\right)$, where $𝔫$ is an ideal in ${𝔽}_q\left[t\right]$. Let $0\to B\to J_0\left(𝔫\right)\to A\to 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0\left(𝔫\right)$ , is stable under the action of the Hecke algebra $𝕋\subset$ End $\left(J_0\left(𝔫\right)\right)$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0\to {\Phi }_{B,\infty }\to {\Phi }_{J,\infty }\to {\Phi }_{A,\infty }\to 0$ of the Néron models of these abelian varieties over ${𝔽}_q\left[\left[\frac{1}{t}\right]\right]$. This criterion is always satisfied when either $A$ or $B$ is one-dimensional. Moreover, we prove that the sequence...

We survey methods to compute three-point branched covers of the projective line, also known as Belyĭ maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and $p$-adic methods. Along the way, we pose several questions and provide numerous examples.

Let $\Lambda$ be a maximal $𝔽{}_q\left[T\right]$-order in a division quaternion algebra over $𝔽{}_q\left(T\right)$ which is split at the place $\infty$. The present article gives an algorithm to compute a fundamental domain for the action of the group of units ${\Lambda }^{*}$ on the Bruhat-Tits tree $𝒯$ associated to ${\mathrm{PGL}}_2\left(𝔽{}_q\left((1/T)\right)\right)$. This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group ${\Lambda }^{*}$ in terms of generators and relations. Moreover we determine an upper bound...

We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a related criterion for primes p such that $16$ divides the class number of the imaginary quadratic field ℚ(√-p)....

Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero...

As a subproduct of the Schoof-Elkies-Atkin algorithm to count points on elliptic curves defined over finite fields of characteristic $p$, there exists an algorithm that computes, for $\ell$ an Elkies prime, $\ell$-torsion points in an extension of degree $\ell -1$ at cost $\tilde{O}(\ell max{(\ell ,logq)}^2)$ bit operations in the favorable case where $\ell \le p/2$.We combine in this work a fast algorithm for computing isogenies due to Bostan, Morain, Salvy and Schost with the $p$-adic approach followed by Joux and Lercier to get an algorithm valid without any limitation...

Rubinstein has produced a substantial amount of data about the even parity quadratic twists of various elliptic curves, and compared the results to predictions from random matrix theory. We use the method of Heegner points to obtain a comparable (yet smaller) amount of data for the case of odd parity. We again see that at least one of the principal predictions of random matrix theory is well-evidenced by the data.