Rank-one Drinfel’d modules on elliptic curves
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- by D. S. Dummit and David Hayes PDF
- Math. Comp. 62 (1994), 875-883 Request permission
Abstract:
The sgn-normalized rank-one Drinfeld modules $\phi$ associated with all elliptic curves E over ${\mathbb {F}_q}$ for $4 \leq q \leq 13$ are computed in explicit form. (Such $\phi$ for $q < 4$ were computed previously.) These computations verify a conjecture of Dorman on the norm of $j(\phi ) = {a^{q + 1}}$ and also suggest some interesting new properties of $\phi$. We prove Dorman’s conjecture in the ramified case. We also prove the formula $\deg N(a) = q({h_k} - 1 + q)$, where $N(a)$ is the norm of a and ${h_k}$ is the class number of $k = {\mathbb {F}_q}(E)$. We describe a remarkable conjectural property of the trace of a in even characteristic that holds in all the examples.References
- David R. Dorman, On singular moduli for rank $2$ Drinfel′d modules, Compositio Math. 80 (1991), no. 3, 235–256. MR 1134255
- D. S. Dummit, Genus two hyperelliptic Drinfel′d modules over $\textbf {F}_2$, The arithmetic of function fields (Columbus, OH, 1991) Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, Berlin, 1992, pp. 117–129. MR 1196515
- Ernst-Ulrich Gekeler, Drinfel′d modular curves, Lecture Notes in Mathematics, vol. 1231, Springer-Verlag, Berlin, 1986. MR 874338, DOI 10.1007/BFb0072692
- Ernst-Ulrich Gekeler, Zur Arithmetik von Drinfel′d-Moduln, Math. Ann. 262 (1983), no. 2, 167–182 (German). MR 690193, DOI 10.1007/BF01455309
- Benedict H. Gross and Don B. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220. MR 772491
- David R. Hayes, Explicit class field theory in global function fields, Studies in algebra and number theory, Adv. in Math. Suppl. Stud., vol. 6, Academic Press, New York-London, 1979, pp. 173–217. MR 535766
- David R. Hayes, Analytic class number formulas in function fields, Invent. Math. 65 (1981/82), no. 1, 49–69. MR 636879, DOI 10.1007/BF01389294
- David R. Hayes, Stickelberger elements in function fields, Compositio Math. 55 (1985), no. 2, 209–239. MR 795715
- David R. Hayes, On the reduction of rank-one Drinfel′d modules, Math. Comp. 57 (1991), no. 195, 339–349. MR 1079021, DOI 10.1090/S0025-5718-1991-1079021-7
- David R. Hayes, A brief introduction to Drinfel′d modules, The arithmetic of function fields (Columbus, OH, 1991) Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, Berlin, 1992, pp. 1–32. MR 1196509
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Math. Comp. 62 (1994), 875-883
- MSC: Primary 11G09; Secondary 11G15
- DOI: https://doi.org/10.1090/S0025-5718-1994-1218342-4
- MathSciNet review: 1218342