## Duke Mathematical Journal

### Exceptional isogenies between reductions of pairs of elliptic curves

François Charles

#### Abstract

Let $E$ and $E'$ be two elliptic curves over a number field. We prove that the reductions of $E$ and $E'$ at a finite place $\mathfrak{p}$ are geometrically isogenous for infinitely many $\mathfrak{p}$, and we draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether–Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.

#### Article information

Source
Duke Math. J., Volume 167, Number 11 (2018), 2039-2072.

Dates
Revised: 8 February 2018
First available in Project Euclid: 26 June 2018

https://projecteuclid.org/euclid.dmj/1530000176

Digital Object Identifier
doi:10.1215/00127094-2018-0011

Mathematical Reviews number (MathSciNet)
MR3843371

Zentralblatt MATH identifier
06941817

#### Citation

Charles, François. Exceptional isogenies between reductions of pairs of elliptic curves. Duke Math. J. 167 (2018), no. 11, 2039--2072. doi:10.1215/00127094-2018-0011. https://projecteuclid.org/euclid.dmj/1530000176

#### References

• [1] P. Autissier, Hauteur des correspondances de Hecke, Bull. Soc. Math. France 131 (2003), 421–433.
• [2] T. Barnet-Lamb, T. Gee, and D. Geraghty, The Sato–Tate conjecture for Hilbert modular forms, J. Amer. Math. Soc. 24 (2011), 411–469.
• [3] T. Barnet-Lamb, T. Gee, D. Geraghty, and R. Taylor, Potential automorphy and change of weight, Ann. of Math. (2) 179 (2014), 501–609.
• [4] T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor, A family of Calabi–Yau varieties and potential automorphy, II, Publ. Res. Inst. Math. Sci. 47 (2011), 29–98.
• [5] R. E. Borcherds, L. Katzarkov, T. Pantev, and N. I. Shepherd-Barron, Families of $K3$ surfaces, J. Algebraic Geom. 7 (1998), 183–193.
• [6] J.-B. Bost, Potential theory and Lefschetz theorems for arithmetic surfaces, Ann. Sci. Éc. Norm. Supér. (4) 32 (1999), 241–312.
• [7] C. Chai and F. Oort, Hypersymmetric abelian varieties, Pure Appl. Math. Q. 2 (2006), 1–27.
• [8] L. Clozel, M. Harris, and R. Taylor, Automorphy for some $\ell$-adic lifts of automorphic mod $\ell$ Galois representations, with Appendix A by the authors and Appendix B by M.-F. Vignéras, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181.
• [9] L. Clozel, H. Oh, and E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math. 144 (2001), 327–351.
• [10] P. Cohen, On the coefficients of the transformation polynomials for the elliptic modular function, Math. Proc. Cambridge Philos. Soc. 95 (1984), 389–402.
• [11] B. Conrad, “Gross–Zagier revisited” in Heegner Points and Rankin $L$-Series, with an appendix by W. R. Mann, Math. Sci. Res. Inst. Publ. 49, Cambridge Univ. Press, Cambridge, 2004, 67–163.
• [12] P. Deligne and M. Rapoport, “Les schémas de modules de courbes elliptiques” in Modular Functions of One Variable, II (Antwerp, 1972), Lecture Notes in Math. 349, Springer, Berlin, 1973, 143–316.
• [13] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Semin. Univ. Hambg. 14 (1941), 197–272.
• [14] D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995.
• [15] N. D. Elkies, Supersingular primes for elliptic curves over real number fields, Compos. Math. 72 (1989), 165–172.
• [16] G. Faltings. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366.
• [17] B. H. Gross, On canonical and quasicanonical liftings, Invent. Math. 84 (1986), 321–326.
• [18] B. H. Gross and D. B. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220.
• [19] B. H. Gross and D. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), 225–320.
• [20] M. Harris, “Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications” in Algebra, Arithmetic, and Geometry: In Honor of Y. I. Manin, Vol. II, Progr. Math. 270, Birkhäuser, Boston, 2009, 1–21.
• [21] M. Harris, N. Shepherd-Barron, and R. Taylor, A family of Calabi–Yau varieties and potential automorphy, Ann. of Math. (2) 171 (2010), 779–813.
• [22] N. Katz, “Serre–Tate local moduli” in Algebraic Surfaces (Orsay, 1976–78), Lecture Notes in Math. 868, Springer, Berlin, 1981, 138–202.
• [23] S. Lang and H. Trotter, Frobenius Distributions in $\mathrm{GL}_{2}$-Extensions: Distribution of Frobenius Automorphisms in ${\mathrm{GL}}_{2}$-Extensions of the Rational Numbers, Lecture Notes in Math. 504, Springer, Berlin, 1976.
• [24] J. Lubin and J. Tate, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France 94 (1966), 49–59.
• [25] K. Oguiso, Local families of $K3$ surfaces and applications, J. Algebraic Geom. 12 (2003), 405–433.
• [26] J. H. Silverman, Hecke points on modular curves, Duke Math. J. 60 (1990), 401–423.
• [27] J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144.
• [28] R. Taylor, Automorphy for some $\ell$-adic lifts of automorphic mod $\ell$ Galois representations, II, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 183–239.
• [29] C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spéc. 10, Soc. Math. France, Paris, 2002.