Results on GL(2) L-Functions: Biases in Coefficients and Gaps Between Zeros

Transcription

1 Results on GL(2) L-Functions: Biases in Coefficients and Gas Between Zeros Steven J. Miller, Williams College With Owen Barrett, Blake Mackall, Brian McDonald, Christina Rati, Patrick Ryan, Caroline Turnage-Butterbaugh & Karl Winsor htt://web.williams.edu/mathematics/sjmiller/ublic_html/ Families of Automorhic Forms and the Trace Formula Banff International Research Station, Dec 1, 2014

2 Gas between Critical Zeros Joint with students Owen Barrett (Yale), Brian McDonald (Rochester), Patrick Ryan (Harvard), Karl Winsor (Michigan) and ostoc Caroline Turnage-Butterbaugh (North Dakota State University) s: and

3 The Random Matrix Theory Connection Philosohy: Critical-zero statistics of L-functions agree with eigenvalue statistics of large random matrices. Montgomery - air-correlations of zeros of ζ(s) and eigenvalues of the Gaussian Unitary Ensemble. Hejhal, Rudnick and Sarnak - Higher correlations and automorhic L-functions. Odlyzko - further evidence through extensive numerical comutations.

4 Consecutive Zero Sacings Consecutive zero sacings of ζ(s) vs. GUE redictions (Odlyzko).

5 5 Gas: Intro Gas: Shifted Moments & Proofs Gas: Pros Bias: Intro Bias: Evidence Bias: Data Bias: Future Refs Large Gas between Zeros Let 0 γ 1 γ 2 γ i be the ordinates of the critical zeros of an L-function. Conjecture Gas between consecutive zeros that are arbitrarily large, relative to the average ga size, aear infinitely often.

6 Large Gas between Zeros Let 0 γ 1 γ 2 γ i be the ordinates of the critical zeros of an L-function. Conjecture Gas between consecutive zeros that are arbitrarily large, relative to the average ga size, aear infinitely often. Letting Λ = lim su n this conjecture is equivalent to Λ =. γ n+1 γ n average sacing, Best unconditional result for the Riemann zeta function is Λ > 2.69.

7 7 Gas: Intro Gas: Shifted Moments & Proofs Gas: Pros Bias: Intro Bias: Evidence Bias: Data Bias: Future Refs Degree 2 Case Higher degree L-functions are mostly unexlored. Theorem (Turnage-Butterbaugh 14) Let T 2, ε > 0, ζ K (s) the Dedekind zeta function attached to a quadratic number field K with discriminant d with d T ε, and S T := {γ 1,γ 2,...,γ N } be the distinct zeros of ζ K ( it, f) in the interval [T, 2T]. Let κ T denote the maximum ga between consecutive zeros in S T. Then κ T π 6 log ( 1+O(d ε log T) 1). d T Assuming GRH, this means Λ

8 A Lower Bound on Large Gas We roved the following unconditional theorem for an L-function associated to a holomorhic cus form f on GL(2). Theorem (BMMRTW 14) Let S T := {γ 1,γ 2,...,γ N } be the set of distinct zeros of L ( it, f) in the interval [T, 2T]. Let κ T denote the maximum ga between consecutive zeros in S T. Then ( ( )) 3π 1 κ T 1+O (log T) δ, log T c f where c f is the residue of the Rankin-Selberg convolution L(s, f f) at s = 1. Assuming GRH, there are infinitely many normalized gas between consecutive zeros at least 3 times the mean sacing, i.e., Λ

9 An Uer Bound on Small Gas Theorem (BMMRTW 14) L in Selberg class rimitive of degree m L. Assume GRH for log L(s) = n=1 b L(n)/n s, n x b L(n) log n 2 = (1+o(1))x log x. Have a comutable nontrivial uer bound on µ L (liminf of smallest average ga) deending on m L. m L uer bound for µ L (m L = 1 due to Carneiro, Chandee, Littmann and Milinovich). Key idea: use air correlation analysis.

10 Results on Gas and Shifted Second Moments

11 Shifted Moment Result To rove our theorem, use a method due to R.R. Hall and the following shifted moment result. Theorem (BMMRTW 14) 2T ( ) ( ) 1 1 L + it +α, f L it +β, f dt T 2 2 = c f T ( 1) n 2 n+1 (α+β) n (log T) n+1 +O ( T(log T) 1 δ), (n+1)! n 0 where α,β C and α, β 1/ log T. Key idea: differentiate wrt arameters, yields formulas for integrals of roducts of derivatives.

12 Shifted Moments Proof Technique Aroximate functional equation: L(s +α, f) = n 1 λ f (n) n n e X s+α + F(s) n X λ f (n) + E(s), n1 s α where λ f (n) are the Fourier coefficients of L(s, f), F(s) is a functional equation term, and E(s) is an error term. We have an analogous exression for L(1 s +β, f).

13 Shifted Moments Proof Technique Analyze roduct L(s +α, f)l(1 s +β, f), where each factor gives rise to four roducts (so sixteen roducts to estimate). Use a generalization of Montgomery and Vaughan s mean value theorem and contour integration to estimate roduct and comute the resulting moments.

14 Shifted Moment Result for Derivatives Shifted moment result yields lower order terms and moments of derivatives of L-functions by differentiation and Cauchy s integral formula. Derive an exression for 2T T L (µ) ( it, f ) L (ν) ( 1 2 it, f ) dt, where T 2 and µ,ν Z +. Use this in Hall s method to obtain the lower bound stated in our theorem. Need (µ,ν) {(0, 0),(1, 0),(1, 1)}; other cases reviously done (Good did (0, 0) and Yashiro did µ = ν).

15 Modified Wirtinger Inequality Using Hall s method, we bound the gas between zeroes. This requires the following result, due to Wirtinger and modified by Bredberg. Lemma (Bredberg) Let y : [a, b] C be a continuously differentiable function and suose that y(a) = y(b) = 0. Then b a ( ) 2 b a b y(x) 2 dx y (x) 2 dx. π a

16 Proving our Result For ρ a real arameter to be determined later, define g(t) := e iρt log T L ( it, f), Fix f and let γ f (k) denote an ordinate zero of L(s, f) on the critical line R(s) = 1 2.

17 Proving our Result For ρ a real arameter to be determined later, define g(t) := e iρt log T L ( it, f), Fix f and let γ f (k) denote an ordinate zero of L(s, f) on the critical line R(s) = 1 2. g(t) has same zeros as L(s, f) (at t = γ f (k)). Use in the modified Wirtinger s inequality.

18 Proving our Result For ρ a real arameter to be determined later, define g(t) := e iρt log T L ( it, f), Fix f and let γ f (k) denote an ordinate zero of L(s, f) on the critical line R(s) = 1 2. g(t) has same zeros as L(s, f) (at t = γ f (k)). Use in the modified Wirtinger s inequality. For adjacent zeros have N 1 γf (n+1) n=1 γ f (n) g(t) 2 dt N 1 n=1 κ 2 γf (n+1) T g (t) 2 dt. π 2 γ f (n) Summing over zeros with n {1,...,N} and trivial estimation yields integrals from T to 2T.

19 Proving our Result g(t) 2 = L(1/2+it, f) 2 and g (t) 2 = L (1/2+it, f) 2 +ρ 2 log 2 T L(1/2+it, f) 2 ( ) + 2ρ log T Re L (1/2+it, f)l(1/2+it, f). Aly sub-convexity bounds to L(1/2+it, f): 2T T g(t) 2 dt κ2 T π 2 2T T ( g (t) 2 dt + O T 2 3 (log T) 5 6 As g(t) and g (t) may be exressed in terms of L ( 1 + it, f) and its derivatives, can write our 2 inequality exlicitly in terms of formula given by our mixed moment theorem. ).

20 Finishing the Proof After substituting our formula, we have κ 2 T π 2 3 3ρ 2 6ρ+4 (log T) 2( 1+O(log T) δ). The olynomial in ρ is minimized at ρ = 1, yielding ( ( )) 3π 1 κ T 1+O (log T) δ. log T c f

21 Essential GL(2) roerties

22 Proerties For rimitive f on GL(2) over Q (Hecke or Maass) with L(s, f) = n=1 a f (n) n s, R(s) > 1, we isolate needed crucial roerties (all are known). 1 L(s, f) has an analytic continuation to an entire function of order 1. 2 L(s, f) satisfies a function equation of the form Λ(s, f) := L(s, f )L(s, f) = ǫ f Λ(1 s, f) s ) s ) with L(s, f ) = Q Γ( s 2 +µ 1 Γ( 2 +µ 2.

23 Proerties (continued) 3 Convolution L-function L(s, f f), a f (n) 2, R(s) > 1, n s n=1 is entire excet for a simle ole at s = 1. 4 The Dirichlet coefficients (normalized so that the critical stri is 0 R(s) 1) satisfy a f (n) 2 x. n x 5 For some small δ > 0, we have a subconvexity bound ( ) 1 L 2 + it, f t 1 2 δ.

24 Proerties (status) Mœglin and Waldsurger rove the needed roerties of L(s, f f) (in greater generality). Dirichlet coefficient asymtotics follow for Hecke forms essentially from the work of Rankin and Selberg, and for Maass by sectral theory. Michel and Venkatesh roved a subconvexity bound for rimitive GL(2) L-functions over Q. Other roerties are standard and are valid for GL(2).

25 Bias Conjecture for Moments of Fourier Coefficients of Ellitic Curve L-functions Joint with students Blake Mackall (Williams), Christina Rati (Bard) and Karl Winsor (Michigan) s: and

26 Families and Moments A one-arameter family of ellitic curves is given by E : y 2 = x 3 + A(T)x + B(T) where A(T), B(T) are olynomials in Z[T]. Each secialization of T to an integer t gives an ellitic curve E(t) over Q. The r th moment of the Fourier coefficients is A r,e () = a E(t) () r. t mod

27 Tate s Conjecture Tate s Conjecture for Ellitic Surfaces Let E/Q be an ellitic surface and L 2 (E, s) be the L-series attached to H 2 ét (E/Q,Q l). Then L 2 (E, s) has a meromorhic continuation to C and satisfies ord s=2 L 2 (E, s) = rank NS(E/Q), where NS(E/Q) is the Q-rational art of the Néron-Severi grou of E. Further, L 2 (E, s) does not vanish on the line Re(s) = 2. Tate s conjecture is known for rational surfaces: An ellitic surface y 2 = x 3 + A(T)x + B(T) is rational iff one of the following is true: 0 < max{3dega, 2degB} < 12; 3degA = 2degB = 12 and ord T=0 T 12 (T 1 ) = 0.

28 Negative Bias in the First Moment A 1,E () and Family Rank (Rosen-Silverman) If Tate s Conjecture holds for E then 1 lim X X X A 1,E () log = rank(e/q). By the Prime Number Theorem, A 1,E () = r + O(1) imlies rank(e/q) = r.

29 Bias Conjecture Second Moment Asymtotic (Michel) For families E with j(t) non-constant, the second moment is A 2,E () = 2 + O( 3/2 ). The lower order terms are of sizes 3/2,, 1/2, and 1.

30 Bias Conjecture Second Moment Asymtotic (Michel) For families E with j(t) non-constant, the second moment is A 2,E () = 2 + O( 3/2 ). The lower order terms are of sizes 3/2,, 1/2, and 1. In every family we have studied, we have observed: Bias Conjecture The largest lower term in the second moment exansion which does not average to 0 is on average negative.

31 Preliminary Evidence and Patterns Let n 3,2, equal the number of cube roots of 2 modulo, [ ( 3 ) and set ] ) ] 2 c 0 () =, c 1 () =, c 3/2 () = x() ( 4x 3 +1 ) + ( 3 ). [ x mod ( x 3 x Family A 1,E () A 2,E () y 2 = x 3 + Sx + T y 2 = x ( 3) 3 (9T + 1) 2 0 y 2 = x 3 ± 4(4T + 2)x 0 { mod mod 3 { mod mod 4 y 2 = x 3 +(T + 1)x 2 + Tx y 2 = x 3 + x 2 + 2T ( ) 3 y 2 = x 3 + Tx n 3,2, 1+c 3/2 () y 2 = x 3 T 2 x + T c 1 () c 0 () y 2 = x 3 T 2 x + T c 1 () c 0 () y 2 = x 3 + Tx 2 (T + 3)x + 1 2c,1;4 2 4c,1;6 1 where c,a;m = 1 if a mod m and otherwise is 0.

32 Preliminary Evidence and Patterns The first family is the family of all ellitic curves; it is a two arameter family and we exect the main term of its second moment to be 3. Note that excet for our family y 2 = x 3 + Tx 2 + 1, all the familiese have A 2,E () = 2 h()+o(1), where h() is non-negative. Further, many of the families have h() = m E > 0. Note c 1 () is the square of the coefficients from an ellitic curve with comlex multilication. It is non-negative and of size for 3 mod 4, and zero for 1 mod 4 (send x x mod and note ( ) 1 = 1). It is somewhat remarkable that all these families have a correction to the main term in Michel s theorem in the same direction, and we analyze the consequence this has on the average rank. For our family which has a 3/2 term, note that on average this term is zero and the term is negative.

33 Lower order terms and average rank 1 N 2N t=n 2 N γ t 2N t=n ( ) log R φ γ t 2π log log R 1 2 φ = φ(0)+φ(0) 2 N 2N t=n ( ) 2 log a t () 2 + O log R ( ) log 1 log R φ log a t () log R ( log log R log R If φ is non-negative, we obtain a bound for the average rank in the family by restricting the sum to) be only over zeros at the central oint. The error O comes from trivial ( log log R log R estimation ) and ignores robable cancellation, and we exect or smaller to be the correct magnitude. For most O( 1 log R families log R log N a for some integer a. ).

34 Lower order terms and average rank (cont) The main term of the first and second moments of the a t () give rφ(0) and 1 2 φ(0). Assume the second moment of a t () 2 is 2 m E + O(1), m E > 0. We have already handled the contribution from 2, and m E contributes S 2 2 ( log N log R φ 2 log ) 1 N log R 2 ( m E) = 2m E log R φ ( 2 log log R Thus there is a contribution of size 1 log R. ) log 2.

35 Lower order terms and average rank (cont) A good choice of test functions (see Aendix A of [ILS]) is the Fourier air φ(x) = sin2 (2π σ 2 x) { σ u (2πx) 2, φ(u) = 4 if u σ 0 otherwise. Note φ(0) = σ2 4, φ(0) = σ 4 = φ(0) σ, and evaluating the rime sum gives S 2 (.986 σ σ 2 log R ) me log R φ(0).

36 Lower order terms and average rank (cont) Let r t denote the number of zeros ) of E t at the central oint (i.e., the analytic rank). Then u to our O errors (which we think should be smaller), we have 1 N 2N t=n ( log log R log R r tφ(0) φ(0) ( σ + r + 1 ) (.986 φ(0)+ 2 Ave Rank [N,2N] (E) 1 σ + r (.986 σ σ 2 log R σ σ 2 log R ) me log R. ) me log R φ(0) σ = 1, m E = 1: for conductors of size 10 12, the average rank is bounded by 1+r = r This is significantly higher than Fermigier s 2 2 observed r σ = 2: lower order correction contributes.02 for conductors of size 10 12, the average rank bounded by 1 + r = r Now in the ballark of Fermigier s bound (already there without the otential correction term!).

37 Interretation: Aroaching semicircle 2nd moment from below Sato-Tate Law for Families without CM For large rimes, the distribution of a E(t) ()/, t {0, 1,..., 1}, aroaches a semicircle on [ 2, 2]. Figure: a E(t) () for y 2 = x 3 + Tx + 1 at the 2014th rime.

38 Imlications for Excess Rank Katz-Sarnak s one-level density statistic is used to measure the average rank of curves over a family. More curves with rank than exected have been observed, though this excess average rank vanishes in the limit. Lower-order biases in the moments of families exlain a small fraction of this excess rank henomenon.

39 Theoretical Evidence

40 Methods for Obtaining Exlicit Formulas For a family E : y 2 = x 3 + A(T)x + B(T), we can write a E(t) () = ( ) x 3 + A(t)x + B(t) ( where ( ) x x mod ) is the Legendre symbol mod given by = 1 if x is a non-zero square modulo 0 if x 0 mod 1 otherwise.

41 Lemmas on Legendre Symbols Linear and Quadratic Legendre Sums x mod x mod ( ) ax + b ( ) ax 2 + bx + c = 0 if a = ( ) a ( ( 1) a ) if b 2 4ac if b 2 4ac

42 Lemmas on Legendre Symbols Linear and Quadratic Legendre Sums x mod x mod ( ) ax + b ( ) ax 2 + bx + c = 0 if a = ( ) a ( ( 1) a ) if b 2 4ac if b 2 4ac Average Values of Legendre Symbols ( ) The value of for x Z, when averaged over all x rimes, is 1 if x is a non-zero square, and 0 otherwise.

43 Rank 0 Families Theorem (MMRW 14): Rank 0 Families Obeying the Bias Conjecture For families of the form E : y 2 = x 3 + ax 2 + bx + ct + d, ( ( ) ( )) 3 a A 2,E () = 2 2 3b The average bias in the size term is 2 or 1, according to whether a 2 3b Z is a non-zero square.

44 Families with Rank Theorem (MMRW 14): Families with Rank For families of the form E : y 2 = x 3 + at 2 x + bt 2, ( ( A 2,E () = 2 1+ ) ( 3 + )) 3a ( ( x() )) x 3 +ax 2. These include families of rank 0, 1, and 2. The average bias in the size terms is 3 or 2, according to whether 3a Z is a non-zero square.

45 Families with Comlex Multilication Theorem (MMRW 14): Families with Comlex Multilication For families of the form E : y 2 = x 3 +(at + b)x, ( ( )) 1 A 2,E () = ( 2 ) 1+. The average bias in the size term is 1. The size 2 term is not constant, but is on average 2, and an analogous Bias Conjecture holds.

46 Families with Unusual Distributions of Signs Theorem (MMRW 14): Families with Unusual Signs For the family E : y 2 = x 3 + Tx 2 (T + 3)x + 1, ( ( )) 3 A 2,E () = The average bias in the size term is 2. The family has an usual distribution of signs in the functional equations of the corresonding L-functions.

47 The Size 3/2 Term Theorem (MMRW 14): Families with a Large Error For families of the form E : y 2 = x 3 +(T + a)x 2 +(bt + b 2 ab + c)x bc, A 2,E () = ( ) cx(x + b)(bx c) x mod The size 3/2 term is given by an ellitic curve coefficient and is thus on average 0. The average bias in the size term is 3.

48 General Structure of the Lower Order Terms The lower order terms aear to always have no size 3/2 term or a size 3/2 term that is on average 0; exhibit their negative bias in the size term; be determined by olynomials in, ellitic curve coefficients, and congruence classes of (i.e., values of Legendre symbols).

49 Numerical Investigations

50 Numerical Methods As comlexity of coefficients increases, it is much harder to find an exlicit formula. We can always just calculate the second moment from the exlicit formula; if E: y 2 = f(x), we have A 2,E () = ( ) f(x) 2. t() x() Takes an hour for the first 500 rimes. Otimizations?

51 Numerical Methods Consider the family y 2 = f(x) = ax 3 +(bt + c)x 2 +(dt + e)x + f. By similar arguments used to rove secial cases, where A 2,E () = C 0 () C 1 () 1+# 1, C 0 () = x() C 1 () = # 1 = x() y(): β(x,y) 0 x(): β(x,x) 0 and β, A, and B are olynomials. ( A(x)A(y) ( ) A(x) 2, y(): A(x) 0 and A(y) 0 ), ( B(x)B(y) ),

52 Numerical Methods C o () ordinarily O( 2 ) to comute. Sum over zeros of β(x, y) mod Fixing an x, β is a quadratic in y. So, with the quadratic formula mod, we know where to look for y to see if there is a zero. Now O(); runs from 6000 th to 7000 th rime in an hour.

53 Potential Counterexamles Families of Rank as Large as 3 E : y 2 = x 3 + ax 2 + bt 2 x + ct 2 with b, c 0: A 2,E () = 2 + ( ) (x 3 + bx)(y 3 + by) P(x,y) 0 + ( ) ax 2 + c 2 ( x 3 + bx x 3 +bx 0 ( 2+ ( )) b x P(x,x) 0 mod ) 2 ( ) x 3 + bx 2 1 where P(x, y) = bx 2 y 2 + c(x 2 + xy + y 2 )+bc(x + y).

54 A Positive Size Term? ( )] 2 ax [ 2 +c x 3 +bx 0 can be +9 on average! Terms such as ( ) x 3 +bx 2, P(x,x) 0 ( ( )) b ( )] 2 x 2+, and [ 3 +bx x mod contribute negatively to the size bias. The term ( ) (x 3 +bx)(y 3 +by) P(x,y) 0 is of size 3/2. ( ) A 2,E () = 2 (x 3 + bx)(y 3 + by) + + P(x,y) 0 ( ) 2 x 3 ( )) + bx b ( 2+ ( P(x,x) 0 x mod where P(x, y) = bx 2 y 2 + c(x 2 + xy + y 2 )+bc(x + y). x 3 +bx 0 ( x 3 + bx ) ax 2 + c 2 ) 2 1

55 55 Gas: Intro Gas: Shifted Moments & Proofs Gas: Pros Bias: Intro Bias: Evidence Bias: Data Bias: Future Refs Analyzing the Size 3/2 Term We averaged ( ) (x 3 +bx)(y 3 +by) P(x,y) 0 over the first 10,000 rimes for several rank 3 families of the form E : y 2 = x 3 + ax 2 + bt 2 x + ct 2. Family Average y 2 = x 3 + 2x 2 4T 2 x + T y 2 = x 3 3x 2 T 2 x + 4T y 2 = x 3 + 4x 2 4T 2 x + 9T y 2 = x 3 + 5x 2 9T 2 x + 4T y 2 = x 3 5x 2 T 2 x + 9T y 2 = x 3 + 7x 2 9T 2 x + T

56 The Right Object to Study ( (x 3 +bx)(y 3 +by) c 3/2 () := P(x,y) 0 to study (for us multily by ). ) is not a natural object An examle distribution for y 2 = x 3 + 2x 3 4T 2 x + T 2. Figure: c 3/2 () over the first 10,000 rimes.

57 57 Gas: Intro Gas: Shifted Moments & Proofs Gas: Pros Bias: Intro Bias: Evidence Bias: Data Bias: Future Refs In Terms of Ellitic Curve Coefficients Comare it to the distribution of a sum of 2 ellitic curve coefficients. Figure: x 10,000 rimes. mod ( ) x 3 +x+1 x mod ( ) x 3 +x+2 over the first

58 More Error Distributions Figure: c 3/2 () for y 2 = 4x 3 + 5x 2 +(4T 2)x + 1, first 10,000 rimes.

59 More Error Distributions Figure: x mod ( ) x 3 +x+1 distribution, first 10,000 rimes.

60 More Error Distributions Figure: c 3/2 () over y 2 = 4x 3 +(4T + 1)x 2 +( 4T 18)x + 49, first 10,000 rimes.

61 More Error Distributions Figure: x mod ( ) x 5 +x 3 +x 2 +x+1 distribution, first 10,000 rimes.

62 Summary of 3/2 Term Investigations In the cases we ve studied, the size 3/2 terms aear to be governed by (hyer)ellitic curve coefficients; may be hiding negative contributions of size ; revent us from numerically measuring average biases that arise in the size terms.

63 Future Directions

64 Questions for Further Study Are the size 3/2 terms governed by (hyer)ellitic curve coefficients? Or at least other L-function coefficients? Does the average bias always occur in the terms of size? Does the Bias Conjecture hold similarly for all higher even moments? What other (families of) objects obey the Bias Conjecture? Kloosterman sums? Cus forms of a given weight and level? Higher genus curves?

65 References

66 References Gas: Gas between zeros of GL(2) L-functions (with Owen Barrett, Brian McDonald, Ryan Patrick, Caroline Turnage-Butterbaugh and Karl Winsor), rerint. htt://arxiv.org/df/ df. Biases: 1- and 2-level densities for families of ellitic curves: evidence for the underlying grou symmetries, Comositio Mathematica 140 (2004), htt://arxiv.org/df/math/ Variation in the number of oints on ellitic curves and alications to excess rank, C. R. Math. Re. Acad. Sci. Canada 27 (2005), no. 4, htt://arxiv.org/abs/math/ Investigations of zeros near the central oint of ellitic curve L-functions, Exerimental Mathematics 15 (2006), no. 3, htt://arxiv.org/df/math/ Lower order terms in the 1-level density for families of holomorhic cusidal newforms, Acta Arithmetica 137 (2009), htt://arxiv.org/df/ v4. Moments of the rank of ellitic curves (with Siman Wong), Canad. J. of Math. 64 (2012), no. 1, htt://web.williams.edu/mathematics/sjmiller/ublic_html/math/aers/ mwmomentsranksec812final.df

67 Funded by NSF Grants DMS , DMS and Williams College.

68 Thank you!