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Journal articles on the topic 'Bloch-Kato conjecture'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Huber, Annette, and Guido Kings. "A cohomological Tamagawa number formula." Nagoya Mathematical Journal 202 (June 2011): 45–75. http://dx.doi.org/10.1215/00277630-1260441.

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AbstractFor smooth linear group schemes over ℤ, we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori, the cohomological and the motivic Tamagawa measures coincide, which proves again the Bloch-Kato conjecture for motives associated to tori.

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2

Huber, Annette, and Guido Kings. "A cohomological Tamagawa number formula." Nagoya Mathematical Journal 202 (June 2011): 45–75. http://dx.doi.org/10.1017/s0027763000010242.

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AbstractFor smooth linear group schemes over ℤ, we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori, the cohomological and the motivic Tamagawa measures coincide, which proves again the Bloch-Kato conjecture for motives associated to tori.

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3

Swinnerton-Dyer, Sir Peter. "Diagonal hypersurfaces and the Bloch-Kato conjecture, I." Journal of the London Mathematical Society 90, no. 3 (October 20, 2014): 845–60. http://dx.doi.org/10.1112/jlms/jdu055.

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4

Guo, Li. "On the Bloch–Kato Conjecture for HeckeL-Functions." Journal of Number Theory 57, no. 2 (April 1996): 340–65. http://dx.doi.org/10.1006/jnth.1996.0053.

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5

Asok, Aravind. "Rationality problems and conjectures of Milnor and Bloch–Kato." Compositio Mathematica 149, no. 8 (June 3, 2013): 1312–26. http://dx.doi.org/10.1112/s0010437x13007021.

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AbstractWe show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number $\ell $ and any integer $n\geq 2$, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree $n$ unramified étale cohomology class with $\ell $-torsion coefficients. When $\ell = 2$, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified étale cohomology class of lower degree.

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6

Tamiozzo, Matteo. "On the Bloch–Kato conjecture for Hilbert modular forms." Mathematische Zeitschrift 299, no. 1-2 (January 30, 2021): 427–58. http://dx.doi.org/10.1007/s00209-020-02689-0.

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AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

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7

Voevodsky, Vladimir. "Motives over simplicial schemes." Journal of K-Theory 5, no. 1 (February 2010): 1–38. http://dx.doi.org/10.1017/is010001030jkt107.

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AbstractThis paper was written as a part of [8] and is intended primarily to provide the definitions and results concerning motives over simplicial schemes, which are used in the proof of the Bloch-Kato conjecture.

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8

Kings, Guido, and Annette Huber. "Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters." Duke Mathematical Journal 119, no. 3 (September 2003): 393–464. http://dx.doi.org/10.1215/s0012-7094-03-11931-6.

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9

DUMMIGAN, NEIL. "RATIONAL TORSION ON OPTIMAL CURVES." International Journal of Number Theory 01, no. 04 (December 2005): 513–31. http://dx.doi.org/10.1142/s1793042105000340.

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Vatsal has proved recently a result which has consequences for the existence of rational points of odd prime order ℓ on optimal elliptic curves over ℚ. When the conductor N is squarefree, ℓ ∤ N and the local root number wp= -1 for at least one prime p | N, we offer a somewhat different proof, starting from an explicit cuspidal divisor on X0(N). We also prove some results linking the vanishing of L(E,1) with the divisibility by ℓ of the modular parametrization degree, fitting well with the Bloch–Kato conjecture for L( Sym2E,2), and with an earlier construction of elements in Shafarevich–Tate groups. Finally (following Faltings and Jordan) we prove an analogue of the result on ℓ-torsion for cuspidal Hecke eigenforms of level one (and higher weight), thereby strengthening some existing evidence for another case of the Bloch–Kato conjecture.

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10

DUMMIGAN, NEIL. "SYMMETRIC SQUARE L-FUNCTIONS AND SHAFAREVICH–TATE GROUPS, II." International Journal of Number Theory 05, no. 07 (November 2009): 1321–45. http://dx.doi.org/10.1142/s1793042109002699.

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We re-examine some critical values of symmetric square L-functions for cusp forms of level one. We construct some more of the elements of large prime order in Shafarevich–Tate groups, demanded by the Bloch–Kato conjecture. For this, we use the Galois interpretation of Kurokawa-style congruences between vector-valued Siegel modular forms of genus two (cusp forms and Klingen–Eisenstein series), making further use of a construction due to Urban. We must assume that certain 4-dimensional Galois representations are symplectic. Our calculations with Fourier expansions use the Eholzer–Ibukiyama generalization of the Rankin–Cohen brackets. We also construct some elements of global torsion which should, according to the Bloch–Kato conjecture, contribute a factor to the denominator of the rightmost critical value of the standard L-function of the Siegel cusp form. Then we prove, under certain conditions, that the factor does occur.

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11

Liu, Yifeng, Yichao Tian, Liang Xiao, Wei Zhang, and Xinwen Zhu. "On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives." Inventiones mathematicae 228, no. 1 (January 21, 2022): 107–375. http://dx.doi.org/10.1007/s00222-021-01088-4.

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12

Brown, Jim. "Saito–Kurokawa lifts and applications to the Bloch–Kato conjecture." Compositio Mathematica 143, no. 02 (March 2007): 290–322. http://dx.doi.org/10.1112/s0010437x06002466.

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13

Spiess, Michael, and Takao Yamazaki. "A counterexample to generalizations of the Milnor-Bloch-Kato conjecture." Journal of K-Theory 4, no. 1 (August 2009): 77–90. http://dx.doi.org/10.1017/is008008014jkt066.

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AbstractWe construct an example of a torus T over a field K for which the Galois symbol K(K;T,T)/nK(K;T,T) → H2(K,T[n] ⊗ T[n]) is not injective for some n. Here K(K;T,T) is the Milnor K-group attached to T introduced by Somekawa. We show also that the motive M(T × T) gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson).

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14

Diamond, Fred, Matthias Flach, and Li Guo. "The Bloch-Kato conjecture for adjoint motives of modular forms." Mathematical Research Letters 8, no. 4 (2001): 437–42. http://dx.doi.org/10.4310/mrl.2001.v8.n4.a4.

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15

Burns, David. "On Artin formalism for the conjecture of Bloch and Kato." Mathematical Research Letters 19, no. 5 (2012): 1155–69. http://dx.doi.org/10.4310/mrl.2012.v19.n5.a16.

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16

Kim, Byoung Du, Riad Masri, and Tong Hai Yang. "Nonvanishing of Hecke L-functions and the Bloch–Kato conjecture." Mathematische Annalen 349, no. 2 (May 1, 2010): 301–43. http://dx.doi.org/10.1007/s00208-010-0521-7.

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17

de Jeu, Rob, and James D. Lewis. "Beilinson's Hodge Conjecture for Smooth Varieties." Journal of K-Theory 11, no. 2 (March 6, 2013): 243–82. http://dx.doi.org/10.1017/is013001030jkt212.

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AbstractLet U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, andclr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

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18

Koya, Yoshihiro. "The Bloch–Kato Conjecture for Good Reduction Curves over Local Fields." K-Theory 18, no. 1 (September 1999): 19–32. http://dx.doi.org/10.1023/a:1007834128416.

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19

DUMMIGAN, NEIL. "CONGRUENCES OF SAITO–KUROKAWA LIFTS AND DENOMINATORS OF CENTRAL SPINOR L-VALUES." Glasgow Mathematical Journal 64, no. 2 (October 14, 2021): 504–25. http://dx.doi.org/10.1017/s0017089521000331.

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AbstractFollowing Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.

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20

DUMMIGAN, NEIL. "CONGRUENCES OF SAITO–KUROKAWA LIFTS AND DENOMINATORS OF CENTRAL SPINOR L-VALUES." Glasgow Mathematical Journal 64, no. 2 (October 14, 2021): 504–25. http://dx.doi.org/10.1017/s0017089521000331.

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AbstractFollowing Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.

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21

Voineagu, Mircea. "Cylindrical homomorphisms and Lawson homology." Journal of K-Theory 8, no. 1 (June 8, 2010): 135–68. http://dx.doi.org/10.1017/is010004024jkt108.

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AbstractWe use the cylindrical homomorphism and a geometric construction introduced by J. Lewis to study the Lawson homology groups of certain hypersurfaces X ⊂ ℙn + 1 of degree d ℙ n + 1. As an application, we compute the rational semi-topological K-theory of generic cubics of dimensions 5, 6 and 8 and, using the Bloch-Kato conjecture, we prove Suslin's conjecture for these varieties. Using generic cubic sevenfolds, we show that there are smooth projective varieties such that the lowest nontrivial step in their s-filtration is infinitely generated and undetected by the Abel-Jacobi map.

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22

Sivatski, A. S. "Applications of the Bloch–Kato conjecture to cohomological invariants and symbol length." Mathematische Zeitschrift 299, no. 1-2 (February 3, 2021): 459–72. http://dx.doi.org/10.1007/s00209-020-02678-3.

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23

Agarwal, Mahesh, and Krzysztof Klosin. "Yoshida lifts and the Bloch–Kato conjecture for the convolution L -function." Journal of Number Theory 133, no. 8 (August 2013): 2496–537. http://dx.doi.org/10.1016/j.jnt.2013.01.009.

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24

Klosin, Krzysztof. "Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture." Annales de l’institut Fourier 59, no. 1 (2009): 81–166. http://dx.doi.org/10.5802/aif.2427.

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25

Hernandez, Valentin. "Families of Picard modular forms and an application to the Bloch–Kato conjecture." Compositio Mathematica 155, no. 7 (June 25, 2019): 1327–401. http://dx.doi.org/10.1112/s0010437x1900736x.

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In this article we construct a p-adic three-dimensional eigenvariety for the group $U$(2,1)($E$), where $E$ is a quadratic imaginary field and $p$ is inert in $E$. The eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta, Iovita and Stevens [$p$-adic families of Siegel modular cuspforms Ann. of Math. (2) 181, (2015), 623–697] by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch–Kato conjecture for some Galois characters of $E$, extending the results of Bellaiche and Chenevier to the case of a positive sign.

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26

Agarwal, Mahesh, and Jim Brown. "On the Bloch–Kato conjecture for elliptic modular forms of square-free level." Mathematische Zeitschrift 276, no. 3-4 (November 1, 2013): 889–924. http://dx.doi.org/10.1007/s00209-013-1226-x.

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27

BROWN, JIM. "SPECIAL VALUES OF L-FUNCTIONS ON GSp4 × GL2 AND THE NON-VANISHING OF SELMER GROUPS." International Journal of Number Theory 06, no. 08 (December 2010): 1901–26. http://dx.doi.org/10.1142/s1793042110003769.

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In this paper, we show how one can use an inner product formula of Heim giving the inner product of the pullback of an Eisenstein series from Sp10 to Sp 2 × Sp 4 × Sp 4 with a new-form on GL2 and a Saito–Kurokawa lift to produce congruences between Saito–Kurokawa lifts and non-CAP forms. This congruence is in part controlled by the L-function on GSp 4 × GL 2. The congruence is then used to produce nontrivial torsion elements in an appropriate Selmer group, providing evidence for the Bloch–Kato conjecture.

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28

Kings, Guido. "The Bloch-Kato conjecture on special values of $L$-functions. A survey of known results." Journal de Théorie des Nombres de Bordeaux 15, no. 1 (2003): 179–98. http://dx.doi.org/10.5802/jtnb.396.

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29

Berger, Tobias. "On the Eisenstein ideal for imaginary quadratic fields." Compositio Mathematica 145, no. 03 (April 15, 2009): 603–32. http://dx.doi.org/10.1112/s0010437x09003984.

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AbstractFor certain algebraic Hecke charactersχof an imaginary quadratic fieldFwe define an Eisenstein ideal in ap-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the specialL-valueL(0,χ). We further prove that its index is bounded from above by thep-valuation of the order of the Selmer group of thep-adic Galois character associated toχ−1. This uses the work of R. Tayloret al. on attaching Galois representations to cuspforms of GL2/F. Together these results imply a lower bound for the size of the Selmer group in terms ofL(0,χ), coinciding with the value given by the Bloch–Kato conjecture.

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30

BENOIS, D., and T. NGUYENQUANGDO. "Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs sur un corps abélien." Annales Scientifiques de l’École Normale Supérieure 35, no. 5 (September 2002): 641–72. http://dx.doi.org/10.1016/s0012-9593(02)01104-7.

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31

KLOSIN, Krzysztof. "The Maass space for $U(2,2)$ and the Bloch–Kato conjecture for the symmetric square motive of a modular form." Journal of the Mathematical Society of Japan 67, no. 2 (April 2015): 797–860. http://dx.doi.org/10.2969/jmsj/06720797.

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32

Sorensen, Claus M. "Level-raising for Saito–Kurokawa forms." Compositio Mathematica 145, no. 4 (July 2009): 915–53. http://dx.doi.org/10.1112/s0010437x09004084.

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AbstractThis paper provides congruences between unstable and stable automorphic forms for the symplectic similitude group GSp(4). More precisely, we raise the level of certain CAP representations Π arising from classical modular forms. We first transfer Π to π on a suitable inner form G; this is achieved by θ-lifting. For π, we prove a precise level-raising result that is inspired by the work of Bellaiche and Clozel and which relies on computations of Schmidt. We thus obtain a $\tilde {\pi }$ congruent to π, with a local component that is irreducibly induced from an unramified twist of the Steinberg representation of the Klingen parabolic. To transfer $\tilde {\pi }$ back to GSp(4), we use Arthur’s stable trace formula. Since $\tilde {\pi }$ has a local component of the above type, all endoscopic error terms vanish. Indeed, by results due to Weissauer, we only need to show that such a component does not participate in the θ-correspondence with any GO(4); this is an exercise in using Kudla’s filtration of the Jacquet modules of the Weil representation. We therefore obtain a cuspidal automorphic representation $\tilde {\Pi }$ of GSp(4), congruent to Π, which is neither CAP nor endoscopic. It is crucial for our application that we can arrange for $\tilde {\Pi }$ to have vectors fixed by the non-special maximal compact subgroups at all primes dividing N. Since G is necessarily ramified at some prime r, we have to show a non-special analogue of the fundamental lemma at r. Finally, we give an application of our main result to the Bloch–Kato conjecture, assuming a conjecture of Skinner and Urban on the rank of the monodromy operators at the primes dividing N.

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33

Seveso, Marco Adamo. "p-adic L-functions and the Rationality of Darmon Cycles." Canadian Journal of Mathematics 64, no. 5 (October 1, 2012): 1122–81. http://dx.doi.org/10.4153/cjm-2011-076-8.

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Abstract Darmon cycles are a higher weight analogue of Stark–Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on Γ0(N) of even weight k0 ≥ 2. They are conjectured to be the restriction of global cohomology classes in the Bloch–Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove p-adic Gross–Zagier type formulas, relating the derivatives of p-adic L-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur– Kitagawa p-adic L-function of the weight variable in terms of a global cycle defined over a quadratic extension of ℚ.

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34

Otsubo, N. "Note on conjectures of Beilinson-Bloch-Kato¶for cycle classes." manuscripta mathematica 101, no. 1 (January 1, 2000): 115–24. http://dx.doi.org/10.1007/s002290050007.

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35

BELLAICHE, J., and G. CHENEVIER. "Formes non temp�r�es pour et conjectures de Bloch?Kato." Annales Scientifiques de l?tcole Normale Sup�rieure 37, no. 4 (August 2004): 611–62. http://dx.doi.org/10.1016/j.ansens.2004.05.001.

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36

Pirutka, Alena. "Invariants birationnels dans la suite spectrale de Bloch-Ogus." Journal of K-theory 10, no. 3 (June 7, 2012): 565–82. http://dx.doi.org/10.1017/is012004021jkt191.

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AbstractFor a field k of cohomological dimension d we prove that the groups , (l, car.k) = 1, are birational invariants of smooth projective geometrically integral varieties over k of dimension n. Using the Kato conjecture, which has been recently established by Kerz and Saito [18], we obtain a similar result over a finite field for the groups . We relate one of these invariants with the cokernel of the l-adic cycle class map , which gives an analogue of a result of Colliot-Thélène and Voisin [5] 3.11 over ℂ for varieties over a finite field.

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37

Berrick, A. J., M. Karoubi, and P. A. Østvær. "Periodicity of hermitianK-groups." Journal of K-theory 7, no. 3 (May 16, 2011): 429–93. http://dx.doi.org/10.1017/is011004009jkt151.

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AbstractBott periodicity for the unitary and symplectic groups is fundamental to topologicalK-theory. Analogous to unitary topologicalK-theory, for algebraicK-groups with finite coefficients, similar results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraicK-groups for any ring implies periodicity for the hermitianK-groups, analogous to orthogonal and symplectic topologicalK-theory.The proofs use in an essential way higherKSC-theories, extending those of Anderson and Green. They also provide an upper bound for the higher hermitianK-groups in terms of higher algebraicK-groups.We also relate periodicity to étale hermitianK-groups by proving a hermitian version of Thomason's étale descent theorem. The results are illustrated in detail for local fields, rings of integers in number fields, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings.

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38

LEI, ANTONIO, DAVID LOEFFLER, and SARAH LIVIA ZERBES. "EULER SYSTEMS FOR HILBERT MODULAR SURFACES." Forum of Mathematics, Sigma 6 (2018). http://dx.doi.org/10.1017/fms.2018.23.

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We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.

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39

Loeffler, David, and Sarah Livia Zerbes. "On the Bloch–Kato conjecture for the symmetric cube." Journal of the European Mathematical Society, July 12, 2022. http://dx.doi.org/10.4171/jems/1256.

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40

Geisser, T., and M. Levine. "The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky." Journal für die reine und angewandte Mathematik (Crelles Journal) 2001, no. 530 (January 12, 2001). http://dx.doi.org/10.1515/crll.2001.006.

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41

Dummigan, Neil. "Twisted adjoint L-values, dihedral congruence primes and the Bloch–Kato conjecture." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, October 29, 2020. http://dx.doi.org/10.1007/s12188-020-00224-w.

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42

BROWN, JIM, and HUIXI LI. "CONGRUENCE PRIMES FOR SIEGEL MODULAR FORMS OF PARAMODULAR LEVEL AND APPLICATIONS TO THE BLOCH–KATO CONJECTURE." Glasgow Mathematical Journal, September 29, 2020, 1–22. http://dx.doi.org/10.1017/s0017089520000439.

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Abstract It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

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43

Lüders, Morten, and Matthew Morrow. "Milnor K-theory of p-adic rings." Journal für die reine und angewandte Mathematik (Crelles Journal), December 9, 2022. http://dx.doi.org/10.1515/crelle-2022-0079.

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Abstract We study the mod p r {p^{r}} Milnor K-groups of p-adically complete and p-henselian rings, establishing in particular a Nesterenko–Suslin-style description in terms of the Milnor range of syntomic cohomology. In the case of smooth schemes over complete discrete valuation rings we prove the mod p r {p^{r}} Gersten conjecture for Milnor K-theory locally in the Nisnevich topology. In characteristic p we show that the Bloch–Kato–Gabber theorem remains true for valuation rings, and for regular formal schemes in a pro sense.

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44

Wake, Preston. "The Eisenstein ideal for weight k and a Bloch–Kato conjecture for tame families." Journal of the European Mathematical Society, June 24, 2022. http://dx.doi.org/10.4171/jems/1251.

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45

Brown, J. "On the Cuspidality of Pullbacks of Siegel Eisenstein Series and Applications to the Bloch-Kato Conjecture." International Mathematics Research Notices, July 8, 2010. http://dx.doi.org/10.1093/imrn/rnq135.

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46

JOHANSSON, CHRISTIAN, and JAMES NEWTON. "PARALLEL WEIGHT 2 POINTS ON HILBERT MODULAR EIGENVARIETIES AND THE PARITY CONJECTURE." Forum of Mathematics, Sigma 7 (2019). http://dx.doi.org/10.1017/fms.2019.23.

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Let $F$ be a totally real field and let $p$ be an odd prime which is totally split in $F$ . We define and study one-dimensional ‘partial’ eigenvarieties interpolating Hilbert modular forms over $F$ with weight varying only at a single place $v$ above $p$ . For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch–Kato conjecture for finite slope Hilbert modular forms with trivial central character (with a technical assumption if $[F:\mathbb{Q}]$ is odd), by reducing to the case of parallel weight $2$ . As another consequence of our results on partial eigenvarieties, we show, still under the assumption that $p$ is totally split in $F$ , that the ‘full’ (dimension $1+[F:\mathbb{Q}]$ ) cuspidal Hilbert modular eigenvariety has the property that many (all, if $[F:\mathbb{Q}]$ is even) irreducible components contain a classical point with noncritical slopes and parallel weight $2$ (with some character at $p$ whose conductor can be explicitly bounded), or any other algebraic weight.

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