User:MPark.Shay/sandbox

Project Title
Below is the outline of Dr. Fried's research on PROJECTNAMEHERE. To read more about this project in full detail, see Dr. Fried's "Research Explanation".

0. Prelude on Topics and Methods
Fried's major work has predominated in the geomety and arithmetic of families of nonsingular projective curve covers. The absolute Galois group of a perfect field K, GK, is the profinite group of automorphisms of an algebraic closure of K.

He often was intent on relating such geometric objects defined over K to GK. The basic geometric objects are normal varieties over a field K. Variety means that there is one component only defined over K , but that allows that there might be several components over, say, the algebraic closure of K. Such a variety has a function field (over K) and therefore the notion of the normalization of that variety in its function field. That normalization is unique. A variety whose singular set has codimension two is already normal. Throughout there is little loss in just assuming all varieties are normal, and usually they are nonsingular. Also, except for considering reductions of covers modulo primes of a number field (and then we must also assume the cover is separable ), we usually assume K is contained in the complex numbers. A cover of algebraic varieties is a finite, flat morphism, &#966;: X &#8594; Y. But. if the domain ( X ) and range ( Y ) are both nonsingular, then flatness of a finite map is automatic. A cover (defined over K) is Galois if the order of the group of automorphisms (also defined over K) of the domain that commute with the map to the range is the same as the degree, n = n &#966; of the cover. The assumption that X and Y are normal, means that the automorphisms of the cover correspond to the automorphisms of the function field of X that fix the function field of Y. The notation Z / t denotes the cyclic group of order t , and the notation Z p, with p a prime denotes the p-adic integers. Arithmetic vs Geometric Monodromy of a cover: Every cover has a (minimal) Galois closure cover. So Galois covers are cofinal in all covers of a given fixed algebraic variety. If &#966; is the covering map. then G &#966; refers to the group of automorphisms over an algebraic closure, K alg. This is the geometric monodromy of &#966;. There is also a Galois closure cover over any definition field K of &#966;: producing the arithmetic monodromy (group, over K ), G&#966;ar. It contains G&#966;as a normal subgroup. There is a canonical permutation representation, T&#966;, attached to G&#966;ar, a faithful, transitive, homomorphism into the symmetric group Sn. Such a representation is primitive if the subgroup stabilizing an integer is maximal in the group. It is typical for applications to start with Y  absolutely irreducible : Irreducible over the algebraic closure, K alg, of K, and usually so is X. Even if both are absolutely irreducible, significantly, we don't expect G&#966;ar and G&#966;to be equal. An example of great significance is the (G&#966;ar, G&#966;) pair [#Modular_curves_form_a_Frattini_system "associated to modular curve covers"] of the j -line. A regular realization of a group G over a field K is a Galois extension, L / K ( x ), with group G, x transcendental over K , where elements of K are the only algebraic elements in L. Said with covers, it is where G&#966;ar = G&#966; and then any quotient of G&#966; is automatically regularly realized over K. In most serious applications, there is no automatic reduction to equality of G&#966;ar and G&#966;. $$G^{ar}_{\phi}$$ Given two groups ( G *, G ), a cover &#966; (over K ), is an  A - G realization of ( G *, G ) (over K ) if ( G *, G ) = (G&#966;ar, G&#966;). This then automatically includes any quotient of G&#966;ar as realized over K, where the part given by the G&#966;ar/G&#966; quotient is stationary under [#VIII.a._Hilberts_Irreducibility: "Hilbert Irreducibility specializations"]. It is easy, therefore, to see that the A-G version of the inverse Galois problem generalizes both the standard and the regular versions of the inverse Galois problem. Covers given by rational functions &#8211; morphisms between copies of the projective line, P 1 &#8211; in one variable have often been the hook that ties Fried's applications to that of many other researchers. Yet, covers beyond these have revealed deeper problems. It can be surprising how classical and many of these problems are, at least partially, answered by producing A-G realizations. The monodromy method and its tools : Fried's monodromy method (MM) uses the categorical equivalence between covers with a fixed range Y and permutation representations of the fundamental group of Y. That fundamental group is the topological fundamental group if the definition fields have characteristic 0. The initial novelty of the MM was its ["#I._Davenports_Problem_guided_early "application to problems on reducing polynomials"] with integer coefficients modulo primes p . Two group theory tools made it possible to describe and create the covers that produced solution to applications.  The description of [#I.a._Primitive_covers_and_Arithmetic_vs " primitive covers "] coming from the finite simple group classification .   The production of the [#VII.a._The_Modular_Tower_conjecture:_ "universal p -Frattini extension"] (cover) of a finite group for which p divides its order.  Three geometric tools related applications to the classical concern with connectedness of moduli. <ul> <li> [#Hurwitz_monodromy_group "Hurwitz monodromy"] (braid) action on [#II.a._Branch_cycles_attached_to_a_cover: " branch cycles "] (BrA) to detect connected components of families of covers in a given <a href="#Nielsen_Class">Nielsen class</a>. </li> <li>The [#II._Full_application_of_the_BCL_and_a "<i>Branch cycle lemma</i>"] (BCL) producing a cyclotomic definition field attached to the whole Nielsen class family. </li> <li>The [#VI.b._Perfect_groups_and_the_LI <i>Fried-Serre lift invariant</i>"] (LI).</li> </ul>

I. Davenport's Problem guided early developments

 * 1) Primitive covers, Arithmetic vs Geometric monodromy and Exceptionality:
 * 2) Davenport and permutation representations vs group representations:
 * 3) The tools for describing Davenport pairs over number fields:

II The B(ranch)C(ycle)L(emma) and Br(aid)A(ction)

 * 1) Branch cycles and Nielsen Classes of covers:
 * 2) Dragging a cover by its branch points:
 * 3) The BCL and Moduli of covers in a Nielsen class:
 * 4) Inner versus absolute Nielsen classes:
 * 5) Geometric/Arithmetic Monodromy and Hurwitz Space covers:

III. The Genus zero problem and covers in positive characteristic

 * 1) The Guralnnick/Thompson Genus 0 Problem:
 * 2) Exceptional positive characteristic covers:
 * 3) Extending Grothendieck's Theorem to Wild Ramification: