Local zeta function: Difference between revisions

Revision as of 14:42, 9 May 2022

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:<math>Z(V, s) = \exp\left(\sum_{m = 1}^\infty \frac{N_m}{m} (q^{-s})^m\right)</math>

where {{mvar|V}} is a [[non-singular]] {{mvar|n}}-dimensional [[projective algebraic variety]] over the field {{math|'''F'''_''q''}} with {{mvar|q}} elements and {{math|''N''_''m''}} is the number of points of {{mvar|''V''}} defined over the finite field extension {{math|'''F'''_''q''^''m''}} of {{math|'''F'''_''q''}}.<ref>Section MakingV.2 the variable transformationof {{math|''u''&nbsp;{{=}}&nbsp;''q''^−''s'',}} givesCitation

| last=Silverman

| first=Joseph H.

| author-link=Joseph H. Silverman

| title=The arithmetic of elliptic curves

| publisher=[[Springer-Verlag]]

| ___location=New York

| series=[[Graduate Texts in Mathematics]]

| isbn=978-0-387-96203-0

| mr=1329092

| year=1992

| volume=106

}}</ref>

Making the variable transformation {{math|''u''&nbsp;{{=}}&nbsp;''q''^−''s'',}} gives

:<math>

\mathit{Z} (V,u) = \exp