Numbers as functions

A. Abdessalam, A. Chandra and G. Guadagni, “Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions,” arXiv:1302.5971.

A. Beilinson, “p-Adic periods and derived De Rham cohomology,” Journ. AMS 25(3), 319–327 (2012); arXiv:1102.1294.

J. Borger, “Lambda-rings and the field with one element,” arXiv:0906.3146.

J. Borger and A. Buium, “Differential forms on arithmetic jet spaces,” Selecta Math. (N.S.) 17(2), 301–335 (2011). arXiv:0908.2512

J. Borger, “The basic geometry of Witt vectors, I: The affine case,” Algebra Number Theory 5(2), 231–285 (2011).

J. Borger, “Basic geometry of Witt vectors. II: Spaces,” Math. Ann. 351(4), 877–933 (2011).

A. Buium, “Differential characters of Abelian varieties over p-adic fields,” Inv. Math. 122, 309–340 (1995).

A. Buium, Arithmetic Differential Equations, AMS Math. Surveys and Monographs 118 (2005).

A. Buium and Y. Manin, “Arithmetic differential equations of Painlevé VI type,” arXiv:1307.3841.

A. Connes and C. Consani, “On the notion of geometry over F1,” J. Algebraic Geom. 20(3), 525–557 (2011).

A. Connes, C. Consani and M. Marcolli, “Fun with F 1,” J. Number Theory 129(6), 1532–1561 (2009); math.AG/0806.2401.

A. Deitmar, “Schemes over F 1,” In: Number Fields and Function Fields — Two Parallel Worlds, Ed. by G. van der Geer, B. Moonen, R. Schoof, Progr. in Math. 239 (2005); math.NT/0404185.

A. Erdogan, “A universal formula for the j-invariant of the canonical lifting,” arXiv:1211.1152.

A. Goncharov and Yu. Manin, “Multiple zeta-motives and moduli spaces,” Compos. Math. 140(1), 1–14 (2004); math.AG/0204102.

G. Faltings, “p-Adic Hodge theory,” J. Amer. Math. Soc. 1, 255–288 (1988).

M. Kapranov and A. Smirnov, “Cohomology determinants and reciprocity laws: number field case,” Unpublished manuscript, 15 pp.

N. Katz, “Serre-Tate local moduli,” Algebraic Surfaces (Orsay, 1976–78), pp. 138–202, Lecture Notes in Math. 868 (Springer, Berlin-New York, 1981).

M. Kontsevich and D. Zagier, “Periods,” In: Mathematics unlimited-2001 and beyond, pp. 771–808 (Springer, Berlin, 2001).

L. Le Bruyn, “Absolute geometry and the Habiro topology,” arXiv:1304.6532.

Yu. Manin, “Reflections on arithmetical physics,” In: Conformal Invariance and String Theory (Poiana Brasov, 1987), pp. 293–303 (Academic Press, Boston, MA, 1989). Reprinted in “Mathematics as Metaphor”, Selected Essays by Yu. I. Manin, pp. 149–155 (AMS 2007).

Yu. Manin, “Lectures on zeta functions and motives (according to Deninger and Kurokawa),” Astérisque 228(4), 121–163 (1995).

Yu. Manin, “Cyclotomy and analytic geometry over F 1,” In: Quanta of Maths. Conference in honour of Alain Connes, Clay Math. Proceedings 11, 385–408 (2010); math.AG/0809.2716.

Yu. Manin, “Renormalization and computation I: motivation and background,” In: Proceedings OPERADS 2009, eds. J. Loday and B. Vallette, Séminaires et Congrès 26, Soc. Math. de France, pp. 181–223 (2012); math.QA/0904.4921.

S. Müller-Stach, S. Weinzierl and R. Zayadeh, “Picard-Fuchs equations for Feynman integrals,” arXiv:1212.4389.

A. L. Smirnov, “Hurwitz inequalities for number fields,” Algebra i Analiz 4(2), 186–209 (1992) [in Russian]; transl. in St. Petersburg Math. J. 4 (2), 357–375 (1993).

A. L. Smirnov, “Absolute determinants and Hilbert symbols,” Preprint MPI 94/72 (Bonn, 1994).

C. Soulé, “Les variétés sur le corps à un élément,” Mosc. Math. J. 4(1), 217–244 (2004).

J. Tits, “Sur les analogues algébriques des groupes semi-simples complexes,” Colloque d’algèbre supérieure, Centre Belge de Recherches Mathématiques, pp. 261–289 (établissement Ceuterick, Louvain, 1957).

B. Toën and M. Vaquié, “Au-dessous de SpecZ,” Au-dessous de SpecZ [in French], (Under SpecZ), J. K-Theory 3(3), 437–500 (2009); math.AG/0509684.

V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Series on Soviet and East European Math. 1 (World Scientific, River Edge, NJ, 1994).

S. Weinzierl, “Periods and Hodge structures in perturbative quantum field theory,” arXiv:1302.0670 [hep-th].