Bryden R Cais

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Journals/Publications

• Cais, B. R., & Booher, J. (2022). Iwasawa theory for p-torsion class group schemes in characteristic p. Nagoya Math J, 54. doi:10.1017/nmj.2022.30
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  This paper summarizes over six months of computations that Jeremy Booher and I have made regarding the growth of the p-torsion group scheme of Jacobians in genus stable Z_p-towers of curves. The computations use code that we developed and wrote, and exhibit a number of striking phenomena. Our paper includes several bold conjectures based on our observations.
• Booher, J., Cais, B., & Booher, J. (2020). A-numbers of curves in Artin–Schreier covers. Algebra & Number Theory, 14(3), 587-641. doi:10.2140/ant.2020.14.587
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  Let $\pi : Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship between the $a$-numbers of $Y$ and $X$ and the ramification of the map $\pi$. This is analogous to the relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$ and the ramification of the cover, so we instead give bounds on the $a$-number of $Y$. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.
• Cais, B. R. (2019). a-Numbers in Artin-Schreier Covers. Algebra & Number Theory, 14(3), 587–641.
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  Let $\pi : Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth,projective, geometrically connected curves over a perfect field ofcharacteristic $p>0$. We investigate the relationship between the $a$-numbersof $Y$ and $X$ and the ramification of the map $\pi$. This is analogous to therelationship between the genus (respectively $p$-rank) of $Y$ and $X$ given theRiemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in specialsituations, the $a$-number of $Y$ is not determined by the $a$-number of $X$and the ramification of the cover, so we instead give bounds on the $a$-numberof $Y$. We provide examples showing our bounds are sharp. The bounds come froma detailed analysis of the kernel of the Cartier operator.[Journal_ref: ]
• Liu, T., & Cais, B. (2019). Corrigendum to “Breuil–Kisin modules via crystalline cohomology”. Transactions of the American Mathematical Society, 373(3), 2251-2252. doi:10.1090/tran/7894
• Cais, B. (2018). The Geometry of Hida families I: $\Lambda$-adic de Rham cohomology. Mathematische Annalen, 372(1), 781--844. doi:10.1007/s00208-017-1608-1
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  Published online in Fall 2017, and in print Fall 2018.
• Cais, B. (2018). The geometry of Hida families I: Λ-adic de Rham cohomology. Mathematische Annalen, 372(1), 781-844. doi:10.1007/s00208-017-1608-1
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  We construct the \(\Lambda \)-adic de Rham analogue of Hida’s ordinary \(\Lambda \)-adic etale cohomology and of Ohta’s \(\Lambda \)-adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \(\mathbf {Q}_p\), we give a purely geometric proof of the expected finiteness, control, and \(\Lambda \)-adic duality theorems. Following Ohta, we then prove that our \(\Lambda \)-adic module of differentials is canonically isomorphic to the space of ordinary \(\Lambda \)-adic cuspforms. In the sequel (Cais, Compos Math, to appear) to this paper, we construct the crystalline counterpart to Hida’s ordinary \(\Lambda \)-adic etale cohomology, and employ integral p-adic Hodge theory to prove \(\Lambda \)-adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and (Cais, Compos Math, to appear), we will be able to provide a “cohomological” construction of the family of \((\varphi ,\Gamma )\)-modules attached to Hida’s ordinary \(\Lambda \)-adic etale cohomology by Dee (J Algebra 235(2), 636–664, 2001), as well as a new and purely geometric proof of Hida’s finiteness and control theorems. We are also able to prove refinements of the main theorems in Mazur and Wiles (Compos Math 59(2):231–264, 1986) and Ohta (J Reine Angew Math 463:49–98, 1995).
• Cais, B. (2018). The geometry of Hida families II: $\Lambda$-adic $(\varphi,\Gamma)$-modules and $\Lambda$-adic Hodge theory. Compositio Mathematica, 154, 719-760. doi:10.1112/S0010437X17007680
• Cais, B., & Liu, T. (2018). Breuil--Kisin modules via crystalline cohomology. Transactions of the AMS., 371(2), 1199--1230.
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  Article electronically published on Sept. 20, 2018
• Cais, B., & Lau, E. (2017). Dieudonn\'e crystals and Wach modules for $p$-divisible groups. Proceedings of the London Mathematical Society, 114(4), 733-763. doi:10.1112/plms.12021
• Liu, T., & Cais, B. (2017). Breuil-Kisin Modules via crystalline cohomology. Transactions of the American Mathematical Society, 371(2), 1199-1230. doi:10.1090/tran/7280
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  For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic etale cohomology $H^i_{\mathrm{et}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.
• Cais, B., & Liu, T. (2016). On $F$-crystalline representations. Documenta Mathematica, 21, 223--270.
• Cais, B., & Liu, T. (2016). On $F$-crystalline representations. Documenta Mathematica, 21, 223-270. doi:10.4171/dm/532
• Lubin, J., Davis, C., & Cais, B. (2016). A characterization of strictly APF extensions. Journal de Theorie des Nombres de Bordeaux, 28(2), 417-430. doi:10.5802/jtnb.946
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  Let K denote a finite extension of Qp. We give necessary and sufficient conditions for an infinite totally wildly ramified extension L/K to be strictly APF in the sense of Fontaine-Wintenberger. Our conditions are phrased in terms of the existence of a certain tower of intermediate subfields. These conditions are well-suited to producing examples of strictly APF extensions, and in particular, our main theorem proves that the phi-iterate extensions previously considered by the first two authors are strictly APF.
• Davis, C., & Cais, B. (2015). Canonical Cohen Rings for Norm Fields. International Mathematics Research Notices, 2015(14), 5473-5517. doi:10.1093/imrn/rnu098
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  Fix $K/\mathbf{Q}_p$ a finite extension and let $L/K$ be an infinite, strictly APF extension in the sense of Fontaine--Wintenberger. Let $X_K(L)$ denote its associated norm field. The goal of this paper is to associate to $L/K$, in a canonical and functorial way, a $p$-adically complete subring $\mathbf{A}_{L/K}^+ \subset \widetilde{\mathbf{A}}^+$ whose reduction modulo~$p$ is contained in the valuation ring of $X_K(L)$. When the extension $L/K$ is of a special form, which we call a $\varphi$-iterate extension, we prove that $X_K(L)$ is (at worst) a finite purely inseparable extension of the fraction field of $\mathbf{A}_{L/K}^+/(p)$. The class of $\varphi$-iterate extensions includes all Lubin--Tate extensions, as well as many other extensions such as the non-Galois ``Kummer" extension occurring in work of Faltings, Breuil, and Kisin. In particular, our work provides a canonical and functorial construction of every characteristic zero lift of the norm fields that have thus far played a foundational role in (integral) $p$-adic Hodge theory, as well as many other cases which have yet to be studied.
• Cais, B. (2014). On the $U_p$ operator in characteristic $p$. Math. Res. Lett., 21(2), 255--260.
• Cais, B., & Davis, C. (2014). Canonical Cohen Rings for Norm Fields. International Mathematics Research Notices.
• Cais, B., Davis, C., & Lubin, J. (2014). A characterization of strictly APF extensions. J. Th\'eorie des Nombres de Bordeaux.
• Cais, B., Ellenberg, J. S., & Zureick-Brown, D. (2013). Random Dieudonn\'e modules, random $p$-divisible groups, and random curves over finite fields. J. Inst. Math. Jussieu, 12(3), 651--676.
• Cais, B. (2010). Canonical extensions of N\'eron models of Jacobians. Algebra Number Theory, 4(2), 111--150.
• Cais, B. (2009). CANONICAL INTEGRAL STRUCTURES ON THE DE RHAM COHOMOLOGY OF CURVES. Annales de l'Institut Fourier, 59(6), 2255-2300. doi:10.5802/aif.2490
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  Soit R un anneau de valuation discrete de corps de fractions K et soit X K une courbe propre et lisse sur K. Nous montrons qu'on peut munir (sous certaines hypotheses faibles) la cohomologie de de Rham de X K sur K d'une structure entiere canonique: c'est-a-dire, d'un sous-R-reseau qui est fonctoriel pour les morphismes finis (et generiquement etales) de courbes sur K, et qui est son propre dual par rapport au cup-produit sur H 1 dR (X K /K). Notre construction de ce reseau utilise une classe de R-modeles normaux et propres de X K et les faisceaux dualisants relatifs. Nous montrons que notre reseau contient le reseau fourni par le complexe de de Rham (tronque) d'un R-modele propre et regulier de X K . L'indice pour cette inclusion est un invariant numerique de X K , qu'on appelle le conducteur de de Rham. Partant d'un travail de Bloch et de Liu-Saito, nous prouvons que le conducteur de de Rham est majore par le conducteur d'Artin, et minore par le conducteur efficace. Nous etudions ensuite comment la position de notre reseau canonique varie sous les extensions finies de scalaires.
• Cais, B. (2009). Canonical integral structures on the de Rham cohomology of curves. Ann. Inst. Fourier (Grenoble), 59(6), 2255--2300.
• Conrad, B., & Cais, B. (2006). Modular curves and Ramanujan's continued fraction. Crelle's Journal, 2006(597), 27-104. doi:10.1515/crelle.2006.063

Proceedings Publications

• Cais, B. R., Bhatt, B., Caraiani, A., Kedlaya, K., Weinstein, J., & Scholze, P. (2019, October, 2019). Perfectoid Spaces: Lectures from the 2017 Arizona Winter School. In 2017 Arizona Winter School.
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  Proceedings volume will be submitted in final form to AMS by March 1, 2019 and should appear in print shortly thereafter.

Presentations

• Cais, B. R. (2019, June 6). Iwasawa theory for function fields. UC Irvine number theory seminar. UC Irvine, Irvine California: UC Irvine Department of Mathematics.
• Cais, B. R. (2019, May 16). Iwasawa theory for function fields. Caltech number theory seminar. Pasadena, Californoa: Caltech Department of Mathematics.
• Cais, B. R. (2019, May 17). Iwasawa theory for function fields. UCLA number theory seminar. Los Angeles, California: UCLA Department of Mathematics.
• Cais, B. R. (2019, May 29). Iwasawa theory for function fields. 49th John H. Barrett Memorial Lecture Series. University of Tennessee, Knoxville, Tennessee: UT Department of Mathematics.
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  I gave one of the plenary invited lectures
• Cais, B. R. (2018, April 12). A Motivic Deuring-Shafarevich formula. UC Irvine number theory seminar. UC Irvine, Irvine California: UC Irvine Department of Mathematics.
• Cais, B. R. (2018, April 30). A Motivic Deuring-Shafarevich formula. Stanford number theory seminar. Stanford University, Palo Alto California: Stanford University Department of Mathematics.
• Cais, B. R. (2018, April 5). Breuil--Kisin modules via crystalline cohomology. 43rd Spring Lecture Series. University of Arkansas, Fayetteville AR: University of Arkansas Department of Mathematics.
• Cais, B. R. (2018, May 18). Motivic class groups in Z_p-towers of function fields. Iwasawa Theory and Related Topics. Heidelberg, Germany: University of Heidelberg.
• Cais, B. R. (2017, April 6). Breuil--Kisin modules via crystalline cohomology. Caltech number theory seminar. Caltech, Pasadena California: Caltech Department of Mathematics.
• Cais, B. R. (2017, December 14). A Motivic Deuring-Shafarevich formula. Quebec-Vermont number theory seminar. Montreal, Quebec Canada: McGill University/ CRM Montreal.
• Cais, B. R. (2017, May 19). Breuil--Kisin modules and crystalline cohomology. University of Padua number theory seminar. University of Padua, Padua Italy: University of Padua.
• Cais, B. R. (2017, May 7). Breuil--Kisin modules and crystalline cohomology. Simons symposium on p-adic Hodge theory. Schloss Elmau, Krun Germany: Simons Foundation.
• Cais, B. R. (2016, February 1). Kisin modules and crystalline cohomology. University of Michigan Group, Lie and Number Theory Seminar.
• Cais, B. R. (2016, January 26). Kisin modules and crystalline cohomology. Stanford University Number Theory Seminar.
• Cais, B. R. (2016, March 17). Kisin modules and crystalline cohomology. CICMA / QVNTS seminar (Montreal, Canada).
• Cais, B. R. (2016, May 18). Differentials on the modular tower and representations of GL2 mod p.. The p-adic Langlands program and related topics. Bloomington, IL.
• Cais, B. R. (2015, December 10, 2015). A Motivic Deuring Shafarevich Formula. Colorado State University FRAGMENT seminar. Fort Collins, CO: Colorado State University.
• Cais, B. R. (2015, January 13, 2015). On the restriction of $F$-crystalline $p$-adic Galois representations. Special Session "Recent Developments in Algebraic Number Theory, I". Joint Mathematics Meetings in San Antonio TX: AMS.
• Cais, B. R. (2015, July 29, 2015). On the restriction of $F$-crystalline $p$-adic Galois representations. AMS Summer Research Institute on Algebraic Geometry. University of Utah, Salt Lake City, Utah: AMS.
• Cais, B. R. (2014, January 23, 2014). Canonical lifts of norm fields and applications. Nagoya University Number Theory seminar. Nagoya, Japan: Nagoya University.
• Cais, B. R. (2014, January 24, 2014). Canonical lifts of norm fields and applications. University of Tokyo Number Theory seminar. Tokyo, Japan: Tokyo University.
• Cais, B. R. (2014, July 24, 2014). On a generalization of the Deuring-Shafarevich formula. Colorado State University FRAGMENT seminar. Fort Collins, CO: Colorado State University.
• Cais, B. R. (2014, November 17, 2014). Ramification in crystalline $p$-adic Galois representations. Northwestern University Number Theory Seminar. Evanston IL: Northwestern university.
• Cais, B. R. (2014, November 18, 2014). Ramification in crystalline $p$-adic Galois representations. University of Chicago Number Theory Seminar. Chicago IL: University of Chicago.
• Cais, B. R. (2014, October 23, 2014). Ramification in crystalline $p$-adic Galois representations. Purdue University Number Theory Seminar. West Lafayette IN: Purdue University.
• Cais, B. R. (2014, September 15, 2014). Crystalline Galois representations. University of Copenhagen Algebra/Topology seminar. Copenhagen, Denmark: University of Copenhagen.