ISSN 2167-5163

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MR3194136 Pending
Dembélé, Lassina(4-WARW-MI)
On the computation of algebraic modular forms on compact inner forms of $\rm {GSp}_4$$\rm {GSp}_4$. (English summary)
Math. Comp. 83 (2014), no. 288, 1931–1950.
11F03 (11Y16)
Publication Year 2014

{A review for this item is in process.}
References
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2. Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265, DOI 10.1006/jsco.1996.0125. Computational algebra and number theory (London, 1993). MR1484478 MR1484478
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5. Clifton Cunningham and Lassina Demb'el'e, Computing genus-2 Hilbert-Siegel modular forms over Q( $\sqrt{5}$$\sqrt{5}$ ) via the Jacquet-Langlands correspondence, Experiment. Math. 18 (2009), no. 3, 337–345. MR2555703 (2010j:11076)
6. Lassina Demb'el'e, Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms, Math. Comp. 76 (2007), no. 258, 1039–1057, DOI 10.1090/S0025-5718-06-01914-4. MR2291849 (2008g:11078)
7. Lassina Demb'el'e and Steve Donnelly, Computing Hilbert modular forms over fields with non- trivial class group, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 371–386, DOI 10.1007/978-3-540-79456-1 25. MR2467859 (2010d:11149)
8. Martin Eichler, On theta functions of real algebraic number fields, Acta Arith. 33 (1977), no. 3, 269–292. MR0563061 (58 #27788) MR0563061 (58 #27788)
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10. Alexandru Ghitza, Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic mod- ular forms, J. Number Theory 106 (2004), no. 2, 345–384, DOI 10.1016/j.jnt.2003.12.009. MR2059479 (2005d:11067)
11. Benedict H. Gross, Algebraic modular forms, Israel J. Math. 113 (1999), 61–93, DOI 10.1007/BF02780173. MR1729443 (2001b:11037)
12. Benedict H. Gross, Groups over Z, Invent. Math. 124 (1996), no. 1-3, 263–279, DOI 10.1007/s002220050053. MR1369418 (96m:20075)
13. Ki-ichiro Hashimoto and Tomoyoshi Ibukiyama, On class numbers of positive definite binary quaternion Hermitian forms. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 695–699 (1982). MR0656045 (83m:10029)
14. Florian Herzig and Jacques Tilouine, Conjecture de type de Serre et formes compagnons pour GSp4, J. Reine Angew. Math. 676 (2013), 1–32, DOI 10.1515/CRELLE.2011.190 (French, with English summary). MR3028753 MR3028753
15. Tomoyoshi Ibukiyama, On symplectic Euler factors of genus two, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 587–614. MR0731520 (85j:11053)
16. Yasutaka Ihara, On certain arithmetical Dirichlet series, J. Math. Soc. Japan 16 (1964), 214–225. MR0179142 (31 #3393) Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1950 LASSINA DEMB'EL'E MR0179142 (31 #3393)
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MR3184337 Pending
Dembélé, Lassina(4-WARW-MI); Voight, John(1-VT)
Explicit methods for Hilbert modular forms. Elliptic curves, Hilbert modular forms and Galois deformations, 135–198,
Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Basel, 2013.
11F80 (11F41)
Publication Year 2013

{A review for this item is in process.}
{For the entire collection see MR3155300.}

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MR3155300 Reviewed
Berger, Laurent(F-ENSLY-PM); Böckle, Gebhard(D-HDBG-SC); Dembélé, Lassina(4-WARW-MI); Dimitrov, Mladen(F-LILL); Dokchitser, Tim(4-BRST); Voight, John(1-VT)
Elliptic curves, Hilbert modular forms and Galois deformations.
Edited by Henri Darmon, Fred Diamond, Luis V. Dieulefait, Bas Edixhoven and Víctor Rotger. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser/Springer, Basel, 2013. xii+249 pp. ISBN: 978-3-0348-0617-6; 978-3-0348-0618-3
11-06 (11F80 11Gxx)
Publication Year 2013 Review Published2014-06-30

Contents:
Part I. Galois deformations: Laurent Berger, On $p$$p$-adic Galois representations (3–19) MR3184334; Gebhard Böckle, Deformations of Galois representations (21–115) MR3184335.
Part II. Hilbert modular forms: Mladen Dimitrov, Arithmetic aspects of Hilbert modular forms and varieties (119–134) MR3184336; Lassina Dembélé and John Voight, Explicit methods for Hilbert modular forms (135–198) MR3184337.
Part III. Elliptic curves: Tim Dokchitser, Notes on the parity conjecture (201–249) MR3184338.

### Citations

MR2801398 (2012g:11087) Reviewed
Dembélé, Lassina(4-WARW-MI); Greenberg, Matthew(3-CALG); Voight, John(1-VT)
Nonsolvable number fields ramified only at 3 and 5. (English summary)
Compos. Math. 147 (2011), no. 3, 716–734.
11F41 (11F80 11G18 11R32)
Publication Year 2011 Review Published2012-03-19

This paper addresses a conjecture, tracing back to B. H. Gross [Internat. Math. Res. Notices 1998, no. 16, 865–875; MR1643625 (99j:11058)], that for every prime there exists a nonsolvable Galois number field ramified only at $p$$p$. The case for $p\geq11$$p\geq11$ was solved via results of J.-P. Serre [in Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, 319–338, Lecture Notes in Math., 317, Springer, Berlin, 1973; MR0466020 (57 #5904a); Abelian $l$$l$-adic representations and elliptic curves, revised reprint of the 1968 original, Res. Notes Math., 7, A K Peters, Wellesley, MA, 1998; MR1484415 (98g:11066)]; for the case $p=2$$p=2$, the first author [C. R. Math. Acad. Sci. Paris 347 (2009), no. 3-4, 111–116; MR2538094 (2010g:11191)] constructed an example by finding a suitable Hilbert modular form over ${\bf Q}(\zeta_{32})$${\bf Q}(\zeta_{32})$. The authors extend this construction to find a number of examples of finite nonsolvable Galois number fields ramified only at $3$$3$ and $5$$5$, respectively.
Starting with a relatively small totally real number field $F$$F$ ramified only at $p$$p$, the authors find a suitable representative $f$$f$ in the space of Hilbert cusp forms associated to $F$$F$ via calculations of the second and third authors [Math. Comp. 80 (2011), no. 274, 1071–1092; MR2772112 (2012c:11103)]. This form gives rise to a representation $\overline{\rho}_f\colon{\rm Gal}(\overline{F}/F)\to{\rm GL}_2 (k)$$\overline{\rho}_f\colon{\rm Gal}(\overline{F}/F)\to{\rm GL}_2 (k)$, where $k$$k$ is a finite field of characteristic $p$$p$. The field with desired ramification is then the field fixed by the kernel of the associated projective representation.
This process requires substantial calculations for the Hecke eigenvalues; such calculations make this line of attack unfeasible for $p=7$$p=7$ using current methods. The authors note, however, that one of their fields ramified only at $p=5$$p=5$ was also discovered by D. P. Roberts [Int. J. Number Theory 7 (2011), no. 2, 289–322; MR2782660 (2012d:11222)] by examining the 5-division field of an elliptic curve over the degree 5 subfield of ${\bf Q}(\zeta_{25})$${\bf Q}(\zeta_{25})$, and posit that the current best hope for $p=7$$p=7$ might lie in that direction.
Reviewed by Spencer Hamblen
References
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This list reflects references listed in the original paper as accurately as possible with no attempt to correct error.

### Citations

From Reviews: 0

MR2555703 (2010j:11076) Reviewed
Cunningham, Clifton(3-CALG); Dembélé, Lassina(D-DUES2-EM)
Computing genus-2 Hilbert-Siegel modular forms over $\Bbb Q(\sqrt 5)$$\Bbb Q(\sqrt 5)$ via the Jacquet-Langlands correspondence. (English summary)
Experiment. Math. 18 (2009), no. 3, 337–345.
11F41 (11F70)
Publication Year 2009 Review Published2010-06-21

Let $F$$F$ be a real quadratic field of narrow class number one and let $B$$B$ be the unique (up to isomorphism) quaternion algebra over $F$$F$ that is ramified at both Archimedean places of $F$$F$ and unramified everywhere else. Let ${\rm GU}_2(B)$${\rm GU}_2(B)$ be the unitary similitude group of $B^{\oplus 2}$$B^{\oplus 2}$. This is the set of ${\Bbb Q}$${\Bbb Q}$-rational points of an algebraic group $G^B$$G^B$ defined over ${\Bbb Q}$${\Bbb Q}$. The group $G^B$$G^B$ is an inner form of $G\coloneq \text{Res}_{F/{\Bbb Q}}(\rm GSp_4)$$G\coloneq \text{Res}_{F/{\Bbb Q}}(\rm GSp_4)$ such that $G^B({\Bbb R})$$G^B({\Bbb R})$ is compact modulo its center.
In the present paper the authors develop an algorithm that computes automorphic forms on $G^B$$G^B$ in the following sense: given an ideal $N$$N$ in ${\scr O}_F$${\scr O}_F$ and an integer $k>2$$k>2$, the algorithm returns the Hecke eigensystems of all automorphic forms $f$$f$ of level $N$$N$ and parallel weight $k$$k$. More precisely, given a prime ${\germ p}$${\germ p}$ in ${\scr O}_F$${\scr O}_F$, the algorithm returns the Hecke eigenvalues of $f$$f$ at ${\germ p}$${\germ p}$, and hence the Euler factor $L_{\germ p}(f, s)$$L_{\germ p}(f, s)$, for each eigenform $f$$f$ of level $N$$N$ and parallel weight $k$$k$. This algorithm is a generalization of the one developed in the second author's paper [L. Dembélé, Experiment. Math. 14 (2005), no. 4, 457–466; MR2193808 (2006h:11050)] to the genus two case.
The idea is to use the Jacquet-Langlands correspondence. Here, the Jacquet-Langlands correspondence refers to the conjectural map $\text{JL}\colon \Pi(G^B) \to \Pi(G)$$\text{JL}\colon \Pi(G^B) \to \Pi(G)$ from automorphic representations of $G^B$$G^B$ to automorphic representations of $G$$G$, which is injective, matches $L$$L$-functions, and enjoys other properties compatible with the principle of functoriality; in particular, the image of the Jacquet-Langlands correspondence is to be contained in the space of holomorphic automorphic representations. If we admit this conjecture, then the algorithm above provides a way to produce examples of cuspidal Hilbert-Siegel modular forms of genus two over $F$$F$ and allows us to compute the $L$$L$-factors of the corresponding automorphic representations for arbitrary finite primes ${\germ p}$${\germ p}$ of $F$$F$.
The authors use these calculations to provide evidence for the Jacquet-Langlands correspondence itself by comparing the Euler factors they find with those of known Hilbert-Siegel modular forms obtained by lifting. In particular, they provide some numerical examples using the quadratic field $F={\Bbb Q}(\sqrt{5})$$F={\Bbb Q}(\sqrt{5})$. It is known that for the Hamilton quaternion algebra $B$$B$ over $F$$F$, the class number of the principal genus of $G^B$$G^B$ is $1$$1$. The authors use their algorithm to compute all the systems of Hecke eigenvalues of Hilbert-Siegel cusp forms of weight $3$$3$ and level $N$$N$ that are defined over real quadratic fields, where $N$$N$ runs over all prime ideals of norm less than $50$$50$. They then determine which of the forms they obtained are possible lifts of Hilbert cusp forms by comparing the Hecke eigenvalues for those primes. They observe that some of the Euler factors they compute match those of lifts of Hilbert modular forms, for the primes they computed. Although this does not definitively establish that these Hilbert-Siegel modular forms are indeed lifts, in principle one can establish equality in this way, using an analogue of the Sturm bound.
Reviewed by Lei Yang
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14. Dinakar Ramakrishnan and Freydoon Shahidi. "Siegel Modular Forms of Genus 2 Attached to Elliptic Curves." Math. Res. Lett 14:2 (2007), 315–332. MR2318628 (2008i:11065)
15. N. C. Ryan. "Computing the Satake $p$$p$-Parameters of Siegel Modular Forms." arXiv:math/0411393, 2006.
16. Nils-Peter Skoruppa. "Computations of Siegel Modular Forms of Genus Two." Math. Comp. 58:197 (1992), 381–398. MR1106982 (92e:11041)
17. Claus M. Sorensen. "Potential Level-Lowering for GSp(4)." arXiv:0804.0588v1, 2008. MR2516307 (2010g:11079)
This list reflects references listed in the original paper as accurately as possible with no attempt to correct error.

### Citations

MR2538094 (2010g:11191) Reviewed
Dembélé, Lassina(D-DUES2-EM)
A non-solvable Galois extension of $\Bbb Q$$\Bbb Q$ ramified at 2 only. (English, French summary)
C. R. Math. Acad. Sci. Paris 347 (2009), no. 3-4, 111–116.
11R32 (11F80 11R29)
Publication Year 2009 Review Published2010-03-29

It is a conjecture of B. H. Gross [Internat. Math. Res. Notices 1998, no. 16, 865–875; MR1643625 (99j:11058)] that for any prime number $p$$p$, there exists a non-solvable Galois extension $K$$K$ of ${\Bbb Q}$${\Bbb Q}$ unramified away from $p$$p$. The conjecture follows for $p\geq 11$$p\geq 11$ as a consequence of the results of J.-P. Serre [in Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, 319–338. Lecture Notes in Math., 317, Springer, Berlin, 1973; MR0466020 (57 #5904a)].
The goal of this paper is to settle Gross' conjecture for $p=2$$p=2$. The author constructs an extension $K/{\Bbb Q}$$K/{\Bbb Q}$ having Galois group $({\rm SL}\sb2({\Bbb F}_{2^8}))\times({\rm SL}_2({\Bbb F}_{2\sp8}))\rtimes C_8$$({\rm SL}\sb2({\Bbb F}_{2^8}))\times({\rm SL}_2({\Bbb F}_{2\sp8}))\rtimes C_8$ of order $2^{19}(3\cdot5\cdot 17\cdot 257)^2$$2^{19}(3\cdot5\cdot 17\cdot 257)^2$, root discriminant $\delta_K<2^{47/8}=58.688\ldots$$\delta_K<2^{47/8}=58.688\ldots$ and such that $K/{\Bbb Q}$$K/{\Bbb Q}$ is imaginary. The root discriminant of a finite extension $E/{\Bbb Q}$$E/{\Bbb Q}$ of degree $n$$n$ is defined as $\delta_E=|d_E|^{1/n}$$\delta_E=|d_E|^{1/n}$ where $d_E$$d_E$ is the discriminant of $E/{\Bbb Q}$$E/{\Bbb Q}$. The construction of $K$$K$ is done using Galois representations attached to Hilbert modular forms over ${\Bbb Q}(\zeta_{32})^+={\Bbb Q}(\zeta _{32})\cap {\Bbb R}$${\Bbb Q}(\zeta_{32})^+={\Bbb Q}(\zeta _{32})\cap {\Bbb R}$.
In a note to this paper, Serre [C. R. Math. Acad. Sci. Paris 347 (2009), no. 3-4, 117–118; MR2538095] lowered the root discriminant of $K$$K$ to $\delta_K\leq 55.394388\ldots$$\delta_K\leq 55.394388\ldots$.
References
1. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (3–4) (1997) 235–265. MR1484478
2. S. Brueggeman, The nonexistence of certain Galois extensions unramified outside 5, J. Number Theory 75 (1999) 47–52. MR1670870 (2000a:11164)
3. C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over $\bold{Q}$$\bold{Q}$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (4) (2001) 843–939. MR1839918 (2002d:11058)
4. H. Carayol, Sur les représentations $l$$l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Ecole Norm. Sup. 19 (1986) 409–468. MR0870690 (89c:11083)
5. F. Diaz y Diaz, Tables minorant la racine $n$$n$-ième du discriminant d'un corps de degré $n$$n$, Publications Mathématiques d'Orsay 80, Université de Paris-Sud, Département de Mathématique, Orsay, 1980, 59 pp. MR0607864 (82i:12007)
6. B. Gross, Modular forms $\pmod \roman{p}$$\pmod \roman{p}$ and Galois representations, Int. Math. Res. Notices 16 (1998) 865–875. MR1643625 (99j:11058)
7. B. Gross, Algebraic modular forms, Israel J. Math. 113 (1999) 61–93. MR1729443 (2001b:11037)
8. F. Hajir, C. Maire, Tamely ramified towers and discriminant bounds for number fields. II, J. Symbolic Comput. 33 (4) (2002) 415–423. MR1890578 (2003h:11137)
9. F. Hajir, C. Maire, Tamely ramified towers and discriminant bounds for number fields, Compositio Math. 128 (1) (2001) 35–53. MR1847664 (2002g:11149)
10. F. Jarvis, On Galois representations associated to Hilbert modular forms of low weight, J. Reine Angew. Math. 491 (1997) 199–216. MR1476093 (98h:11067)
11. C. Khare, J.-P. Wintenberger, On Serre's conjecture for 2-dimensional $\bmod\ \roman{p}$$\bmod\ \roman{p}$ representations of the absolute Galois group of the rationals, Ann. of Math., in press.
12. J. Lansky, D. Pollack, Hecke algebras and automorphic forms, Compositio Math. 130 (1) (2002) 21–48. MR1883690 (2003a:11051)
13. M. Ohta, Hilbert modular forms of weight one and Galois representations, Progr. Math. 46 (1984) 333–353. MR0763021 (86c:11033)
14. J. Rogawski, J. Tunnell, On Artin $\roman{L}$$\roman{L}$-functions associated to Hilbert modular forms of weight 1, Invent. Math. 74 (1983) 1–42. MR0722724 (85i:11044)
15. J.-P. Serre, Sur les représentations modulaires de degré 2 de $\roman{Gal}(\overline{\Bbb{Q}} / \Bbb{Q})$$\roman{Gal}(\overline{\Bbb{Q}} / \Bbb{Q})$, Duke Math. J. 54 (1) (1987) 179–230. MR0885783 (88g:11022)
16. J.-P. Serre, Corps locaux, Deuxième édition, in: Publications de l'Université de Nancago, vol. VIII, Hermann, Paris, 1968, p. 245. MR0354618 (50 #7096)
17. J.-P. Serre, Congruences et formes modulaires [d'après H.P.F. Swinnerton-Dyer], Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, in: Lecture Notes in Math., vol. 317, Springer, Berlin, 1973, pp. 319–338. MR0466020 (57 #5904a)
18. J.-P. Serre, Oeuvres III, Springer-Verlag, 1986, Note 229.2 on p. 710. MR0926691 (89h:01109c)
19. J.-P. Serre, Abelian $l$$l$-Adic Representations and Elliptic Curves, Research Notes in Mathematics, vol. 7, A K Peters, Ltd., Wellesley, MA, 1997. MR1484415 (98g:11066)
20. J. Tate, The non-existence of certain Galois extensions of $\Bbb{Q}$$\Bbb{Q}$ unramified outside 2, Contemp. Math. 174 (1994) 153–156. MR1299740 (95i:11132)
21. R. Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (2) (1989) 265–280. MR1016264 (90m:11176)
22. R. Taylor, On the meromorphic continuation of degree two $L$$L$-functions, Doc. Math. Extra Vol. (2006) 729–779. MR2290604 (2008c:11154)
23. A. Wiles, On ordinary $\lambda$$\lambda$-adic representations associated to modular forms, Invent. Math. 94 (3) (1988) 529–573. MR0969243 (89j:11051)
This list reflects references listed in the original paper as accurately as possible with no attempt to correct error.

### Citations

From Reviews: 0

MR2484426 (2010a:11119) Reviewed
Dembélé, Lassina(D-DUES2-EM)
An algorithm for modular elliptic curves over real quadratic fields. (English summary)
Experiment. Math. 17 (2008), no. 4, 427–438.
11G40 (11F41 11F66 11G05)
Publication Year 2008 Review Published2009-10-14

Let $F$$F$ be a totally real number field of degree $n$$n$ over ${\Bbb{Q}}.$${\Bbb{Q}}.$ We denote by ${\scr{O}}_F$${\scr{O}}_F$ its ring of integers. Let $\germ{n}$$\germ{n}$ be an integral ideal of $F$$F$ and $\germ{p}$$\germ{p}$ a prime ideal of $F$$F$ such that $\germ{p}\nmid \germ{n}.$$\germ{p}\nmid \germ{n}.$ Let $f$$f$ be a Hilbert newform of weight 2 and level $\germ{n}$$\germ{n}$ and $a_{\germ{p}}(f)$$a_{\germ{p}}(f)$ the Fourier coefficient of $f$$f$ at prime $\germ{p}.$$\germ{p}.$ Let $E$$E$ be an elliptic curve defined over the number field $F$$F$ and ${\Bbb{F}}_{\germ{p}}$${\Bbb{F}}_{\germ{p}}$ the residue field ${\scr{O}}_F/{\germ{p}}.$${\scr{O}}_F/{\germ{p}}.$ If $\tilde{E}$$\tilde{E}$ is the reduction of $E\bmod {\germ p}$$E\bmod {\germ p}$ then we set $a_{\germ{p}}(E)=N({\germ{p}})+1-\#E({\Bbb{F}}_{\germ{p}}),$$a_{\germ{p}}(E)=N({\germ{p}})+1-\#E({\Bbb{F}}_{\germ{p}}),$ where $N({\germ{p}})$$N({\germ{p}})$ is the norm of ${\germ{p}}.$${\germ{p}}.$ The $L$$L$-series of $f,$$f,$ $$L(f,s)=\sum_{m\subset {\scr{O}}_F} \frac{a_m(f)}{N(m)^s},$$ and the $L$$L$-series of $E,$$E,$ $$\multline L(E,s)= \prod_{{\germ p} \mid {\rm conductor}(E)}\Big{(}1-\frac{a_{\germ{p}}(E)}{N(\germ{p})}\Big{)}^{-1}\\ \times \prod_{{\germ p} \nmid {\rm conductor}(E)} \left(1-\frac{a_{\germ{p}}(E)}{N(\germ{p})} +\frac{1}{N({\germ{p}})^{2s-1}} \right)^{-1}, \endmultline$$ are equal in the case where $F={\Bbb{Q}}.$$F={\Bbb{Q}}.$ This is the so-called Shimura-Eichler construction. If $[F\colon {\Bbb{Q}}]>1$$[F\colon {\Bbb{Q}}]>1$ then conjecturally there is an elliptic curve defined over $F$$F$ such that $L(f,s)=L(E,s).$$L(f,s)=L(E,s).$ If we suppose that the conjecture is true in the case $[F\colon {\Bbb{Q}}]=2$$[F\colon {\Bbb{Q}}]=2$ (and under the additional assumption that $F$$F$ has narrow class number 1) the author gives an algorithm which produces the possible elliptic curves such that their $L$$L$-series coincides with the $L$$L$-series of $f.$$f.$ The algorithm combines the analytic construction of T. Oda [Periods of Hilbert modular surfaces, Progr. Math., 19, Birkhäuser, Boston, Mass., 1982; MR0670069 (83k:10057)] and the Weierstrass uniformization theorem in order to find the possible equations of the elliptic curves. His method is illustrated by many examples. Finally, the author explains how to use his algorithm in order to compute equations of elliptic curves with everywhere good reduction over $F.$$F.$
Reviewed by Konstantinos Draziotis
References
1. M. Bertolini, H. Darmon, and P. Green. "Periods and Points Attached to Quadratic Algebras." In Proceedings of the MSRI Workshop on Special Values of Rankin $L$$L$-Series, edited by H. Darmon and S. Zhang, pp. 323–367. Cambridge, UK: Cambridge University Press, 2004. MR2083218 (2005e:11062)
2. J. E. Cremona. "Modular Symbols for $\Gamma _{1}(N)$$\Gamma _{1}(N)$and Elliptic Curves with Everywhere Good Reduction." Math. Proc. Cambridge Philos. Soc. 111:2 (1992), 199–218. MR1142740 (93e:11065)
3. J. E. Cremona. Algorithms for Modular Elliptic Curves, second edition. Cambridge: Cambridge University Press, 1997. MR1628193 (99e:11068)
4. J. E. Cremona and M. Lingham. "Finding All Elliptic Curves with Good Reduction outside a Given Set of Primes." To appear in Experiment. Math., 2008. cf. MR2367320 (2008k:11057)
5. H. Darmon. Rational Points on Modular Elliptic Curves, CBMS Regional Conference Series in Mathematics, 101. Providence: American Mathematical Society, 2004. MR2020572 (2004k:11103)
6. Henri Darmon and Adam Logan. "Periods of Hilbert Modular Forms and Rational Points on Elliptic Curves. International Mathematics Research No-tices 2003:40 (2003) 2153–2180. MR1997296 (2005f:11110)
7. Lassina Dembélé. "Explicit Computations of Hilbert Modular Forms on $\Bbb{Q}(sqrt{5})$$\Bbb{Q}(sqrt{5})$. Experiment. Math. 14:4 (2005), 457–466. MR2193808 (2006h:11050)
8. Mladen Dimitrov. "Compactifications arithmétiques des variétés de Hilbert et formes modulaires de Hilbert pour$\Gamma _{1}(c,n)$$\Gamma _{1}(c,n)$." In Geometric Aspects of Dwork Theory, vols. I, II, pp. 527–554. Berlin: Walter de Gruyter, 2004. MR2099078 (2006e:11063)
9. Takayuki Oda. Periods of Hilbert Modular Surfaces, Progress in Mathematics, 19. Bostin: Birkhäuser, 1982. MR0670069 (83k:10057)
10. Takayuki Oda. "Hodge Structures of Shimura Varieties Attached to the Unit Groups of Quaternion Algebras." In Galois Groups and Their Representations (Nagoya, 1981), pp. 15–36, Adv. Stud. Pure Math., 2. Amsterdam: North-Holland, 1983. MR0732459 (86j:11055)
11. R. G. E. Pinch. "Elliptic Curves over Number Fields." D.Phil. thesis, Oxford University, 1982.
12. Jean-Pierre Serre. "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques." Invent. Math. 15:4 (1972), 259–331. MR0387283 (52 #8126)
13. Joseph H. Silverman. The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. New York: Springer-Verlag, 1986. MR0817210 (87g:11070)
14. Jude Socrates and David Whitehouse. "Unramified Hilbert Modular Forms, with Examples Relating to Elliptic Curves." Pacific J. Math. 219:2 (2005), 333–364. MR2175121 (2007c:11059)
15. Shouwu Zhang. "Heights of Heegner Points on Shimura Curves. Ann. of Math. (2) 153:1 (2001), 27–147. MR1826411 (2002g:11081)
This list reflects references listed in the original paper as accurately as possible with no attempt to correct error.

### Citations

MR2467859 (2010d:11149) Reviewed
Dembélé, Lassina(D-DUES2-EM); Donnelly, Steve(5-SYD-SM)
Computing Hilbert modular forms over fields with nontrivial class group. (English summary) Algorithmic number theory, 371–386,
Lecture Notes in Comput. Sci., 5011, Springer, Berlin, 2008.
11Y40 (11F41 68W40)
Publication Year 2008 Review Published2010-01-28

Summary: "We exhibit an algorithm for the computation of Hilbert modular forms over an arbitrary totally real number field of even degree, extending results of the first author. We present some new instances of the conjectural Eichler-Shimura construction for totally real number fields over the fields $\Bbb Q(\sqrt{10})$$\Bbb Q(\sqrt{10})$ and $\Bbb Q(\sqrt{85})$$\Bbb Q(\sqrt{85})$ and their Hilbert class fields, and in particular some new examples of modular abelian varieties with everywhere good reduction over those fields.''

{For the entire collection see MR2467865 (2009h:11002).}

### Citations

MR2291849 (2008g:11078) Reviewed
Dembélé, Lassina(3-CALG-MS)
Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms. (English summary)
Math. Comp. 76 (2007), no. 258, 1039–1057.
11F41 (11F67)
Publication Year 2007 Review Published2008-04-23

The paper under review is concerned with the problem of explicit computations with Hecke eigenforms in the spaces of Hilbert modular forms. It extends in scope and efficiency earlier work of the author [see L. Dembélé, Experiment. Math. 14 (2005), no. 4, 457–466; MR2193808 (2006h:11050)]. This is done by introducing a generalization of modular symbols which is of independent theoretical interest.
The idea is to use the Jacquet-Langlands correspondence in order to move the problem from spaces of Hilbert modular forms to spaces of automorphic forms on a definite quaternion algebra $B$$B$ over a totally real field. (These automorphic forms are special cases of Gross' algebraic modular forms [see B. H. Gross, Israel J. Math. 113 (1999), 61–93; MR1729443 (2001b:11037)].) These are then brought in a form more amenable to computation via (an extension of) the theory of Brandt matrices, and a notion of modular symbols related to the quaternion algebra $B$$B$.
The last two sections describe an algorithm for computing with Hilbert modular forms over real quadratic fields, and give some numerical results from applying this algorithm for weight $(2,2)$$(2,2)$ forms on $\Bbb{Q}(\sqrt{29})$$\Bbb{Q}(\sqrt{29})$ and $\Bbb{Q}(\sqrt{37})$$\Bbb{Q}(\sqrt{37})$.
Reviewed by Alexandru E. Ghitza
References
1. C. Consani and J. Scholten, Arithmetic on a quintic threefold. International Journal of Mathematics 12, No. 8 (2001), pp. 943-972. MR1863287 (2002h:11058)
2. Henri Darmon and Adam Logan, Periods of Hilbert modular forms and rational points on elliptic curves. Int. Math. Res. Not. 2003, no. 40, 2153–2180. MR1997296 (2005f:11110)
3. L. Demb'el'e, Explicit computations of Hilbert modular forms on Q( $\sqrt{5}$$\sqrt{5}$ ). Ph.D. Thesis at McGill University, 2002. MR2620722
4. L. Demb'el'e, Explicit computations of Hilbert modular forms on Q( $\sqrt{5}$$\sqrt{5}$ ). Experiment. Math. 14 (2005), no. 4, 457–466. MR2193808 (2006h:11050)
5. L. Demb'el'e, F. Diamond and Robert; Numerical evidences of the weight part of the Serre conjecture for Hilbert modular forms. (preprint).
6. Stephen S. Gelbart, Automorphic forms on adele groups. Annals of Mathematics Studies, No. 83. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. MR0379375 (52 #280)
7. Benedict H. Gross, Algebraic modular forms. Israel J. Math. 113 (1999), 61–93. MR1729443 (2001b:11037)
8. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2). Lecture Notes in Mathematics, Vol. 114. Springer-Verlag, Berlin-New York, 1970. MR0401654 (53 #5481)
9. Magma algebraic computing system. http://magma.maths.usyd.edu.au
10. Ju. I. Manin, Parabolic points and zeta functions of modular curves. Math. USSR-Izv. 6 (1972), 19–64. MR0314846 (47 #3396)
11. L. Merel, Universal Fourier expansion of modular forms. Lecture Notes in Math. 1585. Springer-Verlag, 1994. MR1322319 (96h:11032)
12. A. Pizer, An algorithm for computing modular forms on $\Gamma$$\Gamma$0(N ). Journal of Algebra 64, (1980), pp.340-390. MR0579066 (83g:10020)
13. D. Pollack, Explicit Hecke action on modular forms. Ph.D. thesis at Havard University 1998. MR2612972
14. G. Shimura, The special values of the zeta functions associated to Hilbert modular forms. Duke Math. J. 45, No. 3 (1978), pp. 637-679. MR0507462 (80a:10043)
15. Jude Socrates and David Whitehouse, Unramified Hilbert modular forms, with examples relating to elliptic curves. Pacific J. Math. 219 (2005), no. 2, 333–364. MR2175121 MR2175121 (2007c:11059)
16. W. Stein, Explicit approaches to Abelian varieties. Ph.D. thesis at University of California at Berkeley, 2000.
17. Richard Taylor, On Galois representations associated to Hilbert modular forms. Invent. Math. 98 (1989), no. 2, 265–280. MR1016264 (90m:11176)
18. Marie France Vign'eras-Gu'eho, Le th'eoreme d'Eichler sur le nombre de classes d'id'eaux d'un corps de quaternions totalement d'efini et la mesure de Tamagawa. Bull. Soc. Math. France Mem., No. 37, Soc. Math. France, Paris, 1974, pp. 107–114. MR0376621 (51 #12796)
This list, extracted from the PDF form of the original paper, may contain data conversion errors, almost all limited to the mathematical expressions.

### Citations

MR2193808 (2006h:11050) Reviewed
Dembélé, Lassina(3-CALG-MS)
Explicit computations of Hilbert modular forms on ${\Bbb Q}(\sqrt{5})$${\Bbb Q}(\sqrt{5})$. (English summary)
Experiment. Math. 14 (2005), no. 4, 457–466.
11F41 (11Y16)
Publication Year 2005 Review Published2006-05-26

The appeal and usefulness of modular forms are due to the fact that they occur in natural and beautiful ways in many areas of mathematics, but also to their highly developed computational aspects. The various generalizations of modular forms appear in an equally beautiful way in mathematical "nature'', yet they suffer from an unfortunate reputation that is largely due to the difficulty to compute them. Interesting examples are hard to come by, and they are often obtained by ad hoc methods.
The present paper addresses these problems in the case of Hilbert modular forms. It gives an algorithm for explicitly computing Hilbert Hecke eigenforms, and lists some specific results of its implementation. More precisely, the algorithm deals with cusp eigenforms of arbitrary weight and level on a totally real field $F$$F$ of narrow class number $1$$1$. The main idea is to use the Jacquet-Langlands correspondence in order to relate Hilbert modular forms on $F$$F$ to automorphic forms on a quaternion algebra $B$$B$ over $F$$F$. This approach is not new, having been used in Pizer's algorithm [see A. Pizer, J. Algebra 64 (1980), no. 2, 340–390; MR0579066 (83g:10020)] and variants (see [C. Consani and J. Scholten, Internat. J. Math. 12 (2001), no. 8, 943–972; MR1863287 (2002h:11058)] or [J. Socrates and D. J. Whitehouse, Pacific J. Math. 219 (2005), no. 2, 333–364; MR2175121]). The novelty in the present paper is the observation that one can work with a maximal order $R$$R$ in $B$$B$, instead of working with a different Eichler order $R_{\germ{c}}$$R_{\germ{c}}$ for each level $\germ{c}$$\germ{c}$. This results in a more efficient algorithm.
As an application, the author lists in Section 4 all modular elliptic curves over $\Bbb{Q}(\sqrt{5})$$\Bbb{Q}(\sqrt{5})$ of prime conductor of norm less than $100$$100$.
Reviewed by Alexandru E. Ghitza
References
1. C. Consani and J. Scholten. "Arithmetic on a Quintic Threefold." International Journal of Mathematics 12:8 (2001), 943–972. MR1863287 (2002h:11058)
2. J. H. Conway and N. Sloane. Sphere Packings, Lattices and Groups, Second edition. New York: Springer Verlag, 1993. MR1194619 (93h:11069)
3. H. Darmon. "Rigid Local Systems, Hilbert Modular Forms, and Fermat's Last Theorem." Duke Math. J. 102:3 (2000), 413–449. MR1756104 (2001i:11071)
4. L. Dembélé. "Explicit Computations of Hilbert Modular Forms on $\Bbb{Q}(\sqrt{5})$$\Bbb{Q}(\sqrt{5})$." PhD diss., McGill University. Available from World Wide Web (www.math.mcgill.ca/$\sim$$\sim$darmon), 2002. MR2620722
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6. G. van der Geer. Hilbert Modular Surfaces. New York: Springer-Verlag, 1988. MR0930101 (89c:11073)
7. S. S. Gelbart. Automorphic Forms on Adele Groups, Annals of Mathematics Studies, 83. Princeton, NJ: Princeton Univ. Press, 1975. MR0379375 (52 #280)
8. H. Jacquet and R. P. Langlands. Automorphic Forms on $GL(2)$$GL(2)$, Lecture Notes in Math., 114. Berlin: Springer-Verlag, 1970. MR0401654 (53 #5481)
9. Kamal Khuri-Makdisi. "On the Curves Associated to Certain Rings of Automorphic Forms." Canad. J. Math. 53:1 (2001), 98–121. MR1814967 (2002i:11045)
10. O. Körner. "Traces of Eichler-Brandt Matrices and Type Numbers of Quaternion Orders." Proc. Indian. Acad. Sci. 97 (1987), 187–199. MR0983614 (90b:11124)
11. Robert P. Langlands. Base Change for GL(2), Annals of Mathematics Studies, 96. Princeton, NJ: Princeton University Press, and Tokyo: University of Tokyo Press, 1980. MR0574808 (82a:10032)
12. Takayuki Oda. Periods of Hilbert Modular Surfaces, Progress in Mathematics, 19. Boston, MA: Birkhuser, 1982. MR0670069 (83k:10057)
13. Arnold Pizer. "An Algorithm for Computing Modular Forms on $\Gamma _{0}(N)$$\Gamma _{0}(N)$." J. Algebra 64:2 (1980), 340–390. MR0579066 (83g:10020)
14. Jean-Pierre Serre. "Sur les représentations modulaires de degré 2 de $\roman{Gal}(\bar{\Bbb{Q}}/\Bbb{Q})$$\roman{Gal}(\bar{\Bbb{Q}}/\Bbb{Q})$." Duke Math. J. 54:1 (1987), 179–230. MR0885783 (88g:11022)
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17. N. Shepherd-Barron and R. Taylor. "Mod 2 and Mod 5 Icosahedral Representations." J.A.M.S. 10 (1997) 283–298. MR1415322 (97h:11060)
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19. G. Shimura. "The Special Values of the Zeta Functions Associated to Hilbert Modular Forms." Duke Math. J. 45:3 (1978), 637–679. MR0507462 (80a:10043)
20. C. M. Skinner and A. J. Wiles. "Residually Reducible Representations and Modular Forms." Inst. Hautes Études Sci. Publ. Math. 89 (1999), 5–126. MR1793414 (2002b:11072)
21. C. M. Skinner and A. J. Wiles. "Nearly Ordinary Deformations of Irreducible Residual Representations." Ann. Fac. Sci. Toulouse Math. 6:1 (2001), 185–215. MR1928993 (2004b:11073)
22. J. T. Socrates and D. Whitehouse. "Unramified Hilbert Modular Forms with Examples Relating to Elliptic Curves." Pacific Journal of Mathematics 219:2 (2005), 333–364. MR2175121 (2007c:11059)
23. R. Taylor. "On Galois Representations Associated to Hilbert Modular Forms." Invent. Math. 98 (1988), 265–280. MR1016264 (90m:11176)
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25. A. Wiles. "Modular Elliptic Curves and Fermat's Last Theorem." Annals of Maths. 141 (1995), 443–551. MR1333035 (96d:11071)
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