Identities Of Mahler Measures Of Certain Polynomials Defining Curves Of Arbitrary Genus

In the late 1950s,Grothendieck introduced K0 group of Abelian category in order to generalize Riemann-Roch theorem.Since then,a new mathematical field-K-theory was born.Subsequently,Bass and Milnor defined the K1 and K2 groups of rings respectively.These K groups are closely related to some other classical groups.For example,K0 group of a commutative ring is its Picard group,K0 group of an algebraic integer ring group is the direct sum of its class group and Z.Quillen creatively defined and studied the higherorder K-groups,which also became the main reason why he won the fields medal in 1978.In short,algebraic K-theory defines a series of Abelian groups…K(-1),K0,K1,K2,…for the algebraic,geometric or arithmetic objects studied,which provides a large amount of important information of the research object,and has been applied to the fields of geometry,topology,ring theory and number theory with great success.K-groups can often reveal the deep relationship between mathematical objects,but it is very difficult to determine the structure of K-group.For example,the famous Birch-Tate conjecture relates the order of the K2 group of the integer ring of a totally real number field to the special value of the Dedekind zeta function of the field.In the beginning of the 1960s,Kurt Mahler became interested in a particular height function of polynomials—a measure.The logarithmic Mahler measure of a nonzero nvariable Laurent polynomial P?C[X1±1,…,Xn±1],denoted by m(P),is defined to be the arithmetic mean of log|P|over the n-dimensional torus.Years later,an original scope of the Mahler measure received a significant expansion after a discovery of its deep links to algebraic geometry and K-theory,in particular,to the famous Bloch-Beilinson's conjectures in algebraic K-theory following Deninger's insight.These interrelations generated a body of challenging problems—relations between Mahler measures and special values of L-functions and identities between Mahler measures of polynomials.Based on Deninger's discovery,Boyd made a systematic numerical study on the relationship between Mahler measures of families of two variable polynomial of genus 1 and some families of reciprocal polynomial of genus 2 and the special values of L-functions,and proposed many concrete conjectures.Some of the conjectures are recently resolved but more remaining open.The relationship between Mahler measure and special value of L-function has been extensively studied.Rodriguez-Villegas proved the relationship between Mahler measure of polynomials defining elliptic curves with complex multiplication and special values of L-functions using Eisenstein-Kronecker series.Brunault,Mellit,Zudilin discovered and proved the formula of the relationship between the regulator integrals of Steinberg symbols given by modular units and the special values of L-functions of modular forms.They applied this formula to prove the relation between the Mahler measures of some polynomials that can be parameterized by modular units and the special values of Lfunctions,which is an important breakthrough in the research of Mahler measure in recent years.At present,many works are also devoted to establishing the identities between Mahler measures of different families of polynomial defining curves with genus less than 3.For example,Rodriguez-Villegas,Rogers,Lalin and others have proved some identities of Mahler measures involving elliptic curves,and Bertin,Zudilin,Lalin and Wu have proved some identities of Mahler measures involving families of genus 2 and genus 3 curve.However,there are few studies on the identities of Mahler measures of families of curves with genus greater than 3.This thesis makes some attempts in this direction.The aim of this thesis is to prove two classes of Mahler measure identities involving families of two-variable polynomials defining curves of arbitrary genus.The first kind of identities contain families of polynomials defining birationally equivalent curves of arbitrary genus.These families display surprising symmetry after birational equivalence.Inspired by the work of Liu Hang and Qin Hourong,the second kind of identities consider the relationship between the Mahler measure of families of polynomials and the Mahler measure of their translation.In the proof of identities,we need to study the relationship between elements of K2 groups of curves,the relationship between closed loops in homology groups,and applying the relationship between regulator integrals of K2 groups and Mahler measures.Some of these intermediate results also have certain theoretical value from the perspective of Ktheory and analysis.For example,we prove some elements in K2 group are nontrivial.As an application,we also obtain the relation between the Mahler measure of non-tempered polynomials defining elliptic curves of conductor 14,15,24,48,54 and corresponding L-values.