Polylogarithms are those multiple polylogarithms that factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers, and Brown, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of [Formula: see text]. To do so, we develop a greatly refined version of the algorithm of Dan-Cohen tailored specifically to this case, and we focus attention on the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. We also discover an interesting consequence of the symmetry-breaking nature of the polylog quotient that forces us to symmetrize our polylogarithmic version of Kim’s conjecture. In this first part of a two-part series, we focus on a specific example, which allows us to verify an interesting new case of Kim’s conjecture.
Given a partition λ of n, consider the subspace E λ of C n where the first λ 1 coordinates are equal, the next λ 2 coordinates are equal, etc. In this paper, we study subspace arrangements X λ consisting of the union of translates of E λ by the symmetric group. In particular, we focus on determining when X λ is Cohen-Macaulay. This is inspired by previous work of the third author coming from the study of rational Cherednik algebras and which answers the question positively when all parts of λ are equal. We show that X λ is not Cohen-Macaulay when λ has at least 4 distinct parts, and handle a large number of cases when λ has 2 or 3 distinct parts. Along the way, we also settle a conjecture of Sergeev and Veselov about the Cohen-Macaulayness of algebras generated by deformed Newton sums. Our techniques combine classical techniques from commutative algebra and invariant theory; in many cases we can reduce an infinite family to a finite check which can sometimes be handled by computer algebra.
This is the second installment in a multi-part series starting with Corwin–Dan-Cohen [arXiv:1812.05707v3]. Building on previous work by Dan-Cohen–Wewers, Dan-Cohen, and F. Brown, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of Spec Z \operatorname {Spec}\mathbb {Z} . To do so, we develop a refined version of the algorithm of Dan-Cohen–Wewers tailored specifically to this case. We also commit ourselves fully to working with the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth- 1 1 part of the mixed Tate Galois group studied extensively by Goncharov. An application was given in Corwin–Dan-Cohen [arXiv:1812.05707v3], where we verified Kim’s conjecture in an interesting new case.
In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL2( Z). In [Gre10], Greicius develops necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius' methods in several directions. First, we consider the analogous problem for elliptic curves with full 2-torsion. Following Greicius, we obtain necessary and sufficient conditions for the associated adelic representation to be maximal and also develop a battery of computationally effective tests that can be used to verify these conditions. We are able to use our tests to construct an infinite family of curves over Q(α) with maximal image, where α is the real root of x 3 + x + 1. Next, we extend Greicius' tests to more general settings, such as nonsemistable elliptic curves over arbitrary cubic number fields. Finally, we give a general discussion concerning such problems for arbitrary torsion subgroups.
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