Explicit Computation of a Galois Representation Attached to an Eigenform Over $${\text {SL}}_3$$ from the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ of a Surface

We describe a method to compute mod $$\ell $$ ℓ Galois representations contained in the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ H e ´ t 2 of surfaces. We apply this method to the case of a representation with values in $${\text {GL}}_3(\mathbb {F}_9)$$ GL 3 ( F 9 ) attached to an eigenform over a congruence subgroup of $${\text {SL}}_3$$ SL 3 . We obtain, in particular, a polynomial with Galois group isomorphic to the simple group $${\text {PSU}}_3(\mathbb {F}_9)$$ PSU 3 ( F 9 ) and ramified at 2 and 3 only.

Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products

We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial $$f(x) = c_0 + c_1 x^{d_1} + \cdots + c_k x^{d_k}$$ f ( x ) = c 0 + c 1 x d 1 + ⋯ + c k x d k by varying its coefficients. If the GCD of the exponents is d, then the polynomial admits the change of variable $$y=x^d$$ y = x d , and its roots split into necklaces of length d. At best we can expect to permute these necklaces, i.e. the Galois group of f equals the wreath product of the symmetric group over $$d_k/d$$ d k / d elements and $${\mathbb {Z}}/d{\mathbb {Z}}$$ Z / d Z . We study the multidimensional generalization of this equality: the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich.

On the p-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

Let E/K be a finite Galois extension of totally real number fields with Galois group G. Let p be an odd prime and let $$r>1$$ r > 1 be an odd integer. The p-adic Beilinson conjecture relates the values at $$s=r$$ s = r of p-adic Artin L-functions attached to the irreducible characters of G to those of corresponding complex Artin L-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the ‘p-part’ of the equivariant Tamagawa number conjecture for the pair $$(h^0(\mathrm {Spec}(E))(r), \mathbb {Z}[G])$$ ( h 0 ( Spec ( E ) ) ( r ) , Z [ G ] ) . If $$r>1$$ r > 1 is even we obtain a similar result for Galois CM-extensions after restriction to ‘minus parts’.

Monogenic trinomials with non-squarefree discriminant

For each integer [Formula: see text], we identify new infinite families of monogenic trinomials [Formula: see text] with non-squarefree discriminant, many of which have small Galois group. Moreover, in certain situations when [Formula: see text] with fixed [Formula: see text] and [Formula: see text], we produce asymptotics on the number of such trinomials with [Formula: see text].

Taylor coefficients of Anderson generating functions and Drinfeld torsion extensions

We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating functions and their hyperderivatives at roots of unity. We also give a direct proof that the image of the Galois representation attached to the [Formula: see text]-adic Tate module lies in the [Formula: see text]-adic points of the motivic Galois group. This is a generalization of the corresponding result of Chang and Papanikolas for the [Formula: see text]-adic case.