Formal groups and congruences

Congruences between Selmer groups ∗

The study of congruences between arithmetically interesting numbers has a long history and plays important roles in several areas of number theory. Examples of such congruences include the Kummer congruences between Bernoulli numbers and congruences between coefficients of modular forms. Many of these congruences could be interpreted as congruences between special values of L-functions of arith...

Congruences via Abelian Groups

Given a group G acting on a set S, Mobius inversion over the lattice of subgroups can be used to obtain congruences relating the number of elements of S stabilized by each subgroup. By taking S to be a set of subsets, partitions, or permutations congruences for binomial and multinomial coefficients. Stirling numbers of both kinds, and various other combinatorial sequences are derived. Congruenc...

Formations of Monoids, Congruences, and Formal Languages

The main goal in this paper is to use a dual equivalence in automata theory started in [RBBCL13] and developed in [BBCLR14] to prove a general version of the Eilenberg-type theorem presented in [BBPSE12]. Our principal results confirm the existence of a bijective correspondence between formations of (non-necessarily finite) monoids, that is, classes of monoids closed under taking epimorphic ima...

Congruences between the Coefficients of the Tate Curve via Formal Groups

Let Eq : Y 2 + XY = X3 + h4X + h6 be the Tate curve with canonical differential, ω = dX/(2Y + X). If the characteristic is p > 0, then the Hasse invariant, H, of the pair (Eq, ω) should equal one. If p > 3, then calculation of H leads to a nontrivial separable relation between the coefficients h4 and h6. If p = 2 or p = 3, Thakur related h4 and h6 via elementary methods and an identity of Raman...

Errata to "Formations of Monoids, Congruences, and Formal Languages"