Lower Bounds on Class Numbers Jose Chavez Mentor: Jeffrey Stopple - PowerPoint PPT Presentation

Lower Bounds on Class Numbers Jose Chavez Mentor: Jeffrey Stopple UC LEADS UCSB-UCLA

Pure Mathematics and the Modern World ● Pure Mathematics has application in many areas : ● Finance ● Stochastic Processes used to understand the system that is “The Market” ● Physics ● Biology ● Yet to be discovered ● Intellectual Pursuit

Number Theory and Our World ● Algebraic Number Theory ● RSA encryption algorithm ● Euler's Theorem, Congruence Classes, difficulty factoring large numbers to secure information ● Analytic Number Theory ● Connections to Physics ● Backwards Heat Equation satisfied by Xi function u t =−Δ u

Binary Quadratic Forms and Equivalence ● Quadratic Form: 2 + bxy + cy 2 { a,b ,c } F ( x , y ) ax ● Examples of BQF's: 2 + 2y 2 { 1,0,2 } F ( x ,y ) x 2 + y 2 { 1,0,1 } F ( x , y ) x ● Equivalence of Forms: If matrix M s.t ∃ ̀ (change of variables) F ( x ,y )= F (( x , y ) M ) ● Example 2 F = 2x ' = 2x 2 + 4xy + 5y 2 + 3y 2 F , ( ( x , y ) [ 1 ] ) = F ( x + y , y )= 2 ( x + y ) 2 + 3y 1 0 ' = 2x 2 + 4xy + 5y 2 = F 2 F 1

Class and Goals ● Forms that are equivalent, form an equivalence class ● Discriminant of form: 2 − 4ac { a ,b ,c }= b ● Class number =h(d): number of classes associated to discriminant d ● Theorem: Suppose is a real non-principal χ mod d character, with . If has no zero in d ≥ 200 L ( s , χ) .35 απ 1 where then √ ∣ d ∣ < h ( d ) [ 1 −α , 1 ] 0 <α< 20ln ( d ) ● Goal-Improve the constant bounding h(d)

Setting Up the Improvement .35 α π ● WANT:Raise lower bound √ ∣ d ∣ < h ( d ) ● Change interval in hypothesis of Hecke's Theorem ● Change lower bound on discriminant ● Introduce Polya-Vinagradov's inequality ∣ ∑ χ D ( m ) ∣ < √ D ln ( D ) to the body of Hecke's Theorem m ≤ x ● Combine techniques ● Pursue clean looking result

Results ● Take new interval 40,000 <α< 1 /( 30ln ( d ))< 0.002 ● Let d>= 10,000,000 ● Employ Polya-Vinagradov Inequality ● Result: .50 α π √ ∣ d ∣ < h ( d )

Conclusions and Future Directions ● Little room for more improvement ● Class numbers are connected to fascinating topics ● Explore the connections between class number and Siegel zeros(zeta-zeros that contradict the Reimann Hypothesis)

Acknowledgments ● UC LEADS ● UCSB ● Jeffrey Stopple ● Those that work in UCSB summer research programs ● Students in UCSB summer research