On Zeta Functions of Weil and Igusa Type

1. On Zeta Functions of Weil and Igusa Type Hillary Sackett Mount Holyoke College REU NSF Grant DMS-0353700

2. Motivation and Research Goals • Understand p-adic numbers • Understand Igusa Zeta function and p-adic integration • Understand Weil Zeta function and count roots of polynomials in finite fields • Examine unramified field extensions • Examine reduction types of elliptic curves • Study the variation in the analogue ofthe Weil Zeta function and compare it with the variation in the rationality of the Igusa Zeta function NSF Grant DMS-0353700

3. p-adic IntegersForming a subring of Qp denoted Zp x = ∑αkpk for kєZ and α0≠ 0 For a prime p, we define the p-adic absolute value: x = pn(a/b) : p†a,b |x|p = p-n convention:|0|p = 0 In general we are looking at things in the form: X = α0 + α1p + α2p2 + … Z3 = (0+3Z3)U(1+3Z3)U(2+3Z3) Where 3Z is an ideal of the ring Z3. (0+3Z3)=(32Z3)U(3+32Z3)U(2(3)+32Z3) (1+3Z3)=(1+32Z3)U(1+3+32Z3)U(1+2(3)+32Z3) (2+3Z3)=(2+32Z3)U(2+3+32Z3)U(2+2(3)+32Z3) We consider numbers in their p-adic expansions. Z3 3Z3 32Z3 2(3)+32Z3 3+32Z3 2+3Z3 1+3Z3 2+32Z3 1+32Z3 1+2(3)+32Z3 2+2(3)+32Z3 1+3+32Z3 2+3+32Z3

4. p-adic Numbers • Each coefficient αtakes a value є {0, 1, 2…p-1} in Qp • Zp/pZp = disjoint union (α= 1 through p-1)α+pZp • We are going to be integrating in the p-adics, so we must understand that the p-adics form a field. NSF Grant DMS-0353700

5. Igusa Local Zeta Function • Let f(x) є Zp[x1, x2, …xn] • Ζ(T) = ∫Zpn|f(x)|s dx = ∑measure({x1, …xn} є Zpn : f(x1, …xn)≡ pe(unit))Te • This measure is called the Haar measure. • T = p-s for s єC • Difficulties arise in computation of polynomials with complicated singularities. NSF Grant DMS-0353700

6. Hensel’s Lemma and Stationary Phase Formula • Lemma (Hensel) Let f є Zp [ x1, …xn] and (a0…an) є (Fpe)n such that f(a) ≡ 0 mod pe and δf/δxi ≠ 0 mod p for some i. Then there exists pn-1 n-tuples, (bo,…bn), such that f(a+peb) ≡ 0 mod pe+1. Helps us understandthe 2nd integral. • Stationary Phase Formula • Introduced by Igusa in 1994 as a systematic technique for representing one p-adic integral as a sum of simpler integrals. Split Fpn for f(x): (i) f(x) ≠ 0 mod p  1st integral (ii) f(x) ≡ 0 mod p and there exists δf/δxi ≠ 0 mod p  2nd integral (iii) f(x) ≡ 0 mod p and all δf/δxi ≡ 0 mod p.  3rd integral NSF Grant DMS-0353700

7. Weil Local Zeta Function • Let f(x1,…xn)єFp[x1…xn] • Zw(T) = exp∑d=1(|Nd|Td)/ d • Diane Meuser has studied a generating function that combines the Weil and Igusa zeta functions into one object. This has a meromorphic continuation to the complex plane, but unfortunately is not rational. NSF Grant DMS-0353700

8. Consider a cubic equation in two unknowns. All elliptic curves (non singular) can be written in this form: F(x,y) = y2 – a1xy + a3y – x3 – a2x2 – a4x – a6 There can be only one singular point modulo p. It is either at the origin, or we can shift it to the origin without changing the zeta function. Q: How do the |Nd,e|’s vary for each reduction type? - Idea: work backwards. Zeta functionPoincaré series|Nd,e| Elliptic Curves in Weierstrass Normal Form Reduction Types of Elliptic Curves I0 In II p|a3,a4,a6 III p|a3,a4,a6 p2†a6 IV p|a3,a4,a6 p2|a6 I0* In* II* III* IV* NSF Grant DMS-0353700

9. Poincaré series The Poincaré series is a generating function associated with a polynomial f(x) for the number of solutions to f(x)≡ 0 modulo powers of the prime p. • f(x) є Z/pZp[x1,…xn] - we can think of this as the finite field with p elements, Fp consider elements modulo p, {0,1,2,…p-1} • Ne = {(x1, …xn) є (Zp/peZp)n | f(x) ≡ 0 mod pe} • Nd = {(x1, …xn) є (Fpd)n | f(x) = 0} • PI(T) = ∑e=0 |Nd,e|p-neTePI(pnT) = ∑e=0 |Nd,e|Te • PW(T) = ∑d=1 |Nd,e|Td • Relating the Poincaré series to the Zeta function: P(T) = 1-T Z(T)/ 1-T NSF Grant DMS-0353700

10. Rationality and our Correction Term • If the Zeta function is rational it implies that the Poincaré series is rational. • The recursive relationship between |Nd,e| and |Nd,e+1| implies rationality of the Poincaré series. • We look at unramified extensions of these finite fields. These are rings of integers Od, where d is the degree of the extension. • We would like to treat the prime p as a variable. • Q: What does the Poincaré series look like in the field extensions? • Idea: A combination (Super) Poincaré series • P(T,W) = ∑d=1∑e=0 |Nd,e| WdTep-nde[ Meuser] • This is not a rational function in general. • Introduce a correction term to ensure rationality of P(T,W) for elliptic curves: P(T,W) = ∑d=1∑e=0 |Nd,e| WdTep-ed NSF Grant DMS-0353700

11. |Nd,e| = { x єOd/peOd : f(x) ≡ 0 mod pe}Od={α0 + α1p + α2p2+…| αiєFpd}Od/peOd = {α+ α1p+ … αe-1pe-1| αiєFpd}  Weil (fix e, vary d)  Igusa (fix d, vary e) NSF Grant DMS-0353700

12. Q: Why does our correction term work? Consider N1,1 = {(x,y)є Fp| f(x,y) = 0} for f(x,y) = E Hasse proved in the 1930’s that |N1,1| = p + error  ||N1,1,|-p| ≤ 2√p. Therefore let us consider |Nd,e| = pde + error Normalizing: |Nd,e| p-de = 1 + error  error ≤ 1 Theorem f(x,y) be any nonsingular curve over Qp. Then P(T,W) єQ(W,T) {δf/δx | (a,b) , δf/δy | (a,b) } one of these is non zero for all points on the curve. For sufficiently large e≥M |Nd,e| = pd(e-M) |Nd,M| NSF Grant DMS-0353700

13. |Nd,e| by Reduction Type of Elliptic Curves

14. |Nd,e| by Reduction Type of Elliptic Curves

15. Super Poincaré Series by Reduction Type of Elliptic Curves We were able to compute the Super Poincaré series for all reduction types of elliptic curves and their sub cases based on their modulo p singularity type. Most importantly, with our correction term, they are all RATIONAL FUNCTIONS! ** Extended results. Email hsackett@email.smith.edu

16. Special Thanks • Margaret Robinson • Mount Holyoke College • National Science Foundation • Diane Meuser • MHC REU students: • -Zebediah Engberg [Hampshire College] • -Neal Lima [UConn] • -Sohini Mahapatra [UMichigan] • -Sam Ruth [Northwestern] NSF Grant DMS-0353700

17. Bibliography [1] Fleischli, Mary. Local zeta functions associated with polynomials of two and three variables. Mount Holyoke College, May 1993. [2] Gouvéa, F. p-adic Numbers: An Introduction. Springer Verlag, 1993. [3] K. Ireland and M. Rosen. A Classical Introduction to Modern Number Theory. New York: Springer-Verlag, 1990. [4] Koblitz, Neal, p-adic Numbers, p-adic Analysis, and Zeta-Functions, New York: Springer-Verlag, 1997. [5] Meuser, Diane. One the poles of a local zeta function for curves. Inventiones. Math, 73 (1983), no. 3, 445-465. [6] Meuser, Diane. The meromorphic continuation of a zeta function of Weil and Igusa Type. Inventiones Math. 85 (1986), 493-514. [7] Meuser, Diane and Robinson, Margaret. The Igusa local zeta function of elliptic curves. [8] Mount Holyoke College REU 1999, Cambell, Mariana and Dubois, Ed. et al. On Igusa local zeta functions of elliptic curves. MathematicsSubject Classification. (1991) Primary 11S40, 11G07. [9] Reuman, Daniel C., The Igusa local zeta function, rationality and the Poincaré series. Harvard University, May 1996. [10] Tate, J.T. The arithmetic of elliptic curves. Inventiones Math. 23 (1974), 179-206. NSF Grant DMS-0353700