Computing zeta functions of nondegenerate hypersurfaces in toric varieties Edgar Costa (Dartmouth College) May 16th, 2018 Presented at ICERM, Birational Geometry and Arithmetic Joint work with David Harvey (UNSW) and Kiran Kedlaya (UCSD) Slides available at edgarcosta.org under Research 1 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate hypersurfaces in toric varieties
Motivation
Riemann zeta function 1 Edgar Costa (Dartmouth College) 2 / 31 • Together with the functional equation • One of the most famous examples of a global zeta function 1 1 Computing zeta functions of nondegenerate hypersurfaces in toric varieties ζ ( s ) = 1 + 1 2 s + 1 3 s + 1 4 s + 1 5 s + 1 6 s + 1 7 s · · · = 1 − 2 − s · 1 − 3 − s · 1 − 5 − s · · · ξ ( s ) := π − s / 2 Γ( s / 2 ) ζ ( s ) = ξ ( 1 − s ) encodes a lot of the arithmetic information of Z . e.g.: Zeros of ζ ( s ) ⇝ precise prime distribution • ζ ( s ) still keeps secret many of its properties
p i t i X p t X p s ? X p t obeys a functional equation and satisfies the Riemann X s ? Hasse–Weil zeta functions Example: X Edgar Costa (Dartmouth College) 3 / 31 • What about hypothesis! • • What arithmetic properties of X can we read from s s , a point, then i t X p 0 i exp p is smooth, then X If X p p Computing zeta functions of nondegenerate hypersurfaces in toric varieties Hasse and Weil generalized an analog of ζ ( s ) for algebraic varieties ∏ ζ X p ( p − s ) ζ X ( s ) :=
X p s ? X p t obeys a functional equation and satisfies the Riemann X s ? Hasse–Weil zeta functions Example: X Edgar Costa (Dartmouth College) 3 / 31 • What about hypothesis! • • What arithmetic properties of X can we read from s s , a point, then Computing zeta functions of nondegenerate hypersurfaces in toric varieties p i Hasse and Weil generalized an analog of ζ ( s ) for algebraic varieties ∏ ζ X p ( p − s ) ζ X ( s ) := If X p := X mod p is smooth, then ∑ ∈ Q ( t ) ζ X p ( t ) := exp # X p ( F p i ) t i i ≥ 0
X p s ? X p t obeys a functional equation and satisfies the Riemann X s ? Hasse–Weil zeta functions i Edgar Costa (Dartmouth College) 3 / 31 • What about hypothesis! • • What arithmetic properties of X can we read from Computing zeta functions of nondegenerate hypersurfaces in toric varieties p Hasse and Weil generalized an analog of ζ ( s ) for algebraic varieties ∏ ζ X p ( p − s ) ζ X ( s ) := If X p := X mod p is smooth, then ∑ ∈ Q ( t ) ζ X p ( t ) := exp # X p ( F p i ) t i i ≥ 0 Example: X = {•} , a point, then ζ {•} ( s ) = ζ ( s )
Hasse–Weil zeta functions i Edgar Costa (Dartmouth College) 3 / 31 hypothesis! Computing zeta functions of nondegenerate hypersurfaces in toric varieties p Hasse and Weil generalized an analog of ζ ( s ) for algebraic varieties ∏ ζ X p ( p − s ) ζ X ( s ) := If X p := X mod p is smooth, then ∑ ∈ Q ( t ) ζ X p ( t ) := exp # X p ( F p i ) t i i ≥ 0 Example: X = {•} , a point, then ζ {•} ( s ) = ζ ( s ) • What arithmetic properties of X can we read from ζ X p ( s ) ? • ζ X p ( t ) obeys a functional equation and satisfies the Riemann • What about ζ X ( s ) ?
L p p E s L E s L E satisfies a functional equation 1 L E s s 1 p s 1 p s 1 s Elliptic curves 1 • a p arithmetic information about E p E . • Modularity theorem • Birch–Swinnerton-Dyer conjecture predicts s E . 4 / 31 Edgar Costa (Dartmouth College) s p Computing zeta functions of nondegenerate hypersurfaces in toric varieties additive reduction; 1 non-split/split multiplicative reduction; p and E an elliptic curve over Q L p ( t ) ∏ ζ E p ( p − s ) ζ E ( s ) := ζ E p ( t ) = ( 1 − t )( 1 − pt ) 1 − a p t + pt 2 , good reduction , a p = p + 1 − # E p ( F p ) L p ( t ) = 1 ± t ,
L E satisfies a functional equation 1 L E s Elliptic curves 1 Edgar Costa (Dartmouth College) 4 / 31 E . s • Birch–Swinnerton-Dyer conjecture predicts • Modularity theorem E . arithmetic information about E p • a p p additive reduction; non-split/split multiplicative reduction; Computing zeta functions of nondegenerate hypersurfaces in toric varieties and p E an elliptic curve over Q L p ( t ) ∏ ζ E p ( p − s ) ζ E ( s ) := ζ E p ( t ) = ( 1 − t )( 1 − pt ) 1 − a p t + pt 2 , good reduction , a p = p + 1 − # E p ( F p ) L p ( t ) = 1 ± t , L p ( p − s ) ( 1 − p − s )( 1 − p − s + 1 ) = ζ ( s ) ζ ( s − 1 ) ∏ ζ E ( s ) = L E ( s )
L E satisfies a functional equation 1 L E s Elliptic curves non-split/split multiplicative reduction; Edgar Costa (Dartmouth College) 4 / 31 E . s • Birch–Swinnerton-Dyer conjecture predicts • Modularity theorem p additive reduction; 1 Computing zeta functions of nondegenerate hypersurfaces in toric varieties p and E an elliptic curve over Q L p ( t ) ∏ ζ E p ( p − s ) ζ E ( s ) := ζ E p ( t ) = ( 1 − t )( 1 − pt ) 1 − a p t + pt 2 , good reduction , a p = p + 1 − # E p ( F p ) L p ( t ) = 1 ± t , L p ( p − s ) ( 1 − p − s )( 1 − p − s + 1 ) = ζ ( s ) ζ ( s − 1 ) ∏ ζ E ( s ) = L E ( s ) • a p ⇝ arithmetic information about E p ⇝ E .
Elliptic curves non-split/split multiplicative reduction; Edgar Costa (Dartmouth College) 4 / 31 p additive reduction; 1 Computing zeta functions of nondegenerate hypersurfaces in toric varieties and p E an elliptic curve over Q L p ( t ) ∏ ζ E p ( p − s ) ζ E ( s ) := ζ E p ( t ) = ( 1 − t )( 1 − pt ) 1 − a p t + pt 2 , good reduction , a p = p + 1 − # E p ( F p ) L p ( t ) = 1 ± t , L p ( p − s ) ( 1 − p − s )( 1 − p − s + 1 ) = ζ ( s ) ζ ( s − 1 ) ∏ ζ E ( s ) = L E ( s ) • a p ⇝ arithmetic information about E p ⇝ E . • Modularity theorem = ⇒ L E satisfies a functional equation • Birch–Swinnerton-Dyer conjecture predicts ord s = 1 L E ( s ) = rk( E ) .
X p t for an arbitrary X p i t i X p t Computing zeta functions of nondegenerate hypersurfaces in toric varieties Edgar Costa (Dartmouth College) 5 / 31 t i X p 0 i exp Problem Given an explicit description of X , compute • extremely difficult to calculate s • easy to explicitly write down Major difference • surfaces ? numerically • genus 2 curves ? ζ ( s ) vs ζ X ( s ) We always expect ζ X ( s ) to satisfy a functional equation. • zero-dimensional varieties (number fields) ✓ • elliptic curves over Q ✓
X p t for an arbitrary X p i t i X p t Computing zeta functions of nondegenerate hypersurfaces in toric varieties Edgar Costa (Dartmouth College) 5 / 31 t i X p 0 i exp Given an explicit description of X , compute Problem • extremely difficult to calculate s • easy to explicitly write down Major difference • surfaces ? ζ ( s ) vs ζ X ( s ) We always expect ζ X ( s ) to satisfy a functional equation. • zero-dimensional varieties (number fields) ✓ • elliptic curves over Q ✓ • genus 2 curves ? numerically ✓
X p t for an arbitrary X p i t i X p t Computing zeta functions of nondegenerate hypersurfaces in toric varieties Edgar Costa (Dartmouth College) 5 / 31 t i X p 0 i exp Given an explicit description of X , compute Problem • extremely difficult to calculate s • easy to explicitly write down Major difference • surfaces ? ζ ( s ) vs ζ X ( s ) We always expect ζ X ( s ) to satisfy a functional equation. • zero-dimensional varieties (number fields) ✓ • elliptic curves over Q ✓ • genus 2 curves ? numerically ✓
p i t i X p t exp Edgar Costa (Dartmouth College) 5 / 31 t i X p 0 i Computing zeta functions of nondegenerate hypersurfaces in toric varieties Given an explicit description of X , compute Problem Major difference • surfaces ? ζ ( s ) vs ζ X ( s ) We always expect ζ X ( s ) to satisfy a functional equation. • zero-dimensional varieties (number fields) ✓ • elliptic curves over Q ✓ • genus 2 curves ? numerically ✓ • easy to explicitly write down ζ ( s ) • extremely difficult to calculate ζ X p ( t ) for an arbitrary X
Computing zeta functions of nondegenerate hypersurfaces in toric varieties Problem Edgar Costa (Dartmouth College) 5 / 31 i • surfaces ? Major difference Given an explicit description of X , compute ζ ( s ) vs ζ X ( s ) We always expect ζ X ( s ) to satisfy a functional equation. • zero-dimensional varieties (number fields) ✓ • elliptic curves over Q ✓ • genus 2 curves ? numerically ✓ • easy to explicitly write down ζ ( s ) • extremely difficult to calculate ζ X p ( t ) for an arbitrary X ∑ ∈ Q ( t ) ζ X p ( t ) := exp # X p ( F p i ) t i i ≥ 0
X is bounded by the geometry of X , and we can then q i for enough i to pinpoint The zeta function problem Theoretically this is “trivial”. Edgar Costa (Dartmouth College) 6 / 31 low genus curves and p small. This approach is only practical for very few classes of varieties, e.g., X . enumerate X The degree of Computing zeta functions of nondegenerate hypersurfaces in toric varieties Problem i consider Let X be a smooth variety over a finite field F q of characteristic p , ∑ ζ X ( t ) := exp # X ( F q i ) t i i ≥ 1 Compute ζ X from an explicit description of X .
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