Serre’s Constant of Elliptic Curves Over the Rationals

References

• Adelmann, C. (2001). The Decomposition of Primes in Torsion Point Fields. Lecture Notes in Mathematics. 1761. Berlin: Springer-Verlag, p. vi+142.
   Google Scholar
• Baker, M. H., Gonzalez-Jimenez, E., Gonzalez, J., Poonen, B. (2005). Finiteness results for modular curves of genus at least 2. Amer. J. Math. 127(6): 1325–1387.
   Web of Science ®Google Scholar
• Balakrishnan, J. S., Dogra, N., Müller, J. S., Tuitman, J., Vonk, J. (2019). Explicit Chabauty-Kim for the Split Cartan Modular Curve of level 13. Ann. Math. 189: 885–944.
   Web of Science ®Google Scholar
• Baran, B. (2010). Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem. J. Number Theory. 130(12): 2753–2772.
   Web of Science ®Google Scholar
• Baran, B. (2014). An exceptional isomorphism between modular curves of level 13. J. Number Theory. 145: 273–300.
   Web of Science ®Google Scholar
• Banwait, B. S., Cremona, J. E. (2014). Tetrahedral elliptic curves and the local-global principle for isogenies. Algebra Number Theory. 8(5): 1201–1229.
   Web of Science ®Google Scholar
• Bilu, Y., Parent, P. (2011). Serre’s uniformity problem in the split Cartan case. Ann. Math. 173(1): 569–584.
   Web of Science ®Google Scholar
• Bilu, Y., Parent, P., and Rebolledo, M. (2013). Rational points on X0+(pr)(Q). Ann. Inst. Fourier (Grenoble). 63: 957–984.
   Web of Science ®Google Scholar
• Birch, B. J., Kuyk, W. (eds.). (1975). Modular Functions of One Variable IV. Lecture Notes in Mathematics 476. New York, NY: Springer.
   Google Scholar
• Bosma, W., Cannon, J., Fieker, C., and Steel, A. (eds.). (2018). Handbook of Magma Functions, Edition 2.23. http://magma.maths.usyd.edu.au/magma.
   Google Scholar
• Bruin, N., Stoll, M. (2009). Two-cover descent on hyperelliptic curves. Math. Comp. 78(268): 2347–2370.
   Web of Science ®Google Scholar
• Camacho-Navarro, C., Li, W., Morrow, J., Petok, J., Zureick-Brown, D. (In preparation). Modular curves of low composite level and genus zero subgroups.
   Google Scholar
• Cojocaru, A. C., Kani, E. (2005). On the surjectivity of the galois representations associated to non-cm elliptic curves (with an Appendix by Ernst Kani). Can. Math. Bull. 48(1): 16–31.
   Web of Science ®Google Scholar
• Cremona, J. E. (2016). Elliptic curve data for conductors up to 400.000. Available at: http://www.warwick.ac.uk/∼masgaj/ftp/data.
   Google Scholar
• Daniels, H. B. (2015). An infinite family of Serre curves. J. Number Theory. 155: 226–247.
   Web of Science ®Google Scholar
• Daniels, H. B., González-Jiménez, E. Magma scripts and electronic transcript of computations for the paper “Serre’s constant of elliptic curves over the rationals”. Available at: http://matematicas.uam.es/∼enrique.gonzalez.jimenez.
   Google Scholar
• Daniels, H. B., González-Jiménez, E., Xarles, X. (In preparation). Serre’s constant of elliptic curves over the rationals II.
   Google Scholar
• Daniels, H. B., Lozano-Robledo, Á., Najman, F., and Sutherland, A. V. (2017). Torsion subgroups of rational elliptic curves over the compositum of all cubic fields. Math. Comp. 87(309): 425–458.
   Web of Science ®Google Scholar
• Diamond, F., Shurman, J. (2005). A first course in modular forms. In: Graduate Texts in Mathematics, vol. 228. New York: Springer-Verlag, pp. 436, xvi+.
   Google Scholar
• Dokchitser, T., Dokchitser, V. (2012). Surjectivity of mod 2n representations of elliptic curves. Math. Z. 272(3–4): 961–964.
   Web of Science ®Google Scholar
• Duke, W. D. (1997). Elliptic curves with no exceptional primes. C. R. Math. Acad. Sci. Paris Sér. I. 325(8): 813–818.
   Google Scholar
• Elkies, N. D. (2006). Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9. arXiv:math/0612734.
   Google Scholar
• Faltings, G. (1983). Endlichkeitssätze für abelsche Varietten über Zahlkörpern. Invent. Math. 73(3): 349–366.
   Google Scholar
• Jones, N. (2009). Almost all elliptic curves are Serre curves. Trans. Amer. Math. Soc. 362 (3): 1547–1570.
   Web of Science ®Google Scholar
• Kenku, M. A. (1979). The modular curve X0(39) and rational isogeny. Math. Proc. Camb. Phil. Soc. 85: 21–23.
   Web of Science ®Google Scholar
• Kenku, M. A. (1980a). The modular curves X0(65) and X0(91) and rational isogeny. Math. Proc. Camb. Phil. Soc. 87: 15–20.
   Web of Science ®Google Scholar
• Kenku, M. A. (1980b). The modular curve X0(169) and rational isogeny. J. London Math. Soc. 22(2): 239–244.
   Google Scholar
• Kenku, M. A. (1981). The modular curve X0(125),X1(25) and X1(49). J. London Math. Soc. 23(2): 415–427.
   Google Scholar
• Lemos, P. (2018). Serre’s uniformity conjecture for elliptic curves with rational cyclic isogenies. Trans. Amer. Math. Soc. 371(1): 137–146.
   Web of Science ®Google Scholar
• Lemos, P. (2019). Some cases of Serre’s uniformity problem. Math. Z. 292(1–2): 739–762.
   Web of Science ®Google Scholar
• LMFDB Collaboration. The L-functions and modular forms database. Available at: http://www.lmfdb.org. Accessed 6 February, 2019.
   Google Scholar
• Mavrides, L. (2011). Modular curves and surjectivity of galois representations attached to elliptic curves over Q. Master thesis. University of Warwick.
   Google Scholar
• Mazur, B. (1977). Modular curves and the Eisenstein ideal. Publications Mathématiques de L’institut. Des. Hautes. Scientifiques. 47(1): 33–186.
   Google Scholar
• Mazur, B., Goldfeld, D. (1978). Rational isogenies of prime degree. Invent. Math. 44(2): 129–162.
   Web of Science ®Google Scholar
• Morrow, J. S. (2019). Composite images of galois for elliptic curves over Q & entanglement fields. Math. Comp. 88: 2389–2421. .
   Web of Science ®Google Scholar
• Rouse, J., Zureick-Brown, D. (2015). Elliptic curves over Q and 2-adic images of Galois. Res. Number Theory. 1: 34. (data files and subgroup descriptions available at: http://users.wfu.edu/rouseja/2adic/).
   Web of Science ®Google Scholar
• Serre, J.-P. (1971). Propriétès galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4): 259–331.
   Google Scholar
• Serre, J.-P. (1981). Quelques applications du théorème de densité de. Chebotarev. Inst. Hautes Études Sci. Publ. Math. 54: 323–401.
   Google Scholar
• Serre, J.-P. (1998). Abelian l-adic Representations and Elliptic Curves. Research Notes in Mathematics, vol. 7. Wellesley, MA: A K Peters, Ltd.
   Google Scholar
• Sutherland, A. V. (2016). Computing images of galois representations attached to elliptic curves. Forum Math. Sigma. 4: e4, 79.
   Web of Science ®Google Scholar
• Sutherland, A. V., Zywina, D. (2017). Modular curves of prime power-level with infinitely many rational points. Alg. Number Th. 11(5): 1199–1229.
   Web of Science ®Google Scholar
• Zywina, D. (2015). On the possible images of the mod ℓ representations associated to elliptic curves over Q. arXiv:1508.07660.
   Google Scholar
• Zywina, D. (2015). On the surjectivity of mod ℓ representations associated to elliptic curves. arXiv:1508.07661.
   Google Scholar