Robert Pollack

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• Emerton, M., Pollack, R., & Weston, T. (2025). EXPLICIT RECIPROCITY LAWS AND IWASAWA THEORY FOR MODULAR FORMS. Duke Mathematical Journal, 174(Issue 11). doi:10.1215/00127094-2024-0073
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  We prove that the Mazur–Tate elements of an eigenform f sit inside the Fitting ideals of the corresponding dual Selmer groups along the cyclotomic ℤp-extension (up to scaling by a single constant). Our method begins with a construction of local cohomology classes built via the p-adic local Langlands correspondence. From these classes, we build algebraic analogues of the Mazur–Tate elements which we directly verify sit in the appropriate Fitting ideals. Using Kato’s Euler system and explicit reciprocity laws, we prove that these algebraic elements divide the corresponding Mazur–Tate elements, implying our theorem.
• Pollack, R., & Wake, P. (2025). Iwasawa invariants in residually reducible Hida families. Tunisian Journal of Mathematics, 7(Issue 3). doi:10.2140/tunis.2025.7.755
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  We study the variation of μ-invariants of modular forms in a cuspidal Hida family in the case that the family intersects an Eisenstein family. We allow for intersections that occur because of “trivial zeros” (that is, because p divides an Euler factor) as in Mazur’s Eisenstein ideal paper, and pay special attention to the case of the 5-adic family passing through the elliptic curve X0(11).
• Bergdall, J., & Pollack, R. (2022). SLOPES OF MODULAR FORMS AND REDUCIBLE GALOIS REPRESENTATIONS, AN OVERSIGHT IN THE GHOST CONJECTURE. Proceedings of the American Mathematical Society, Series B, 9(Issue). doi:10.1090/bproc/136
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  The ghost conjecture, formulated by this article’s authors, predicts the list of p-adic valuations of the non-zero ap-eigenvalues (“slopes”) for overconvergent p-adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the “ghost series”). The prediction is restricted to eigenforms whose Galois representation modulo p is reducible on a decomposition group at p. It has been discovered, however, that the conjecture is not formulated correctly. Here we explain the issue and propose a salvage.
• Bergdall, J., & Pollack, R. (2019). Slopes of modular forms and the ghost conjecture, II. Transactions of the American Mathematical Society, 372(1). doi:10.1090/tran/7549
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  In a previous article we constructed an entire power series over p-adic weight space (the ghost series) and conjectured, in the Γ0(N)-regular case, that this series encodes the slopes of overconvergent modular forms of any p-adic weight. In this paper, we construct abstract ghost series which can be associated to various natural subspaces of overconvergent modular forms. This abstraction allows us to generalize our conjecture to, for example, the case of slopes of overconvergent modular forms with a fixed residual representation that is locally reducible at p. Ample numerical evidence is given for this new conjecture. Further, we prove that the slopes computed by any abstract ghost series satisfy a distributional result at classical weights (consistent with conjectures of Gouvêa) while the slopes form unions of arithmetic progressions at all weights not in Zp.
• Bergdall, J., & Pollack, R. (2019). Slopes of modular forms and the ghost conjecture. International Mathematics Research Notices, 2019(4). doi:10.1093/imrn/rnx141
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  We formulate a conjecture on slopes of overconvergent p-adic cusp forms of any padic weight in the-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes at the boundary of weight space.
• Pollack, R., & Bellaïche, J. (2019). Congruences with Eisenstein series and invariants. Compositio Mathematica, 155(5). doi:10.1112/s0010437x19007127
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  We study the variation of -invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the -adic zeta function. This lower bound forces these -invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the -adic -function is simply a power of up to a unit (i.e. ). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.
• Kim, C., Pollack, R., & Weston, T. (2017). On the freeness of anticyclotomic Selmer groups of modular forms. International Journal of Number Theory, 13(6). doi:10.1142/s1793042117500804
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  We establish the freeness of certain anticyclotomic Selmer groups of modular forms. The freeness of these Selmer groups plays a key role in the Euler system arguments introduced by Bertolini and Darmon in their work on the anticyclotomic main conjecture for modular forms. In particular, our result fills some implicit gaps which appeared in generalizations of the Bertolini-Darmon result to the case where the associated residual representation is not minimally ramified. The removal of such a minimal ramification hypothesis is essential for applications involving congruences of modular forms.
• Bergdall, J., & Pollack, R. (2016). Arithmetic properties of Fredholm series for p-adic modular forms. Proceedings of the London Mathematical Society, 113(4). doi:10.1112/plms/pdw031
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  We study the relationship between recent conjectures on slopes of overconvergent $p$-adic modular forms 'near the boundary' of $p$-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the $U-p$ operator never vanish modulo $p$, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo $p$ using a modular interpretation of the mod $p$ reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.
• Dummit, E., Hablicsek, M., Harron, R., Jain, L., Pollack, R., & Ross, D. (2016). Explicit computations of Hida families via overconvergent modular symbols. Research in Number Theory, 2(1). doi:10.1007/s40993-016-0052-8
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  In Pollack and Stevens (Ann Sci Éc Norm Supér 44(1):1–42, 2011), efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of p-adic L-functions and have further been applied to compute rational points on elliptic curves (e.g. Darmon and Pollack in Israel J Math 153:319–354, 2006, Trifkovic in Duke Math J 135(3):415–453, 2006). In this paper, we generalize these algorithms to the case of families of overconvergent modular symbols. As a consequence, we can compute p-adic families of Hecke-eigenvalues, two-variable p-adic L-functions, L-invariants, as well as the shape and structure of ordinary Hida–Hecke algebras.
• Pollack, R., & Stevens, G. (2013). Critical slope p-adic L-functions. Journal of the London Mathematical Society, 87(2). doi:10.1112/jlms/jds057
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  Let g be an eigenform of weight k+2 on Γ0(p) ∩Γ1(N) with p | 2224; N. If g is non-critical (that is, of slope less than k+1), using the methods of Amice-Vélu and Višik, one can attach ['Distributions p-adiques associées aux séries de Hecke', Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), Astérisque 24-25 (Soc. Math. France, Paris, 1975) 119-131 (French)] and Višik [Mat. Sb. (N.S.) 99 (1976) 248-260], then one can attach a p-adic L-function to g which is uniquely determined by its interpolation property together with a bound on its growth. However, in the critical slope case, the corresponding growth bound is too large to uniquely determine the p-adic L-function with its standard interpolation property.In this paper, using the theory of overconvergent modular symbols, we give a natural definition of p-adic L-functions in this critical slope case. If, moreover, the modular form is not in the image of theta, then the p-adic L-function satisfies the standard interpolation property. © 2012 London Mathematical Society.
• Dasgupta, S., Darmon, H., & Pollack, R. (2011). Hilbert modular forms and the Gross-Stark conjecture. Annals of Mathematics, 174(1). doi:10.4007/annals.2011.174.1.12
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  Let F be a totally real field and χ an abelian totally odd character of F. In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt's conjecture holds, assuming that either there are at least two primes above p in F, or that a certain condition relating the L-invariants of χ and χ-1 holds. This condition on L-invariants is always satisfied when χ is quadratic.
• Pollack, R., & Stevens, G. (2011). Overconvergent modular symbols and p-adic L-functions. Annales Scientifiques de l'Ecole Normale Superieure, 44(1). doi:10.24033/asens.2139
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  This paper is a constructive investigation of the relationship between classical modular symbols and overconvergent p-adic modular symbols. Specifically, we give a constructive proof of a control theorem (Theorem1.1) due to the second author [19] proving existence and uniqueness of overconvergent eigenliftings of classical modular eigensymbols of non-critical slope. As an application we describe a polynomial-time algorithm for explicit computation of associated p-adic L-functions in this case. In the case of critical slope, the control theorem fails to always produce eigenliftings (see Theorem 5.14 and [16] for a salvage), but the algorithm still "succeeds" at producing p-adic L-functions. In the final two sections we present numerical data in several critical slope examples and examine the Newton polygons of the associated p-adic L-functions.
• Pollack, R., & Weston, T. (2011). Mazur-tate elements of nonordinary modular forms. Duke Mathematical Journal, 156(3). doi:10.1215/00127094-2010-214
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  We establish formulae for the Iwasawa invariants of Mazur-Tate elements of cuspidal eigenforms, generalizing known results in weight 2. Our first theorem deals with forms of "medium" weight, and our second deals with forms of small slope. We give examples illustrating the strange behavior which can occur in the high-weight, high-slope case. © 2011.
• Pollack, R., & Weston, T. (2011). On anticyclotomic μ-invariants of modular forms. Compositio Mathematica, 147(5). doi:10.1112/s0010437x11005318
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  We prove the μ-part of the main conjecture for modular forms along the anticyclotomic Zp-extension of a quadratic imaginary field. Our proof consists of first giving an explicit formula for the algebraic μ-invariant, and then using results of Ribet and Takahashi showing that our formula agrees with Vatsal’s formula for the analytic μ-invariant. © 2011, Foundation Compositio Mathematica. All rights reserved.
• Pollack, D., & Pollack, R. (2009). A Construction of Rigid Analytic Cohomology Classes for Congruence Subgroups of SL3 (ℤ). Canadian Journal of Mathematics, 61(3). doi:10.4153/cjm-2009-036-0
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  We give a constructive proof, in the special case of GL3, of a theorem of Ash and Stevens which compares overconvergent cohomology to classical cohomology. Namely, we show that every ordinary classical Hecke-eigenclass can be lifted uniquely to a rigid analytic eigenclass. Our basic method builds on the ideas of M. Greenberg; we first form an arbitrary lift of the classical eigenclass to a distribution-valued cochain. Then, by appropriately iterating the Up-operator, we produce a cocycle whose image in cohomology is the desired eigenclass. The constructive nature of this proof makes it possible to perform computer computations to approximate these interesting overconvergent eigenclasses. © Canadian Mathematical Society 2009.
• Emerton, M., Pollack, R., & Weston, T. (2006). Variation of Iwasawa invariants in Hida families. Inventiones Mathematicae, 163(3). doi:10.1007/s00222-005-0467-7
• Pollack, R. (2005). An algebraic version of a theorem of Kurihara. Journal of Number Theory, 110(1). doi:10.1016/j.jnt.2003.10.008
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  Let E/Q be an elliptic curve and let p be an odd supersingular prime for E. In this article, we study the simplest case of Iwasawa theory for elliptic curves, namely when E(Q) is finite, III (E/Q) has no p-torsion and the Tamagawa factors for E are all prime to p. Under these hypotheses, we prove that E(Qn) is finite and make precise statements about the size and structure of the p-power part of III (E/Qn). Here Qn is the n-th step in the cyclotomic Zp-extension of Q. © 2004 Elsevier Inc. All rights reserved.
• Pollack, R., & Rubin, K. (2004). The main conjecture for CM elliptic curves at supersingular primes. Annals of Mathematics, 159(1). doi:10.4007/annals.2004.159.447
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  At a prime of ordinary reduction, the Iwasawa "main conjecture" for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the p-adic L-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case. Namely, Kobayashi's conjecture relates modified Selmer groups, which he defined, with modified p-adic L-functions defined by the first author. In this paper we prove Kobayashi's conjecture for elliptic curves with complex multiplication.
• Pollack, R. (2003). On the p-adic L-function of a modular form at a supersingular prime. Duke Mathematical Journal, 118(3). doi:10.1215/s0012-7094-03-11835-9
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  In this paper we study the two p-adic L-functions attached to a modular form f = ∑ anqn at a supersingular prime p. When ap = 0, we are able to decompose both the sum and the difference of the two unbounded distributions attached to f into a bounded measure and a distribution that accounts for all of the growth. Moreover, this distribution depends only upon the weight of f (and the fact that ap vanishes). From this description we explain how the p-adic L-function is controlled by two Iwasawa functions and by two power series with growth which have a fixed infinite set of zeros (Theorem 5.1). Asymptotic formulas for the p-part of the analytic size of the Tate-Shafarevich group of an elliptic curve in the cyclotomic direction are computed using this result. These formulas compare favorably with results established by M. Kurihara in [11] and B. Perrin-Riou in [23] on the algebraic side. Moreover, we interpret Kurihara's conjectures on the Galois structure of the Tate-Shafarevich group in terms of these two Iwasawa functions.