L -functions via deformations: from hyperelliptic curves to hypergeometric motives Kiran S. Kedlaya Department of Mathematics, University of California, San Diego kedlaya@ucsd.edu http://kskedlaya.org/slides/ Arithmetic of Hyperelliptic Curves Abdus Salam International Centre for Theoretical Physics (ICTP) September 8, 2017 K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 1 / 18
Overview Contents Overview 1 Hyperelliptic curves 2 Hypergeometric motives 3 A demonstration 4 K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 2 / 18
Overview Zeta functions of algebraic varieties For X an algebraic variety over a finite field F q , the zeta function � ∞ � # X ( F q n ) T n (1 − T [ κ ( x ): F q ] ) − 1 = exp � � ζ ( X , T ) = ∈ Z � T � n x ∈ X ◦ n =1 is a rational function of T . That is because it is possible to a spectral interpretation of ζ ( X , T ) consisting of a field K of characteristic 0; finite-dimensional K -vector spaces V i for i = 0 , 1 , . . . , 2 dim( X ); and K -linear endomorphisms F i on V i satisfying the Lefschetz trace formula : 2 dim( X ) ( − 1) i trace( F n � # X ( F q n ) = i , V i ) ( n = 1 , 2 , . . . ) . i =0 This then implies that 2 dim( X ) det(1 − F i T , V i ) ( − 1) i +1 . � ζ ( X , T ) = i =0 K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 3 / 18
Overview Zeta functions of algebraic varieties For X an algebraic variety over a finite field F q , the zeta function � ∞ � # X ( F q n ) T n (1 − T [ κ ( x ): F q ] ) − 1 = exp � � ζ ( X , T ) = ∈ Z � T � n x ∈ X ◦ n =1 is a rational function of T . That is because it is possible to a spectral interpretation of ζ ( X , T ) consisting of a field K of characteristic 0; finite-dimensional K -vector spaces V i for i = 0 , 1 , . . . , 2 dim( X ); and K -linear endomorphisms F i on V i satisfying the Lefschetz trace formula : 2 dim( X ) ( − 1) i trace( F n � # X ( F q n ) = i , V i ) ( n = 1 , 2 , . . . ) . i =0 This then implies that 2 dim( X ) det(1 − F i T , V i ) ( − 1) i +1 . � ζ ( X , T ) = i =0 K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 3 / 18
Overview Weil cohomology: ℓ -adic versus p -adic Such data are provided in a systematic way by Weil cohomology constructions, of which there are two general types. Grothendieck’s formalism of ´ etale cohomology produces one Weil cohomology theory with coefficients in Q ℓ for each prime ℓ other than p , the characteristic of F q . This theory is quite rich, and has been the basis for most new developments on geometric zeta functions. Building on Dwork’s original proof of rationality (predating ´ etale cohomology!), Berthelot introduced rigid cohomology with coefficients in a finite extension 1 of Q p . (This relates explicitly to crystalline cohomology for smooth proper varieties or Monsky-Washnitzer cohomology for smooth affine varieties.) Recently, most formalism of ´ etale cohomology has been replicated for rigid cohomology. 1 The residue field must contain F q . K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 4 / 18
Overview Weil cohomology: ℓ -adic versus p -adic Such data are provided in a systematic way by Weil cohomology constructions, of which there are two general types. Grothendieck’s formalism of ´ etale cohomology produces one Weil cohomology theory with coefficients in Q ℓ for each prime ℓ other than p , the characteristic of F q . This theory is quite rich, and has been the basis for most new developments on geometric zeta functions. Building on Dwork’s original proof of rationality (predating ´ etale cohomology!), Berthelot introduced rigid cohomology with coefficients in a finite extension 1 of Q p . (This relates explicitly to crystalline cohomology for smooth proper varieties or Monsky-Washnitzer cohomology for smooth affine varieties.) Recently, most formalism of ´ etale cohomology has been replicated for rigid cohomology. 1 The residue field must contain F q . K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 4 / 18
Overview Weil cohomology: ℓ -adic versus p -adic Such data are provided in a systematic way by Weil cohomology constructions, of which there are two general types. Grothendieck’s formalism of ´ etale cohomology produces one Weil cohomology theory with coefficients in Q ℓ for each prime ℓ other than p , the characteristic of F q . This theory is quite rich, and has been the basis for most new developments on geometric zeta functions. Building on Dwork’s original proof of rationality (predating ´ etale cohomology!), Berthelot introduced rigid cohomology with coefficients in a finite extension 1 of Q p . (This relates explicitly to crystalline cohomology for smooth proper varieties or Monsky-Washnitzer cohomology for smooth affine varieties.) Recently, most formalism of ´ etale cohomology has been replicated for rigid cohomology. 1 The residue field must contain F q . K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 4 / 18
Overview Factorization of zeta functions and varieties Suppose X is smooth proper over F q . By Deligne’s analogue of the Riemann hypothesis, there exists a unique factorization 2 dim( X ) P i ( T ) ( − 1) i +1 � ζ ( X , T ) = i =0 in which P i ( T ) ∈ 1 + T Z [ T ] has all C -roots of absolute value q − i / 2 . More precisely, Deligne (1974, 1980) showed that for the data F i , V i arising from ℓ -adic ´ etale cohomology, the polynomial P i ( T ) = det(1 − F i T , V i ) has all C -roots of absolute value q − i / 2 . A variant of the second proof can be executed with rigid cohomology (K, 2006). There is a formal process for “factoring” X into pieces that account for the individual P i ; this is the theory of motives . K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 5 / 18
Overview Factorization of zeta functions and varieties Suppose X is smooth proper over F q . By Deligne’s analogue of the Riemann hypothesis, there exists a unique factorization 2 dim( X ) P i ( T ) ( − 1) i +1 � ζ ( X , T ) = i =0 in which P i ( T ) ∈ 1 + T Z [ T ] has all C -roots of absolute value q − i / 2 . More precisely, Deligne (1974, 1980) showed that for the data F i , V i arising from ℓ -adic ´ etale cohomology, the polynomial P i ( T ) = det(1 − F i T , V i ) has all C -roots of absolute value q − i / 2 . A variant of the second proof can be executed with rigid cohomology (K, 2006). There is a formal process for “factoring” X into pieces that account for the individual P i ; this is the theory of motives . K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 5 / 18
Overview Factorization of zeta functions and varieties Suppose X is smooth proper over F q . By Deligne’s analogue of the Riemann hypothesis, there exists a unique factorization 2 dim( X ) P i ( T ) ( − 1) i +1 � ζ ( X , T ) = i =0 in which P i ( T ) ∈ 1 + T Z [ T ] has all C -roots of absolute value q − i / 2 . More precisely, Deligne (1974, 1980) showed that for the data F i , V i arising from ℓ -adic ´ etale cohomology, the polynomial P i ( T ) = det(1 − F i T , V i ) has all C -roots of absolute value q − i / 2 . A variant of the second proof can be executed with rigid cohomology (K, 2006). There is a formal process for “factoring” X into pieces that account for the individual P i ; this is the theory of motives . K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 5 / 18
Overview Zeta functions and L -functions Suppose now that X is a smooth proper variety over a number field K . Then for the motive of weight i associated to X , one gets an L -function by taking an Euler product � p L p ( s ) in which for almost all prime ideals p of o K , we have L p ( s ) = P i (Norm( p ) − s ) where P i is the corresponding factor of the zeta function of the reduction of (an integral model of) X modulo p . This may be familiar for X = E an elliptic curve. Over F q , we have P 1 ( T ) = 1 − a E T + qT 2 , P 0 ( T ) = 1 − T , P 2 ( T ) = 1 − qT . Over K , for i = 0 , 1 , 2, the resulting L -functions are ζ K ( s ) , L ( E , s ) , ζ K ( s − 1) p (1 − a E , p q − s + q 1 − 2 s ) − 1 for q = Norm( p ). where L ( E , s ) is (almost) � Similar considerations apply when X is a hyperelliptic (or arbitrary) curve. K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 6 / 18
Overview Zeta functions and L -functions Suppose now that X is a smooth proper variety over a number field K . Then for the motive of weight i associated to X , one gets an L -function by taking an Euler product � p L p ( s ) in which for almost all prime ideals p of o K , we have L p ( s ) = P i (Norm( p ) − s ) where P i is the corresponding factor of the zeta function of the reduction of (an integral model of) X modulo p . This may be familiar for X = E an elliptic curve. Over F q , we have P 1 ( T ) = 1 − a E T + qT 2 , P 0 ( T ) = 1 − T , P 2 ( T ) = 1 − qT . Over K , for i = 0 , 1 , 2, the resulting L -functions are ζ K ( s ) , L ( E , s ) , ζ K ( s − 1) p (1 − a E , p q − s + q 1 − 2 s ) − 1 for q = Norm( p ). where L ( E , s ) is (almost) � Similar considerations apply when X is a hyperelliptic (or arbitrary) curve. K.S. Kedlaya L -functions via deformations Trieste, August 24, 2017 6 / 18
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