Some effective estimates for André–Oort in Y(1)
                  n

Gal Binyamini; Emmanuel Kowalski

doi:10.1515/crelle-2019-0028

Article

Some effective estimates for André–Oort in Y(1)ⁿ

Gal Binyamini and Emmanuel Kowalski

Published/Copyright: September 11, 2019

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2020 Issue 767

Abstract

Let X ⊂ Y ⁢ ( 1 ) n be a subvariety defined over a number field 𝔽 and let ( P 1 , … , P n ) ∈ X be a special point not contained in a positive-dimensional special subvariety of X. We show that if a coordinate P i corresponds to an order not contained in a single exceptional Siegel–Tatuzawa imaginary quadratic field K * , then the associated discriminant | Δ ⁢ ( P i ) | is bounded by an effective constant depending only on deg ⁡ X and [ 𝔽 : ℚ ] . We derive analogous effective results for the positive-dimensional maximal special subvarieties.

From the main theorem we deduce various effective results of André–Oort type. In particular, we define a genericity condition on the leading homogeneous part of a polynomial, and give a fully effective André–Oort statement for hypersurfaces defined by polynomials satisfying this condition.

Funding statement: Gal Binyamini is the incumbent of the Dr. A. Edward Friedmann career development chair in mathematics. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 802107). This research was supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1167/17) and by funding received from the MINERVA Stiftung with the funds from the BMBF of the Federal Republic of Germany.

A Duke’s theorem avoiding Tatuzawa fields (by Emmanuel Kowalski at Zurich)

Duke’s theorem [9, Theorem 1] for CM points states that as d → + ∞ , the CM points Λ d of discriminant - d become equidistributed in the modular curve Y ⁢ ( 1 ) , in the quantitative form

| Λ d ∩ Ω | | Λ d | = μ ⁢ ( Ω ) + O ⁢ ( | d | - δ )

for some δ > 0 depending only on the domain Ω ⊂ Y ⁢ ( 1 ) , which is assumed to be convex with piecewise smooth boundary. Here μ is the hyperbolic area measure normalized so that μ ⁢ ( Y ⁢ ( 1 ) ) = 1 . The constant δ > 0 and the implied constant depend only on Ω; the former is effective (and explicit), but the latter is not, due to the use of Siegel’s lower bound for class numbers in the proof.

Tatuzawa [28, Theorem 1] has given a form of Siegel’s bound that, for a given ε > 0 , gives an effective (explicit) constant c ⁢ ( ε ) > 0 (in fact, proportional to ε) such that

| Λ d | > c ⁢ ( ε ) ⁢ | d | 1 2 - ε

for all fundamental discriminants d with at most one exception, which may depend on ε. We write d ε * for this exception.

Let F be the standard fundamental domain of SL 2 ⁡ ( ℤ ) ∖ ℍ . We view Λ d as a subset of F. We consider the regions

Ω R = { z ∈ F : 3 2 < Im ⁡ ( z ) < R } .

Note a minor issue even in a direct application of Duke’s theorem, without consideration of effectivity: this region is not convex in the hyperbolic sense.

The following is however a simple deduction from the principles of the proof, combined with Tatuzawa’s theorem.

Proposition 9.

Let 0 < ε < 1 16 be fixed. Let d ε * be the corresponding exceptional discriminant. Let m ≥ 1 be an integer. There exists an effective constant c ⁢ ( m , ε ) > 0 such that for d > c ⁢ ( m , ε ) and for d ≠ d ε * , we have

1 | Λ d | ⁢ | { z ∈ Λ d : z ∈ Ω 8 ⁢ m } | ≥ 1 - 1 4 ⁢ m .

Remark 10.

(1) The choice R = 8 ⁢ m is to have the hyperbolic area of Ω R equal to 1 - 1 R .

(2) We sketch the proof with fundamental discriminants, but proceeding as in [7], it extends to all discriminants.

Sketch of proof.

Recall that

| Λ d | = 1 2 ⁢ π ⁢ w d ⁢ | d | 1 2 ⁢ L ⁢ ( 1 , χ d ) ,

where w d is the number of roots of unity in the quadratic field with discriminant d, by Dirichlet’s Class Number Formula (see, e.g., [16, (22.59)]).

Let R = 8 ⁢ m . Consider a smooth compactly supported function ψ : Y ⁢ ( 1 ) → ℂ which satisfies 0 ≤ ψ ≤ 1 , is equal to 1 for z ∈ F with imaginary part ≤ R 2 (hence on Ω R / 2 ) and vanishes for z with imaginary part ≥ R , and such that the partial derivatives of ψ are bounded (by constants depending only on the order of the derivative).

Note that the choice of such a function depends on m. Observe that

1 | Λ d | ⁢ | { z ∈ Λ d : z ∈ Ω R } | ≥ 1 | Λ d | ⁢ ∑ z ∈ Λ d ψ ⁢ ( z ) .

Now by the spectral decomposition in L 2 ⁢ ( Y ⁢ ( 1 ) ) (see, e.g., [15, Theorem 4.7, Theorem 7.3]), we have

ψ ⁢ ( z ) = μ ψ + ∫ 0 + ∞ 〈 ψ , E ⁢ ( ⋅ , 1 2 + i ⁢ t ) 〉 ⁢ E ⁢ ( z , 1 2 + i ⁢ t ) ⁢ 𝑑 t + ∑ j 〈 ψ , u j 〉 ⁢ u j ⁢ ( z ) ,

where

μ ψ = ∫ F ψ ⁢ ( z ) ⁢ 𝑑 μ ⁢ ( z ) ,

the functions E ⁢ ( z , s ) are the Eisenstein series for SL 2 ⁡ ( ℤ ) and ( u j ) runs over an orthonormal basis of the cuspidal subspace of L 2 ⁢ ( Y ⁢ ( 1 ) ) , which we may assume consists of Hecke eigenforms.

We have

μ ψ ≥ μ ⁢ ( Ω R / 2 ) = 1 - 2 R ,

hence

1 | Λ d | ⁢ ∑ z ∈ Λ d ψ ⁢ ( z ) ≥ 1 - 1 4 ⁢ m + ℛ ,

where

ℛ = ∫ 0 + ∞ 〈 ψ , E ⁢ ( ⋅ , 1 2 + i ⁢ t ) 〉 ⁢ 1 | Λ d | ⁢ ∑ z ∈ Λ d E ⁢ ( z , 1 2 + i ⁢ t ) ⁢ d ⁢ t + ∑ j 〈 ψ , u j 〉 ⁢ 1 | Λ d | ⁢ ∑ z ∈ Λ d u j ⁢ ( z ) .

A classical formula (see references in [9, p. 88] or [16, (22.45)]) computes

1 | Λ d | ⁢ ∑ z ∈ Λ d E ⁢ ( z , 1 2 + i ⁢ t ) = w d ⁢ ζ ⁢ ( 1 2 + i ⁢ t ) ⁢ L ⁢ ( χ d , 1 2 + i ⁢ t ) | d | 1 4 - i ⁢ t 2 ⁢ L ⁢ ( 1 , χ d ) ,

where w d is the number of roots of unity in the quadratic field. Combining an old result of Weyl for ζ ⁢ ( s ) and a result of Heath-Brown [14], whose proof is effective, yields upper bounds

ζ ⁢ ( 1 2 + i ⁢ t ) ≪ ( 1 + | t | ) 1 6 , L ⁢ ( χ d , 1 2 + i ⁢ t ) ≪ | d | 1 6 + η ⁢ ( 1 + | t | ) 1 6 + η

for any η > 0 , where the implied constant is effective and depends only on η.

On the other hand, using the Waldspurger formula (see the discussion of Michel and Venkatesh [21, (2.5)]), one finds a formula of the (similar) type

| 1 | Λ d | ⁢ ∑ z ∈ Λ d u j ⁢ ( z ) | 2 = α ⁢ L ⁢ ( u j , 1 2 ) ⁢ L ⁢ ( u j × χ d , 1 2 ) | d | 1 2 ⁢ L ⁢ ( 1 , χ d ) 2

(where α is a constant) in terms of central values of twisted L-functions. We use the subconvexity estimate of Blomer and Harcos [5, Theorem 2] (although we could use also that of Michel and Venkatesh [22], or indeed any subconvex bound that has polynomial control in terms of the eigenvalue of u j would suffice, and there are many more versions): we have

L ⁢ ( u j × χ d , 1 2 ) ≪ | d | 3 8 ⁢ ( 1 + | t j | ) 3 , L ⁢ ( u j , 1 2 ) ≪ ( 1 + | t j | ) 3 ,

where the implied constants are effective and 1 4 + t j 2 is the Laplace eigenvalue of the cusp form u j (we have t j ∈ ℝ since it is known that there are no eigenvalues < 1 4 for Y ⁢ ( 1 ) ).

Using “integration by parts”, namely writing

〈 ψ , u j 〉 = 1 ( 1 4 + t j 2 ) A ⁢ 〈 ψ , Δ A ⁢ u j 〉 = 1 ( 1 4 + t j 2 ) A ⁢ 〈 Δ A ⁢ ψ , u j 〉 ,

we obtain for any A ≥ 1 the bound

| 〈 ψ , u j 〉 | ≤ 1 ( 1 4 + t j 2 ) A ⁢ ∥ Δ A ⁢ ψ ∥ ≪ A R 2 ⁢ A + 1 ( 1 + | t j | ) 2 ⁢ A

(since Δ A ⁢ ψ ⁢ ( z ) vanishes unless R 2 ≤ Im ⁡ ( z ) ≤ R , and the derivatives are bounded). Similarly, one gets

〈 ψ , E ⁢ ( ⋅ , 1 2 + i ⁢ t ) 〉 ≪ R 2 ⁢ A + 1 ( 1 + | t | ) 2 ⁢ A

for any A > 0 , where the implied constant depends on A and is effective.

Taking A fixed and large enough to make the integral and series converge absolutely (e.g., A = 3 ), we derive the lower bound

1 | Λ d | ⁢ | { z ∈ Λ d : z ∈ Ω 8 ⁢ m } | ≥ 1 - 2 R + O ⁢ ( R 2 ⁢ A + 1 ⁢ | d | 1 2 - 1 16 ⁢ | Λ d | - 1 ) ,

where the implied constant is effective, and hence for d ≠ d ε * , we obtain

1 | Λ d | ⁢ | { z ∈ Λ d : z ∈ Ω R } | ≥ 1 - 2 R + O ⁢ ( R 2 ⁢ A + 1 ⁢ | d | ε - 1 16 ) ,

where the implied constant is effective. The result now follows. ∎

Acknowledgements

I would like to express my gratitude to Jonathan Pila for discussions regarding effectivity issues surrounding André–Oort, and in particular for pointing me in the direction of Tatuzawa’s result; to Elon Lindenstrauss for suggesting Duke’s equidistribution result as a potential remedy for the compactness condition in my paper [4]; and to Gabriel Dill and the anonymous referee for some corrections and suggestions on the initial version of the manuscript. I also thank the Tokyo Institute of Technology for their hospitality during a visit in which some of this work was carried out.

References

[1] Y. Bilu and L. Kühne, Linear equations in singular moduli, Int. Math. Res. Not. IMRN, to appear. 10.1093/imrn/rny216Search in Google Scholar

[2] Y. Bilu, D. Masser and U. Zannier, An effective “theorem of André” for CM-points on a plane curve, Math. Proc. Cambridge Philos. Soc. 154 (2013), no. 1, 145–152. 10.1017/S0305004112000461Search in Google Scholar

[3] G. Binyamini, Multiplicity estimates: a Morse-theoretic approach, Duke Math. J. 165 (2016), no. 1, 95–128. 10.1215/00127094-3165220Search in Google Scholar

[4] G. Binyamini, Density of algebraic points on Noetherian varieties, Geom. Funct. Anal. 29 (2019), no. 1, 72–118. 10.1007/s00039-019-00475-7Search in Google Scholar

[5] V. Blomer and G. Harcos, Hybrid bounds for twisted L-functions, J. reine angew. Math. 621 (2008), 53–79. 10.1515/CRELLE.2008.058Search in Google Scholar

[6] E. Bombieri and U. Zannier, Heights of algebraic points on subvarieties of abelian varieties, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996), no. 4, 779–792. Search in Google Scholar

[7] L. Clozel and E. Ullmo, Équidistribution des points de Hecke, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins University Press, Baltimore (2004), 193–254. Search in Google Scholar

[8] H. Cohn, Introduction to the construction of class fields. Corrected reprint of the 1985 original, Dover Publications, New York 1994. Search in Google Scholar

[9] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), no. 1, 73–90. 10.1007/BF01393993Search in Google Scholar

[10] J. Freitag and T. Scanlon, Strong minimality and the j-function, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 1, 119–136. 10.4171/JEMS/761Search in Google Scholar

[11] A. Gabrielov, Multiplicity of a zero of an analytic function on a trajectory of a vector field, The Arnoldfest Toronto 1997, Fields Inst. Commun. 24, American Mathematical Society, Providence (1999), 191–200. 10.1090/fic/024/13Search in Google Scholar

[12] D. M. Goldfeld, The conjectures of Birch and Swinnerton-Dyer and the class numbers of quadratic fields, Journées arithmétiques de Caen, Astérisque 41–42, Société mathématique de France, Paris (1977), 219–227. Search in Google Scholar

[13] B. H. Gross and D. B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320. 10.1007/BF01388809Search in Google Scholar

[14] D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions. II, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 122, 157–167. 10.1093/qmath/31.2.157Search in Google Scholar

[15] H. Iwaniec, Introduction to the spectral theory of automorphic forms, Revista Matemática Iberoamericana, Madrid 1995. Search in Google Scholar

[16] H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence 2004. 10.1090/coll/053Search in Google Scholar

[17] L. Kühne, An effective result of André–Oort type II, Acta Arith. 161 (2013), no. 1, 1–19. 10.4064/aa161-1-1Search in Google Scholar

[18] L. Kühne, Intersections of class fields, preprint (2017), https://arxiv.org/abs/1709.00998. 10.4064/aa180717-9-6Search in Google Scholar

[19] S. Lang, Algebra, 3rd ed., Grad. Texts in Math. 211, Springer, New York 2002. 10.1007/978-1-4613-0041-0Search in Google Scholar

[20] D. Masser, Heights, transcendence, and linear independence on commutative group varieties, Diophantine approximation Cetraro 2000, Lecture Notes in Math. 1819, Springer, Berlin (2003), 1–51. 10.1007/3-540-44979-5_1Search in Google Scholar

[21] P. Michel and A. Venkatesh, Equidistribution, L-functions and ergodic theory: On some problems of Yu. Linnik, International Congress of Mathematicians. Vol. II, European Mathematical Society, Zürich (2006), 421–457. 10.4171/022-2/19Search in Google Scholar

[22] P. Michel and A. Venkatesh, The subconvexity problem for GL 2 , Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171–271. 10.1007/s10240-010-0025-8Search in Google Scholar

[23] Y. V. Nesterenko, Estimates for the number of zeros of certain functions, New advances in transcendence theory Durham 1986, Cambridge University Press, Cambridge (1988), 263–269. 10.1017/CBO9780511897184.017Search in Google Scholar

[24] J. Pila, On the algebraic points of a definable set, Selecta Math. (N.S.) 15 (2009), no. 1, 151–170. 10.1007/s00029-009-0527-8Search in Google Scholar

[25] J. Pila, O-minimality and the André–Oort conjecture for ℂ n , Ann. of Math. (2) 173 (2011), no. 3, 1779–1840. 10.4007/annals.2011.173.3.11Search in Google Scholar

[26] J. Pila and A. J. Wilkie, The rational points of a definable set, Duke Math. J. 133 (2006), no. 3, 591–616. 10.1215/S0012-7094-06-13336-7Search in Google Scholar

[27] T. Scanlon, Counting special points: Logic, Diophantine geometry, and transcendence theory, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 1, 51–71. 10.1090/S0273-0979-2011-01354-4Search in Google Scholar

[28] T. Tatuzawa, On a theorem of Siegel, Jap. J. Math. 21 (1951), 163–178. 10.4099/jjm1924.21.0_163Search in Google Scholar

[29] S.-W. Zhang, Equidistribution of CM-points on quaternion Shimura varieties, Int. Math. Res. Not. IMRN 2005 (2005), no. 59, 3657–3689. 10.1155/IMRN.2005.3657Search in Google Scholar

Received: 2018-10-09

Revised: 2019-05-15

Published Online: 2019-09-11

Published in Print: 2020-10-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/crelle-2019-0028

Some effective estimates for André–Oort in Y(1) n

Article

Abstract

A Duke’s theorem avoiding Tatuzawa fields (by Emmanuel Kowalski at Zurich)

Proposition 9.

Remark 10.

Sketch of proof.

Acknowledgements

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

Some effective estimates for André–Oort in Y(1)ⁿ