Celebratio Mathematica

B. H. Gross: “Intersection triangles and block intersection numbers of Steiner systems,” Math. Z. 139 (June 1974), pp. 87–104. MR 357146 Zbl 0276.05018 article

A Steiner system \( S(t,k,v) \) is a collection of \( k \)-subsets, called blocks, of a \( v \)-set of points with the property that any t-subset of points is contained in a unique block. For any block \( B \) of \( S(t,k,v) \) and for \( i \) in the range \( 0 \leq \) \( i \) \( \leq t - 1 \) we let \( x_i \) denote the number of blocks of \( S \) meeting \( B \) in precisely \( i \) points. Mendelsohn [1970] demonstrated that the values of the \( x_i \) depend only on the parameters \( (t,k,v) \) of \( S \), and not on the particular block \( B \) chosen. In this paper we continue an investigation begun by Noda [1972] and determine all the possible parameter sets \( (t,k,v) \) for a Steiner system with the property that \( x_0 = 0 \), that is–that any two blocks of \( S(t,k,v) \) have non-trivial intersection. They are:

\( (t,k,v) = (2,n+1,n^2+n+1) \) and \( S \) is a projective plane,
\( (t,k,v) = (4,7,23) \) and \( S \) is the unique system with these parameters first discovered by Witt [1937], or
\( (t,k,v) = (t,t+1,2t+3) \) and \( t+3 \) is a prime number.

We then extend this result slightly to characterize those Steiner systems with \( x_i = 0 \) for any \( i < t \).

B. H. Gross: Arithmetic on elliptic curves with complex multiplication. Ph.D. thesis, Harvard University, 1978. Advised by J. Tate, Jr. A version of this was published in two parts: part I as a monograph in 1980; part II in Invent. Math. 79 (1985). MR 2940819 phdthesis

B. H. Gross: “On the periods of abelian integrals and a formula of Chowla and Selberg,” Invent. Math. 45 : 2 (1978), pp. 193–211. With an appendix by David E. Rohrlich. MR 480542 Zbl 0418.14023 article

B. H. Gross and D. E. Rohrlich: “Some results on the Mordell–Weil group of the Jacobian of the Fermat curve,” Invent. Math. 44 : 3 (1978), pp. 201–224. MR 491708 Zbl 0369.14011 article

Let \( p \) be an odd prime. Let \( F(p) \) denote the \( p \)th Fermat curve \[ X^p + Y^p = 1 \] and \( J(p) \) its Jacobian. Faddeev [1961] proved that when \( p\leq 7 \) the Mordell–Weil group of \( J(p) \) over \( \mathbf{Q} \) is finite. Here we show that this group contains points of infinite order whenever \( p > 7 \); for example, \( J(11) \) has rank 6 over \( \mathbf{Q} \). More precisely, following Faddeev we consider an isogeny of \( J(p) \) onto the product of \( p-2 \) abelian varieties, and we show that when \( p > 7 \) all but possibly 3 of the factors in this product have infinite Mordell–Weil group. It seems likely that all \( p-2 \) factors have rational points of infinite order when \( p > 11 \).

These factors come from subfields of the Fermat function field, and we begin our analysis in §1 with the associated quotient curves–particularly with the rational torsion in their Jacobians. In §2 we apply these results to prove the existence of a point of infinite order. In §3 we compute the root number for the \( L \)-function associated to each factor; by the conjectures of Birch and Swinnerton-Dyer [1967], this should determine the parity of the rank of the Mordell–Weil group. Then, in §4 we compare our results with the rank estimates obtained by Faddeev [1961] (see also Bashmakov [1972]). Finally, in §5, we give some diophantine applications of the ideas of the preceding sections.

B. H. Gross and N. Koblitz: “Gauss sums and the \( p \)-adic \( \Gamma \)-function,” Ann. Math. (2) 109 : 3 (May 1979), pp. 569–581. MR 534763 Zbl 0406.12010 article

B. H. Gross: “On an identity of Chowla and Selberg,” pp. 344–348 in S. Chowla anniversary issue, published as J. Number Theory 11 : 3. Issue edited by R. P. Bambah. Academic Press (San Diego, CA), August 1979. MR 544262 Zbl 0418.14024 incollection

B. H. Gross: “Ramification in \( p \)-adic Lie extensions,” pp. 81–102 in Journées de géométrie algébrique de Rennes: Groupes formels, représentations galoisiennes et cohomologie des variétés de caractéristique positive, III [Algebraic geometry proceedings from Rennes: Formal groups, Galois representations and the cohomology of varieties of positive characteristic, III] (Rennes, France, July 1978). Edited by P. Berthelot and L. Breen. Astérisque 65. Société Mathématique de France (Paris), 1979. MR 563473 Zbl 0423.14030 incollection

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B. H. Gross: Arithmetic on elliptic curves with complex multiplication. Lecture Notes in Mathematics 776. Springer (Cham, Switzerland), 1980. With an appendix by B. Mazur. MR 563921 Zbl 0433.14032 book

B. H. Gross: “On the factorization of \( p \)-adic \( L \)-series,” Invent. Math. 57 : 1 (1980), pp. 83–95. MR 564185 Zbl 0472.12011 article

Let \( p \) be a rational prime and \( K = \mathbb{Q}(\sqrt{-D}) \) an imaginary quadratic field in which \( p \) splits. By using a measure constructed by Katz, we can attach a \( p \)-adic \( L \)-function \( L_p(s,\chi_K) \) to any continuous \( p \)-adic character \( \chi_K \) of \( \mathrm{Gal}(\overline{K}/K) \). When \( \chi_K \) is the restriction of a character of \( \mathrm{Gal}(\overline{K}/\mathbb{Q}) \), this \( L \)-function may be factored.

Let \( \chi \) be a continuous \( p \)-adic character of \( \mathrm{Gal}(K(\mu_{p^{\infty}})/\mathbb{Q}) \) which is trivial on complex conjugation, and let \( \chi_K \) be its restriction to \( \mathrm{Gal}(K(\mu_{p^{\infty}})/K) \). Then \[ L_p(s,\chi_K) = L_p(s,\chi\varepsilon\omega)L_p(1-s,\chi^{-1}). \]

The terms on the right are Kubota–Leopoldt \( p \)-adic \( L \)-series for \( \mathbb{Q} \), \( \varepsilon \) is the quadratic character mod \( D \) associated to \( K \), and \( \omega \) is the “Teichmuller” character.

By interpreting this theorem as a result on measures, one reduces to the case where \( \chi \) is of finite order and \( s = 0 \). There one uses the explicit formulas of Leopoldt and Katz to evaluate each side, and the classical formulas of Dirichlet and Kronecker for the complex \( L \)-functions to compare them.

B. H. Gross and D. Zagier: “On the critical values of Hecke \( L \)-series,” Mém. Soc. Math. France (N.S.) 2 (1980), pp. 49–54. MR 608638 Zbl 0462.14015 article

L. J. Federer and B. H. Gross: “Regulators and Iwasawa modules,” Invent. Math. 62 : 3 (1981), pp. 443–457. With an appendix by Warren Sinnott. MR 604838 Zbl 0468.12005 article

Let \( K \) be a CM field with maximal totally real subfield \( k \). Let \( p \) be an odd rational prime and let \( r(K) \) be the number of primes dividing \( p \) in \( k \) which split in \( K \).

Iwasawa [1973] and Greenberg [1973] have shown that the characteristic polynomial \( f(T) \) of the \( \Lambda \)-module \( X^- \) of minus ideal classes in the \( p \)th cyclotomic tower over \( K \) is divisible by \( T^{r(K)} \). Let \[ g(T) = f(T)/T^{r(K)} ;\] Greenberg has also shown that \( g(0) \) is non-zero whenever \( K \) is abelian over \( \mathbb{Q} \) [Greenberg 1973].

Using similar methods–but working with an arbitrary CM field \( K \)–we give a criterion for the non-vanishing of \( g(0) \), and a formula for its \( p \)-adic valuation whenever this criterion is met. We then discuss the implications of this result in the theory of \( p \)-adic \( L \)-functions.

B. H. Gross and J. Harris: “Real algebraic curves,” Ann. Sci. École Norm. Sup. (4) 14 : 2 (1981), pp. 157–182. MR 631748 Zbl 0533.14011 article

In this paper we investigate the geometry and topology of real algebraic curves. After some introductory material on real abelian varieties in paragraph 1, we take up the study of a real curve \( X \) and its Picard scheme \( \operatorname{Pic} X \) in paragraphs 2 and 3. In paragraph 4 we show how the topological invariants of a real curve X are determined by the action of complex conjugation on the group \[ H_1(X(\mathbb{C}),\mathbb{Z}/2) \] and in paragraph 5 we show how the topological invariants of \( X \) determine the number of real theta-characteristics of each parity in \( \operatorname{Pic} X \). We illustrate the general theory with a discussion of real hyper-elliptic curves in paragraph 6, real plane curves in paragraph 7, and real trigonal curves in paragraph 8, and end with some remarks on real moduli.

B. H. Gross: “\( p \)-adic \( L \)-series at \( s = 0 \),” J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 : 3 (1981), pp. 979–994. To the memory of Takuro Shintani. MR 656068 Zbl 0507.12010 article

The \( p \)-adic \( L \)-series we will consider in this paper are analogous to the complex \( L \)-series of Artin: they are associated to finite dimensional linear representations of the Galois group of a totally real number field. These \( L \)-series are known to be meromorphic functions on \( \mathbb{Z}_p \); we will give a conjectural formula for the leading term in their Taylor expansions at \( s = 0 \). This conjecture, which expresses the leading term as the product of a \( p \)-adic regulator and an algebraic number, was inspired by Tate’s formulation of Stark’s conjectures for Artin \( L \)-series [Stark 1976, 1980; Tate 1981].

Following Stark and Tate, we will also present a stronger conjecture for the first derivative of abelian \( L \)-series at \( s = 0 \). One consequence of this refinement would be the explicit construction of classfields using special values of \( p \)-adic analytic functions.

Finally, we will prove that all of our conjectures are true for the \( p \)-adic \( L \)-series of Kubota and Leopoldt: those associated to 1-dimensional representations of the Galois group of \( \mathbf{Q} \). The ingredients of the proof are: an analytic formula of Ferrero and Greenberg [1978], a transcendence result of Brumer [1967], and some results from the \( p \)-adic theory of Gauss sums [Gross and Koblitz 1979].

B. H. Gross: “The annihilation of divisor classes in Abelian extensions of the rational function field” in Séminaire de Théorie des Nombres, 1981–1982 (Bordeaux, France, 1980–1981). Edited by M. J. Bertin. Séminaire de Théorie des Nombres, Université Bordeaux I. Université Bordeaux, 1981. Exposé no. 3, 5 pages. Zbl 0507.14020 incollection

B. H. Gross: “Topological invariants of real algebraic curves” (Grenoble, France, January 1981). Séminaire de Théorie des Nombres de Grenoble 9: 1980–1981. Université Scientifique et Médicale de Grenoble, 1981. Exposé no. 2, 2 pages. Zbl 0508.14022 incollection

B. H. Gross: “Algebraic Hecke characters for function fields,” pp. 87–90 in Theorie des nombres: Seminaire Delange–Pisot–Poitou (Paris, 1980–1981). Progress in Mathematics 22. Birkhäuser (Cham, Switzerland), 1982. MR 693312 Zbl 0512.12012 incollection

B. H. Gross: “Heegner points on \( X_0(11) \)” in Séminaire de Théorie des Nombres, 1981–1982 (Bordeaux, France, 1981–1982). Edited by M. J. Bertin. Séminaire de Théorie des Nombres, Université Bordeaux I. Université Bordeaux, 1982. Exposé no. 34, 5 pages. MR 695347 Zbl 0514.14017 incollection

B. H. Gross: “Minimal models for elliptic curves with complex multiplication,” Compositio Math. 45 : 2 (1982), pp. 155–164. MR 651979 Zbl 0541.14010 article

Let \( R \) be the ring of integers in an algebraic number field \( F \). An abelian variety \( A \) of dimension \( g \) over \( F \) determines an element \( c_A \) in the ideal class group \( R \) in the following manner. Let \( N \) denote the Néron model of \( A \) over \( R \) [Néron 1964]; the space \( \underline{\omega}_{N/R} \) of invariant differentials on \( N \) is a projective \( R \)-module of rank \( g \). We may define \( c_A \) to be the class of \( \bigwedge^g\underline{\omega}_{N/R} \) in \( \operatorname{Pic}(R) \).

When \( \dim A = 1 \) Tate has given an alternate description of the class \( c_A \) in terms of minimal Weierstrass models [Tate 1975]. We use this formulation, and some classical results of Deuring [1958] and Hasse, to calculate \( c_A \) for some elliptic curves with complex multiplication.

B. H. Gross: “On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication,” pp. 219–236 in Number theory related to Fermat’s last theorem: Proceedings of the conference sponsored by the Vaughn Foundation (Cambridge, MA, September 1981). Edited by N. Koblitz. Progress in Mathematics 26. Birkhäuser (Boston), 1982. MR 685298 Zbl 0506.14040 incollection

B. Gross and D. Zagier: “Points de Heegner et dérivées de fonctions \( L \)” [Heegner points and derivatives of \( L \)-functions], C. R. Acad. Sci., Paris, Sér. I 297 (1983), pp. 85–87. With English summary. An expanded version of this was published in Invent. Math. 84 (1986). MR 720914 Zbl 0538.14023 article

B. H. Gross: “Heegner points on \( X_0(N) \),” pp. 87–105 in Modular forms: Contributions to a symposium on modular forms of one and several variables (Durham, UK, 30 June–10 July 1983). Edited by R. A. Rankin. Ellis Horwood Series in Mathematics and its Applications. Ellis Horwood Limited (Chichester, UK), 1984. MR 803364 Zbl 0559.14011 incollection

In his work on the class-number problem for imaginary quadratic fields, Heegner [1952] introduced a remarkable collection ofpoints on certain modular curves. These points always form a subset of the singular moduli; on the curve \( X_0(N) \) they correspond to the moduli of \( N \)-isogenous elliptic curves with the same ring of complex multiplication. Birch [1970] was the first to recognize the significance of the divisor classes supported on these points in the arithmetic of the Jacobian \( J_0(N) \). Using them, he was able to construct points of infinite order in certain elliptic quotients of \( J_0(N) \) possessing a cuspidal group of even order [Birch 1975]. Mazur [1979] later found an interesting method to construct points of infinite order in Eisenstein quotients of \( J_0(N) \), when \( N \) is prime.

In this paper, I would like to show how to obtain some ofthe above results via the theory of modular, elliptic, and circular units. This method will be exposed in Section II, after a review of the basic material on Heegner points in Section I. These theoretical results, although fragmentary, fit in nicely with the extensive computations which Birch and Stephens have made on this subject [1984]. On the basis of this evidence, I was led to conjecture a simple identity relating the height ofa Heegner divisor class to the first derivative at \( s = 1 \) of the \( L \)-series of an automorphic form on \( \mathrm{PGL}(2) \times \mathrm{GL}(2) \). Zagier and I have obtained a proof of this identity in many cases: I will discuss this work briefly in Section III. In Section IV I will present a general program of work on other modular curves.

B. H. Gross and D. B. Zagier: “On singular moduli,” J. Reine Angew. Math. 355 (1985), pp. 191–220. Dedicated to J.-P. Serre. MR 772491 Zbl 0545.10015 article

The values of the modular function \( j(\tau) \) at imaginary quadratic in the upper half plane are known as singular noduii. They are all algebraic integers. In this paper we will study the prime factorizations of the differences of two singular moduli. These differences turn out to be highly divisible numbers. For instance, we will determine the set of primes dividing the absolute norm of \[ j(\tau) - 1728 = j(\tau) - j(i) \] (and the multiplicities with which they occur); they turn out to be contained among the prime divisors of the positive integers of the form \( d - x^2 \) , where \( d \) is the discriminant of \( \tau \), and hence smaller than or equal to \( d \), e.g.: \[ j\Bigl(\frac{1+i\sqrt{163}}{2}\Bigr) - 1728 = - 2^6 3^6 7^2 11^2 19^2 127^2 163. \]

J. P. Buhler and B. H. Gross: “Arithmetic on elliptic curves with complex multiplication, II,” Invent. Math. 79 : 1 (1985), pp. 11–29. Part I was Gross’s 1980 LNM monograph. Both parts appear to be based on his 1978 PhD thesis. MR 774527 Zbl 0584.14027 article

J. P. Buhler, B. H. Gross, and D. B. Zagier: “On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of \( \operatorname{rank} 3 \),” Math. Comput. 44 : 170 (April 1985), pp. 473–481. MR 777279 Zbl 0606.14021 article

B. H. Gross and J. Lubin: “The Eisenstein descent on \( J_0(N) \),” Invent. Math. 83 : 2 (June 1986), pp. 303–319. MR 818355 Zbl 0594.14027 article

B. H. Gross and D. B. Zagier: “Heegner points and derivatives of \( L \)-series,” Invent. Math. 84 (1986), pp. 225–320. To John Tate. This work expands on a short note published in C. R. Acad. Sci., Paris 297 (1983). Part II was published in Math. Ann. 278 (1987). MR 833192 Zbl 0608.14019 article

The main theorem of this paper gives a relation between the heights of Heegner divisor classes on the Jacobian of the modular curve \( X_0(N) \) and the first derivatives at \( s = 1 \) of the Rankin \( L \)-series of certain modular forms. In the first six sections of this chapter, we will develop enough background material on modular curves, Heegner points, heights, and \( L \)-functions to be able to state one version of this identity precisely. In §7 we will discuss some applications to the conjecture of Birch and Swinnerton-Dyer for elliptic curves. For example, we will show that any modular elliptic curve over \( \mathbb{Q} \) whose \( L \)-function has a simple zero at \( s = 1 \) contains rational points of infinite order. Combining our work with that of Goldfeld [1976], one obtains an effective lower bound for the class numbers of imaginary quadratic fields as a function of their discriminants (§8). In §9 we will describe the plan of proof and the contents of the remaining chapters.

Many of the results of this paper were announced in our Comptes Rendus note [1983]. A more leisurely introduction to Heegner points and Rankin \( L \)-series may be found in our earlier paper [1984].

B. H. Gross: “On canonical and quasi-canonical liftings,” Invent. Math. 84 : 2 (June 1986), pp. 321–326. MR 833193 Zbl 0597.14044 article

B. H. Gross: “Local heights on curves,” pp. 327–339 in Arithmetic geometry (Storrs, CT, 30 July–10 August 1984). Edited by G. Cornell and J. H. Silverman. Springer (New York), 1986. MR 861983 Zbl 0605.14027 incollection

B. H. Gross: “Heegner points and the modular curve of prime level,” J. Math. Soc. Japan 39 : 2 (April 1987), pp. 345–362. MR 879933 Zbl 0623.14010 article

B. H. Gross: “Representation theory and the cuspidal group of \( X(p) \),” Duke Math. J. 54 : 1 (1987), pp. 67–75. To Yu. I. Manin. MR 885776 Zbl 0629.14021 article

Let \( p \) be a prime number, and let \( X(p) \) denote the modular curve which classifies elliptic curves with a level \( p \) structure. Then \( X(p) \) is a normal covering of the moduli \( X(1) \) of elliptic curves, with Galois group isomorphic to \[ \mathrm{GL}_2(\mathbb{Z}/p\mathbb{Z})/\langle \pm 1 \rangle .\] Hecke used the complex representation theory of the geometric Galois group \[ \mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})/\langle \pm 1 \rangle \] to study certain holomorphic differentials on \( X(p) \) [Hecke 1939, §7, 8]. In this note we will use the representation theory of \( \mathrm{GL}_2(\mathbb{Z}/p\mathbb{Z}) \) in characteristic \( p \) to study the cuspidal subgroup \( C \) in the Jacobian.

A general argument due to Manin and Drinfeld shows that \( C \) is a finite group [Manin 1972, Cor. 3.6], and Kubert and Lang have obtained a formula for its order [Kubert and Lang 1981, p. 118]. Here we will determine the structure of \( C\otimes \mathbb{Z}_p \) as a module over the Galois group of \( X(p) \). In particular, using the non semi-simplicity of certain modular representations of \( \mathrm{GL}_2(\mathbb{Z}/p\mathbb{Z}) \), we will show that for \( p \geq 5 \) the quotient \( C/pC \) has dimension \( \geq (p-5)(p-1)/4 \) over \( \mathbb{Z}/p\mathbb{Z} \), with equality holding if and only if the prime \( p \) is regular.

B. H. Gross: “Heights and the special values of \( L \)-series,” pp. 115–187 in Number theory (Montreal, 17–29 June 1985). Edited by H. Kisilevsky and J. Labute. CMS Conference Proceedings 7. American Mathematical Society (Providence, RI), 1987. MR 894322 Zbl 0623.10019 incollection

B. Gross, W. Kohnen, and D. Zagier: “Heegner points and derivatives of \( L \)-series, II,” Math. Ann. 278 (1987), pp. 497–562. Part I was published in Inventiones Mathematicae 84 (1986). MR 909238 Zbl 0641.14013 article

B. H. Gross: “Heights and \( L \)-series,” pp. 425–433 in Proceedings of the International Congress of Mathematicians (Berkeley, CA, 3–11 August 1986), vol. 1. Edited by A. M. Gleason. American Mathematical Society (Providence, RI), 1987. MR 934242 Zbl 0665.14010 incollection

One prominent aspect of current research in number theory has been the attempt to relate the \( L \)-series of certain automorphic forms to the \( L \)-series of motives. Once established, such relations usually lead to a fruitful interplay between the analytic and arithmetic sides of the subject, in the spirit of Dirichlet’s class number formula and Kronecker’s limit formula.

Over the last thirty years great advances have been made in establishing a correspondence between the \( L \)-series of holomorphic modular forms for \( \mathrm{GL}(2) \) and the \( L \)-series of 2-dimensional \( l \)-adic Galois representations. Zagier and I have recently found a limit formula in this setting, which I will discuss in a simple case below. This formula connects the heights of special divisor classes on modular curves to the behavior of certain Rankin \( L \)-series in the center of their critical strip. This paper is expository in nature, and is intended as an introduction to the main results and the steps in the proofs. For the details, we refer the reader to the papers in our bibliography.

B. H. Gross: “On the values of abelian \( L \)-functions at \( s = 0 \),” J. Fac. Sci., Univ. Tokyo, Sect. I A 35 : 1 (1988), pp. 177–197. MR 931448 Zbl 0681.12005 article

B. Gross and M. Rosen: “Fourier series and the special values of \( L \)-functions,” Adv. Math. 69 : 1 (May 1988), pp. 1–31. MR 937316 Zbl 0649.12009 article

B. H. Gross: “Local orders, root numbers, and modular curves,” Am. J. Math. 110 : 6 (December 1988), pp. 1153–1182. MR 970123 Zbl 0675.12011 article

R. F. Coleman and B. H. Gross: “\( p \)-adic heights on curves,” pp. 73–81 in Algebraic number theory: In honor of K. Iwasawa (Berkeley, CA, 20–24 January 1987). Edited by J. Coates. Advanced Studies in Pure Mathematics 17. Academic Press (Boston), 1989. MR 1097610 Zbl 0758.14009 incollection

B. H. Gross: “Group representations and lattices,” J. Am. Math. Soc. 3 : 4 (October 1990), pp. 929–960. MR 1071117 Zbl 0745.11035 article

This paper began as an attempt to understand the Euclidean lattices that were recently constructed (using the theory of elliptic curves) by Elkies [1989] and Shioda [1989a, 1989b], from the point of view of group representations. The main idea appears in a note of Thompson [1976]: if one makes strong irreducibility hypotheses on a rational representation \( V \) of a finite group \( G \), then the \( G \)-stable Euclidean lattices in \( V \) are severely restricted. Unfortunately, these hypotheses are rarely satisfied when \( V \) is absolutely irreducible over \( \mathbf{Q} \). But there are many examples where the ring \( \mathrm{End}_G(V) \) is an imaginary quadratic field or a definite quaternion algebra. These representations allow us to construct some of the Mordell–Weil lattices considered by Elkies and Shioda, as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves.

In §1 we discuss lattices and Hermitian forms on \( V \), and in §§2–4 the strong irreducibility hypotheses we wish to make. In §5 we show how our hypotheses imply the existence of a finite number (up to isomorphism) of Euclidean \( \mathbf{Z}[G] \)-lattices \( L \) in \( V \) with \( \mathrm{End}_G(L) \) a maximal order in \( \mathrm{End}_G(V) \). We give some examples with \( \dim V \leq 8 \) in §6, and in §§7–9 discuss the invariants of \( L \), such as the dual lattice and theta function. The rest of the paper is devoted to examples: in most, \( G \) is a finite group of Lie type and \( V \) is obtained as an irreducible summand of the Weil representation of \( G \).

Some of the representation theoretic problems left open by this paper are: to find all examples of pairs \( (G,V) \) satisfying the strong irreducibility hypotheses of §§2–4, and to determine the invariants (shortest nonzero vector, theta function, Thompson series, …) of the \( G \)-lattices \( L \) so effortlessly constructed inside \( V \).

B. H. Gross: “A tameness criterion for Galois representations associated to modular forms mod \( p \),” Duke Math. J. 61 : 2 (1990), pp. 445–517. MR 1074305 Zbl 0743.11030 article

B. H. Gross: “Some applications of Gel’fand pairs to number theory,” Bull. Am. Math. Soc., New Ser. 24 : 2 (1991), pp. 277–301. MR 1074028 Zbl 0733.11018 article

B. H. Gross: “Kolyvagin’s work on modular elliptic curves,” pp. 235–256 in \( L \)-functions and arithmetic (Durham, UK, 30 June–11 July 1989). Edited by J. Coates and M. J. Taylor. London Mathematical Society Lecture Note Series 153. Cambridge University Press, 1991. MR 1110395 Zbl 0743.14021 incollection

John Henry Coates	Related
Martin John Taylor	Related

B. H. Gross and D. Prasad: “Test vectors for linear forms,” Math. Ann. 291 : 2 (1991), pp. 343–355. MR 1129372 Zbl 0768.22004 article

B. H. Gross and S. S. Kudla: “Heights and the central critical values of triple product \( L \)-functions,” Compos. Math. 81 : 2 (1992), pp. 143–209. MR 1145805 Zbl 0807.11027 article

B. H. Gross: “Elliptic curves and modular forms,” pp. 85–88 in Mathematics into the twenty-first century (Providence, RI, 8–12 August 1988). Edited by F. E. Browder. American Mathematical Society (Providence, RI), 1992. MR 1184615 Zbl 1011.11503 incollection

B. H. Gross and D. Prasad: “On the decomposition of a representation of \( \mathrm{SO}_n \) when restricted to \( \mathrm{SO}_{n-1} \),” Can. J. Math. 44 : 5 (1992), pp. 974–1002. MR 1186476 Zbl 0787.22018 article

B. H. Gross and K. Keating: “On the intersection of modular correspondences,” Invent. Math. 112 : 2 (1993), pp. 225–245. MR 1213101 Zbl 0811.11026 article

M. J. Hopkins and B. H. Gross: “The rigid analytic period mapping, Lubin–Tate space, and stable homotopy theory,” Bull. Am. Math. Soc., New Ser. 30 : 1 (1994), pp. 76–86. MR 1217353 Zbl 0857.55003 article

B. H. Gross: “A remark on tube domains,” Math. Res. Lett. 1 : 1 (1994), pp. 1–9. MR 1258484 Zbl 0873.32032 article

M. J. Hopkins and B. H. Gross: “Equivariant vector bundles on the Lubin–Tate moduli space,” pp. 23–88 in Topology and representation theory (Evanston, IL, 1–5 May 1992). Edited by E. M. Friedlander and M. E. Mahowald. American Mathematical Society (Providence, RI), 1994. MR 1263712 Zbl 0807.14037 incollection

B. H. Gross: “\( L \)-functions at the central critical point,” pp. 527–535 in Motives (Seattle, WA, 20 July–2 August 1991). Edited by J.-P. Serre, S. Kleiman, and U. Jannsen. American Mathematical Society (Providence, RI), 1994. MR 1265543 Zbl 0807.14015 incollection

B. H. Gross and D. Prasad: “On irreducible representations of \( \mathrm{SO}_{2n+1}\times \mathrm{SO}_{2m} \),” Can. J. Math. 46 : 5 (1994), pp. 930–950. MR 1295124 Zbl 0829.22031 article

B. H. Gross and N. R. Wallach: “A distinguished family of unitary representations for the exceptional groups of real rank \( = 4 \),” pp. 289–304 in Lie theory and geometry: In honor of Bertram Kostant on the occasion of his 65th birthday (Cambridge, MA, May 1993). Edited by J.-L. Brylinski, R. Brylinski, V. Guillemin, and V. Kac. Birkhäuser (Boston), 1994. MR 1327538 Zbl 0839.22006 incollection

B. H. Gross and C. Schoen: “The modified diagonal cycle on the triple product of a pointed curve,” Ann. Inst. Fourier 45 : 3 (1995), pp. 649–679. MR 1340948 Zbl 0822.14015 article

B. H. Gross: “Groups over \( \mathbb{Z} \),” Invent. Math. 124 : 1–3 (1996), pp. 263–279. To Reinhold Remmert. MR 1369418 Zbl 0846.20049 article

Let \( G \) be a connected, reductive algebraic group over \( \mathbb{Q} \). We say that \( \underline{G} \) is a model for \( G \) over \( \mathbb{Z} \) if \( \underline{G} \) is a smooth affine group scheme over \( \mathbb{Z} \) with general fibre \( G \), and the special fibre \( \underline{G}\otimes\mathbb{Z}/p\mathbb{Z} \) is reductive for all primes \( p \). This condition is fairly restrictive; for example, among tori only split groups admit a model over \( \mathbb{Z} \). However, for semi-simple groups there are some nonsplit examples: a necessary and sufficient condition is that \( G \) be split over \( \mathbb{Q}_p \) for all primes \( p \). We enumerate all the possibilities in Section 1. In Sect. 2 we prove an analog of Harder’s theorem on the Euler–Poincaré characteristic of a Chevalley group over \( \mathbb{Z} \), using recent results of Kottwitz and Chernousov. For groups \( G \) over \( \mathbb{Q} \) with \( G(\mathbb{R}) \) compact, this gives a “mass formula” for the different models over \( \mathbb{Z} \).

Section 3 is devoted to the construction of some models over \( \mathbb{Z} \) for the classical groups \( G \) of rank \( \leq 8 \) with \( G(\mathbb{R}) \) compact, and Sect. 4 to the construction of models for \( G_2 \), \( F_4 \), and \( E_6 \). In Sect. 5 we use the mass formula to show that the examples given exhaust the integral models for the groups of these types. In Sect. 6 we consider the adjoint representation for some of these groups, and construct two models for the anisotropic form of \( E_8 \).

N. D. Elkies and B. H. Gross: “The exceptional cone and the Leech lattice,” Int. Math. Res. Not. 1996 : 14 (July 1996), pp. 665–698. MR 1411589 Zbl 0863.11027 article

B. H. Gross and N. R. Wallach: “On quaternionic discrete series representations, and their continuations,” J. Reine Angew. Math. 1996 : 481 (1996), pp. 73–123. MR 1421947 Zbl 0857.22012 article

B. H. Gross and G. Savin: “The dual pair \( \mathrm{PGL}_3 \times \mathrm{G}_2 \),” Can. Math. Bull. 40 : 3 (1997), pp. 376–384. MR 1464847 Zbl 0881.22010 article

B. H. Gross: “On the motive of a reductive group,” Invent. Math. 130 : 2 (1997), pp. 287–313. MR 1474159 Zbl 0904.11014 article

B. H. Gross: “On the motive of \( G \) and the principal homomorphism \( \mathrm{SL}_2 \to \hat{G} \),” Asian J. Math. 1 : 1 (1997), pp. 208–213. MR 1480995 Zbl 0942.20031 article

Let \( k \) be a field, and let \( G \) be a connected reductive group over \( k \). Let \( \hat{G} \) be the Langlands dual group, which is a reductive group over \( \mathbf{C} \). In [1997] we attached a motive \( M \) of Artin–Tate type to \( G \). In this paper, we relate \( M \) to the principal homomorphism \[ \mathrm{SL}_2 \to \hat{G} ,\] which was introduced by de Siebenthal [1950] and Dynkin [1957], and studied extensively by Kostant [1959]. As a corollary, we relate the \( L \)-function of the dual motive \( M^{\vee} \), when \( k \) is a local non-Archimedean field, to the Langlands \( L \)-function of the Steinberg representation of \( G(k) \), with respect to the adjoint representation of the \( L \)-group. We also construct an involution \( \theta \) of \( \hat{\mathbf{\mathfrak{g}}} = \mathrm{Lie}(\hat{G}) \) when \( k = \mathbf{R} \).

N. Elkies and B. H. Gross: “Embeddings into the integral octonions,” pp. 147–158 in Olga Taussky-Todd: In memoriam, published as Pac. J. Math. 181 : 3. Issue edited by M. Aschbacher, D. Blasius, and D. Ramakrishnan. International Press (Cambridge, MA), 1997. MR 1610847 Zbl 0981.11036 incollection

B. Gross, B. Kostant, P. Ramond, and S. Sternberg: “The Weyl character formula, the half-spin representations, and equal rank subgroups,” Proc. Natl. Acad. Sci. USA 95 : 15 (July 1998), pp. 8441–8442. MR 1639139 Zbl 0918.17002 article

Let \( B \) be a reductive Lie subalgebra of a semi-simple Lie algebra \( F \) of the same rank both over the complex numbers. To each finite dimensional irreducible representation \( V_{\lambda} \) of \( F \) we assign a multiplet of irreducible representations of \( B \) with \( m \) elements in each multiplet, where \( m \) is the index of the Weyl group of \( B \) in the Weyl group of \( F \). We obtain a generalization of the Weyl character formula; our formula gives the character of \( V_{\lambda} \) as a quotient whose numerator is an alternating sum of the characters in the multiplet associated to \( V_{\lambda} \) and whose denominator is an alternating sum of the characters of the multiplet associated to the trivial representation of \( F \).

B. H. Gross: “Modular forms mod \( p \) and Galois representations,” Int. Math. Res. Not. 1998 : 16 (1998), pp. 865–875. MR 1643625 Zbl 0978.11018 article

B. H. Gross and G. Savin: “Motives with Galois group of type \( \mathrm{G}_2 \): An exceptional theta-correspondence,” Compos. Math. 114 : 2 (1998), pp. 153–217. MR 1661756 Zbl 0931.11015 article

B. H. Gross: “On the Satake isomorphism,” pp. 223–237 in Galois representations in arithmetic algebraic geometry (Durham, UK, 9–18 July 1996). Edited by A. J. Scholl and R. L. Taylor. Cambridge University Press, 1998. MR 1696481 Zbl 0996.11038 incollection

In this paper, we present an expository treatment of the Satake transform. This gives an isomorphism between the spherical Hecke algebra of a split reductive group \( G \) over a local field and the representation ring of the dual group \( \hat{G} \).

If one wants to use the Satake isomorphism to convert information on eigenvalues for the Hecke algebra to local \( L \)-functions, it has to be made quite explicit. This was done for \( G = \mathrm{GL}_n \) by Tamagawa, but the results in this case are deceptively simple, as all of the fundamental representations of the dual group are minuscule. Lusztig discovered that, in the general case, certain Kazhdan–Lusztig polynomials for the affine Weyl group appear naturally as matrix coefficients of the transform. His results were extended by S. Kato.

We will explain some of these results, with several examples.

B. H. Gross and W. T. Gan: “Haar measure and the Artin conductor,” Trans. Am. Math. Soc. 351 : 4 (1999), pp. 1691–1704. MR 1458303 Zbl 0991.20033 article

B. H. Gross and W. T. Gan: “Commutative subrings of certain non-associative rings,” Math. Ann. 314 : 2 (1999), pp. 265–283. MR 1697445 Zbl 0990.11018 article

B. H. Gross: “Algebraic modular forms,” Israel J. Math. 113 (December 1999), pp. 61–93. MR 1729443 Zbl 0965.11020 article

B. H. Gross: “On simply-connected groups over \( \mathbf{Z} \), with \( G(\mathbf{R}) \) compact,” pp. 113–118 in Integral quadratic forms and lattices: Dedicated to the memory of Dennis Ray Estes (Seoul, 15–19 June 1998). Edited by M.-H. Kim, J. S. Hsia, Y. Kitaoka, and R. Schulze-Pillot. Contemporary Mathematics 249. American Mathematical Society (Providence, RI), 1999. MR 1732354 Zbl 0969.20023 incollection

This paper is a continuation of [1996], and studies simply-connected, semi-simple groups \( G \) over \( \mathbf{Q} \) which are split over \( \mathbf{Q}_p \) for all finite primes \( p \) and anisotropic over \( \mathbf{R} \). We may then find a smooth group scheme \( \underline{G} \) over \( \mathbf{Z} \), with general fibre \( G \) and with good reduction mod \( p \) for all primes \( p \) [1996, Prop. 1.1]. We want to clarify the relationship between elements of the (finite) double coset space \[ G(\mathbf{Q})\backslash G(\hat{\mathbf{Q}})/\underline{G}(\hat{\mathbf{Z}}), \] other models \( \underline{G}^{\prime} \) over \( \mathbf{Z} \) in the same genus as \( \underline{G} \), and lattices \( L^{\prime} \) in globally irreducible representations \( V \) of \( G \) over \( \mathbf{Q} \).

B. Gross and N. Wallach: “Restriction of small discrete series representations to symmetric subgroups,” pp. 255–272 in The mathematical legacy of Harish-Chandra: A celebration of representation theory and harmonic analysis (Baltimore, MD, 9–10 January 1998). Edited by R. S. Doran and V. S. Varadarajan. Proceedings of Symposia in Pure Mathematics 68. American Mathematical Society (Providence, RI), 2000. MR 1767899 Zbl 0960.22008 incollection

W. T. Gan and B. H. Gross: “Integral embeddings of cubic norm structures,” J. Algebra 233 : 1 (November 2000), pp. 363–397. To Nathan Jacobson, in memoriam. MR 1793601 Zbl 0990.11017 article

B. H. Gross: “On minuscule representations and the principal \( \mathrm{SL}_2 \),” Represent. Theory 4 (2000), pp. 225–244. MR 1795753 Zbl 0986.22011 article

N. D. Elkies and B. H. Gross: “Cubic rings and the exceptional Jordan algebra,” Duke Math. J. 109 : 2 (2001), pp. 383–409. MR 1845183 Zbl 1028.11041 article

W. T. Gan, B. Gross, and G. Savin: “Fourier coefficients of modular forms on \( G_2 \),” Duke Math. J. 115 : 1 (2002), pp. 105–169. MR 1932327 Zbl 1165.11315 article

B. H. Gross and C. T. McMullen: “Automorphisms of even unimodular lattices and unramified Salem numbers,” J. Algebra 257 : 2 (November 2002), pp. 265–290. Dedicated to J. G. Thompson on his 70th birthday. MR 1947324 Zbl 1022.11016 article

P. Deligne and B. H. Gross: “On the exceptional series, and its descendants,” C. R. Math. Acad. Sci. Paris 335 : 11 (2002), pp. 877–881. With French summary. MR 1952563 Zbl 1017.22008 article

Many of the striking similarities which occur for the adjoint representation of groups in the exceptional series (cf. [Cohen and de Man 1996; Deligne 1996; Deligne and de Man 1996]) also occur for certain representations of specific reductive subgroups. The tensor algebras on these representations are easier to describe (cf. [Gross and Wallach 1994; Schwarz 1988; Wenzl 2002]), and may offer clues to the original situation.

The subgroups which occur form a Magic Triangle, which extends Freudenthal’s Magic Square of Lie algebras. We describe these groups from the perspective of dual pairs, and their representations from the action of the dual pair on an exceptional Lie algebra.

B. H. Gross: “Unramified reciprocal polynomials and Coxeter decompositions,” Mosc. Math. J. 2 : 4 (October–December 2002), pp. 681–692. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1986086 Zbl 1136.11313 article

B. H. Gross: “Some remarks on signs in functional equations,” Ramanujan J. 7 : 1–3 (2003), pp. 91–93. In memory of Robert A. Rankin. MR 2035794 Zbl 1057.11052 article

B. H. Gross and G. Nebe: “Globally maximal arithmetic groups,” J. Algebra 272 : 2 (February 2004), pp. 625–642. MR 2028074 Zbl 1113.20040 article

B. H. Gross and J. Harris: “On some geometric constructions related to theta characteristics,” pp. 279–311 in Contributions to automorphic forms, geometry, and number theory: Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday (Baltimore, MD, 14–17 May 2002). Edited by H. Hida, D. Ramakrishnan, and F. Shahidi. Johns Hopkins University Press (Baltimore, MD), 2004. MR 2058611 Zbl 1072.14032 incollection

The theory of quadratic forms over the field of 2 elements has many mathematical applications, from finite group theory to algebraic topology. Here we pursue a connection discovered by Mumford, relating theta characteristics on an algebraic curve to quadratic forms on the vector space of 2-torsion points in its Jacobian.

We develop the algebraic and combinatorial aspects of quadratic forms in the first three sections, then review some of the theory of theta characteristics in §4. The last three sections use this theory to investigate some classical geometric constructions on curves of genus 2 and 3.

Some of the material in sections 2 and 3 appears in the 19th century literature (cf. for example [Coble 1913, 1929; Weber 1896]), and has been abstracted in expository articles (cf. [Saavedra-Rivano 1976]). Similarly, versions of the geometric constructions in sections 5–7 have appeared in several excellent modern expositions (cf. [Griffiths and Harris 1978; Dolgachev and Ortland 1988]).

B. Birch and B. Gross: “Correspondence,” pp. 11–23 in Heegner points and Rankin \( L \)-series (Berkeley, CA, December 2001). Edited by H. Darmon and S.-W. Zhang. MSRI Publications 49. Cambridge University Press, 2004. MR 2083208 Zbl 1073.11002 incollection

B. H. Gross: “Heegner points and representation theory,” pp. 37–65 in Heegner points and Rankin \( L \)-series (Berkeley, CA, December 2001). Edited by H. Darmon and S.-W. Zhang. MSRI Publications 49. Cambridge University Press, 2004. MR 2083210 Zbl 1126.11032 incollection

B. H. Gross and J. Harris: The magic of numbers. Prentice Hall (Upper Saddle River, NJ), 2004. Zbl 1121.00004 book

B. H. Gross: “An elliptic curve test for Mersenne primes,” J. Number Theory 110 : 1 (January 2005), pp. 114–119. MR 2114676 Zbl 1074.11065 article

Let \( l\geq 3 \) be a prime, and let \( p = 2^l - 1 \) be the corresponding Mersenne number. The Lucas–Lehmer test for the primality of \( p \) goes as follows. Define the sequence of integers \( x_k \) by the recursion \begin{align*} x_0 &= 4,\\ x_k &= x_{k-1}^2 - 2. \end{align*} Then \( p \) is a prime if and only if each \( x_k \) is relatively prime to \( p \), for \( 0 \leq \) \( k \) \( \leq l-3 \), and \( \gcd(x_{l-2},p) > 1 \). We show, in the Section 1, that this test is based on the successive squaring of a point on the one-dimensional algebraic torus \( T \) over \( \mathbb{Q} \), associated to the real quadratic field \( k = \mathbb{Q}(\sqrt{3}) \). This suggests that other tests could be developed, using different algebraic groups. As an illustration, we will give a second test involving the sucessive squaring of a point on an elliptic curve.

If we define the sequence of rational numbers \( x_k \) by the recursion \begin{align*} x_0 &= -2,\\ x_k &= \frac{(x_{k-1}^2 + 12)^2}{4\cdot x_{k-1}\cdot (x_{k-1}^2 - 12)}, \end{align*} then we show that \( p \) is prime if and only if \( x_k\cdot \) \( (x_{k-1}^2 - 12) \) is relatively prime to \( p \), for \( 0 \leq \) \( k \) \( \leq l-2 \), and \( \gcd(x_{l-1},p) > 1 \). This test involves the successive squaring of a point on the elliptic curve \( E \) over \( \mathbb{Q} \) defined by \[ y^2 = x^3 - 12x. \] We provide the details in Section 2.

The two tests are remarkably similar. For example, both take place on groups with good reduction away from 2 and 3. Can one be derived from the other?

B. H. Gross and D. Pollack: “On the Euler characteristic of the discrete spectrum,” J. Number Theory 110 : 1 (January 2005), pp. 136–163. MR 2114678 Zbl 1080.11041 article

This paper, which is largely expository in nature, seeks to illustrate some of the advances that have been made on the trace formula in the past 15 years. We review the basic theory of the trace formula, then introduce some ideas of Arthur and Kottwitz that allow one to calculate the Euler characteristic of the \( S \)-cohomology of the discrete spectrum. This Euler characteristic is first expressed as a trace of a certain test function on the space of automorphic forms, and then, by the stable trace formula, is converted into a sum of orbital integrals. A result on global measures allows us to calculate these integrals in terms of the values of certain Artin \( L \)-functions at negative integers.

Our intention is to show how advances in the theory have allowed one to render such calculations completely explicit. As a byproduct of this calculation, we obtain the existence of automorphic representations with certain local behavior at the places in \( S \).

B. H. Gross: “On the centralizer of a regular, semi-simple, stable conjugacy class,” Represent. Theory 9 (2005), pp. 287–296. MR 2133761 Zbl 1107.20034 article

A. Pacetti and F. Rodriguez Villegas: “Computing weight 2 modular forms of level \( p^2 \),” Math. Comput. 74 : 251 (2005), pp. 1545–1557. With an appendix by B. Gross. MR 2137017 Zbl 1093.11027 article

B. H. Gross: “On the values of Artin \( L \)-functions,” Q. J. Pure Appl. Math. 1 : 1 (2005), pp. 1–13. MR 2154331 Zbl 1169.11050 article

I wrote this paper in 1979, as an attempt to extend the results of Borel [1977] on zeta functions at negative integers to Artin \( L \)-functions. The conceptual framework was provided by Tate’s formulation [1984] of Stark’s conjectures. What I needed was a workable definition of the regulator homomorphism in complex \( K \)-theory. I discussed this with Borel at the Institute, first over lunch and then in his office. It is an honor to dedicate this paper to his memory.

Using results of Bloch and Thurston, I was able to treat the special case of Dirichlet \( L \)-series at \( s = -1 \). I had the hope of treating Dirichlet \( L \)-series at all negative integers, where the order of vanishing is either zero or one, but was unable to construct the required “cyclotomic classes” in \( K \)-theory. This was done by Beilinson [1985], who also found the generalization of my conjecture, and the conjectures of Deligne [1979] on special values, to all motivic \( L \)-functions.

I didn’t publish this paper, but it has circulated as a preprint for 25 years. For reasons of historical interest, I decided to publish it in its original form here. I have updated the references, and added some comments on the recent literature at the end of the paper.

B. H. Gross and M. Reeder: “From Laplace to Langlands via representations of orthogonal groups,” Bull. Am. Math. Soc. (N.S.) 43 : 2 (2006), pp. 163–205. MR 2216109 Zbl 1159.11047 article

B. H. Gross, E. Hironaka, and C. T. McMullen: “Cyclotomic factors of Coxeter polynomials,” J. Number Theory 129 : 5 (May 2009), pp. 1034–1043. MR 2516970 Zbl 1183.11065 article

B. H. Gross and M. W. Lucianovic: “On cubic rings and quaternion rings,” J. Number Theory 129 : 6 (2009), pp. 1468–1478. MR 2521487 Zbl 1166.11034 article

E. Frenkel and B. Gross: “A rigid irregular connection on the projective line,” Ann. Math. (2) 170 : 3 (2009), pp. 1469–1512. MR 2600880 Zbl 1209.14017 article

In this paper we construct a connection \( \nabla \) on the trivial \( G \)-bundle on \( \mathbb{P}^1 \) for any simple complex algebraic group \( G \), which is regular outside of the points 0 and \( \infty \), has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point \( \infty \), with slope \( 1/h \), the reciprocal of the Coxeter number of \( G \). The connection \( \nabla \), which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic 0 counterpart of a hypothetical family of \( l \)-adic representations, which should parametrize a specific automorphic representation under the global Langlands correspondence. These \( l \)-adic representations, and their characteristic 0 counterparts, have been constructed in some cases by Deligne and Katz. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. It provides an example of the geometric Langlands correspondence with wild ramification. We compute the de Rham cohomology of our connection with values in a representation \( V \) of \( G \), and describe the differential Galois group of \( \nabla \) as a subgroup of \( G \).

B. H. Gross: “The arithmetic of elliptic curves — an update,” Arab. J. Sci. Eng. ASJE. Math. 34 : 1D (2009), pp. 95–103. MR 2792223 Zbl 1229.11088 article

B. H. Gross: “Rigid local systems on \( \mathbb{G}_m \) with finite monodromy,” Adv. Math. 224 : 6 (August 2010), pp. 2531–2543. MR 2652215 Zbl 1193.22001 article

B. H. Gross and M. Reeder: “Arithmetic invariants of discrete Langlands parameters,” Duke Math. J. 154 : 3 (September 2010), pp. 431–508. MR 2730575 Zbl 1207.11111 article

The local Langlands correspondence can be used as a tool for making verifiable predictions about irreducible complex representations of \( p \)-adic groups and their Langlands parameters, which are homomorphisms from the local Weil–Deligne group to the \( L \)-group. In this article, we refine a conjecture of Hiraga, Ichino, and Ikeda which relates the formal degree of a discrete series representation to the value of the local gamma factor of its parameter. We attach a rational function in \( x \) with rational coefficients to each discrete parameter, which specializes at \( x = q \), the cardinality of the residue field, to the quotient of this local gamma factor by the gamma factor of the Steinberg parameter. The order of this rational function at \( x = 0 \) is also an important invariant of the parameter — it leads to a conjectural inequality for the Swan conductor of a discrete parameter acting on the adjoint representation of the \( L \)-group. We verify this conjecture in many cases. When we impose equality, we obtain a prediction for the existence of simple wild parameters and simple supercuspidal representations, both of which are found and described in this article.

B. H. Gross: “Irreducible cuspidal representations with prescribed local behavior,” Am. J. Math. 133 : 5 (2011), pp. 1231–1258. MR 2843098 Zbl 1228.22017 article

B. H. Gross: “Lectures on the conjecture of Birch and Swinnerton-Dyer,” pp. 169–209 in Arithmetic of \( L \)-functions (Park City, UT, 28 June–18 July 2009). Edited by C. Popescu, K. Rubin, and A. Silverberg. IAS/Park City Mathematics Series 18. American Mathematical Society (Providence, RI), 2011. MR 2882691 Zbl 1285.11096 incollection

B. H. Gross and N. R. Wallach: “On the Hilbert polynomials and Hilbert series of homogeneous projective varieties,” pp. 253–263 in Arithmetic geometry and automorphic forms: Festschrift dedicated to Stephen Kudla on the occasion of his 60th birthday. Edited by J. Cogdell, J. Funke, M. Rapoport, and T. Yang. Advanced Lectures in Mathematics 19. International Press (Somerville, MA), 2011. MR 2906911 Zbl 1310.14044 incollection

B. H. Gross: “On Bhargava’s representation and Vinberg’s invariant theory,” pp. 317–321 in Frontiers of mathematical sciences. Edited by B. Gu and S.-T. Yau. International Press (Somerville, MA), 2011. MR 3050830 incollection

B. H. Gross and J. A. Parson: “On the local divisibility of Heegner points,” pp. 215–241 in Number theory, analysis and geometry: In memory of Serge Lang. Edited by D. Goldfeld, J. Jorgenson, P. Jones, D. Ramakrishnan, K. Ribet, and J. Tate. Springer (Berlin), 2012. MR 2867919 Zbl 1276.11091 incollection

B. H. Gross: “The classes of singular moduli in the generalized Jacobian,” pp. 137–141 in Geometry and arithmetic (Schiermonnikoog, Netherlands, 20–24 September 2010). Edited by C. Faber, G. Farkas, and R. de Jong. EMS Series of Congress Reports. European Mathematical Society (Zürich), 2012. MR 2987658 Zbl 1317.11047 incollection

M. Reeder, P. Levy, J.-K. Yu, and B. H. Gross: “Gradings of positive rank on simple Lie algebras,” Transform. Groups 17 : 4 (2012), pp. 1123–1190. MR 3000483 Zbl 1310.17017 ArXiv 1307.5765 article

Sur les conjectures de Gross et Prasad, I. Edited by W. T. Gan, B. H. Gross, D. Prasad, and J.-L. Waldspurger. Astérisque 346. Société Mathématique de France (Paris), 2012. MR 3052279 Zbl 1257.22001 book

B. H. Gross: “Hanoi lectures on the arithmetic of hyperelliptic curves,” Acta Math. Vietnam. 37 : 4 (2012), pp. 579–588. MR 3058664 Zbl 1294.11107 article

Manjul Bhargava and I have recently proved a result on the average order of the 2-Selmer groups of the Jacobians of hyperelliptic curves of a fixed genus \( n\geq 1 \) over \( \mathbb{Q} \), with a rational Weierstrass point [Bhargava and Gross 2012, Thm. 1]. A surprising fact which emerges is that the average order of this finite group is equal to 3, independent of the genus \( n \). This gives us a uniform upper bound of \( 3/2 \) on the average rank of the Mordell–Weil groups of their Jacobians over \( \mathbb{Q} \). As a consequence, we can use Chabauty’s method to obtain a uniform bound on the number of points on a majority of these curves, when the genus is at least 2.

We will state these results more precisely below, after some general material on hyperelliptic curves with a rational Weierstrass point. We end with a short discussion of hyperelliptic curves with two rational points at infinity.

W. T. Gan, B. H. Gross, and D. Prasad: “Symplectic local root numbers, central critical \( L \) values, and restriction problems in the representation theory of classical groups,” pp. 1–109 in Sur les conjectures de Gross et Prasad, I [The conjectures of Gross and Prasad, I]. Edited by W. T. Gan, B. H. Gross, D. Prasad, and J.-L. Waldspurger. Astérisque 346. Société Mathématique de France (Paris), 2012. MR 3202556 Zbl 1280.22019 ArXiv 0909.2999 incollection

It has been almost 20 years since two of us proposed a rather speculative approach to the problem of restriction of irreducible representations from \( \mathrm{SO}_n \) to \( \mathrm{SO}_{n-1} \) [Gross and Prasad 1992, 1994]. Our predictions depended on the Langlands parametrization of irreducible representations, using \( L \)-packets and \( L \)-parameters. Since then, there has been considerable progress in the construction of local \( L \)-packets, as well as on both local and global aspects of the restriction problem. We thought it was a good time to review the precise conjectures which remain open, and to present them in a more general form, involving restriction problems for all of the classical groups.

Wee Teck Gan	Related
Dipendra Prasad	Related

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}

W. T. Gan, B. H. Gross, and D. Prasad: “Restrictions of representations of classical groups: Examples,” pp. 111–170 in Sur les conjectures de Gross et Prasad, I [The conjectures of Gross and Prasad, I]. Edited by W. T. Gan, B. H. Gross, D. Prasad, and J.-L. Waldspurger. Astérisque 346. Société Mathématique de France (Paris), 2012. with French summary. MR 3202557 Zbl 1279.22023 ArXiv 0909.2993 incollection

In an earlier paper, we considered several restriction problems in the representation theory of classical groups over local and global fields. Assuming the Langlands–Vogan parameterization of irreducible representations, we formulated precise conjectures for the solutions of these restriction problems. In the local case, our conjectural answer is given in terms of Langlands parameters and certain natural symplectic root numbers associated to them. In the global case, the conjectural answer is expressed in terms of the central critical value or derivative of a global \( L \)-function. In this paper, using methods of base change and the theta correspondence, we test our conjectures for depth zero supercuspidal representations of unitary groups, and for more general representations of groups of low rank.

Wee Teck Gan	Related
Dipendra Prasad	Related

B. H. Gross: “Trivial \( L \)-functions for the rational function field,” J. Number Theory 133 : 3 (March 2013), pp. 970–976. MR 2997780 Zbl 1318.11147 article

M. Bhargava and B. H. Gross: “The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point,” pp. 23–91 in Automorphic representations and \( L \)-functions (Mumbai, 3–11 January 2012). Edited by D. Prasad, C. S. Rajan, A. Sankaranarayanan, and J. Sengupta. Tata Institute of Fundamental Research Studies in Mathematics 22. Tata Institute of Fundamental Research (Mumbai), 2013. MR 3156850 Zbl 1303.11072 ArXiv 1208.1007 incollection

M. Bhargava and B. H. Gross: “Arithmetic invariant theory,” pp. 33–54 in Symmetry: Representation theory and its applications. In honor of Nolan R. Wallach. Edited by R. E. Howe, M. Hunziker, and J. F. Willenbring. Progress in Mathematics 257. Springer (Cham, Switzerland), 2014. Part II was published in Progress in Mathematics 312 (2015). MR 3363006 Zbl 1377.11045 ArXiv 1206.4774 incollection

Let \( k \) be a field, let \( G \) be a reductive algebraic group over \( k \), and let \( V \) be a linear representation of \( G \). Geometric invariant theory involves the study of the \( k \)-algebra of \( G \)-invariant polynomials on \( V \), and the relation between these invariants and the \( G \)-orbits on \( V \), usually under the hypothesis that the base field \( k \) is algebraically closed. In favorable cases, one can determine the geometric quotient \[ V\mathbin{\!/\!\!/\!} G = \operatorname{Spec}(\textrm{Sym}^*(V^{\vee})^G) \] and can identify certain fibers of the morphism \( V \to V\mathbin{\!/\!\!/\!} G \) with certain \( G \)-orbits on \( V \). In this paper we study the analogous problem when \( k \) is not algebraically closed. The additional complexity that arises in the orbit picture in this scenario is what we refer to as arithmetic invariant theory. We illustrate some of the issues that arise by considering the regular semisimple orbits — i.e., the closed orbits whose stabilizers have minimal dimension — in three arithmetically rich representations of the split odd special orthogonal group \( G = \mathrm{SO}_{2n+1} \).

B. H. Gross: “The work of Manjul Bhargava,” pp. 56–63 in Proceedings of the International Congress of Mathematicians (Seoul, 13–21 August 2014), vol. 1: Plenary lectures and ceremonies. Edited by S. Y. Jang, Y. R. Kim, D.-W. Lee, and I. Yie. Kyung Moon Sa (Seoul), 2014. MR 3728462 Zbl 1373.11003 incollection

Y. K. Leong: “Benedict Gross: Elliptic curves, Millennium Problem,” published as Imprints 26 (2015). Reprinted in The art and practice of mathematics (2021). incollection

M. Bhargava, B. H. Gross, and X. Wang: “Arithmetic invariant theory, II: Pure inner forms and obstructions to the existence of orbits,” pp. 139–171 in Representations of reductive groups: In honor of the 60th birthday of David A. Vogan, Jr. (Cambridge, MA, 19–23 May 2014). Edited by M. Nevins and P. E. Trapa. Progress in Matematics 312. Springer (Cham, Switzerland), 2015. Part I was published in Progress in Mathematics 257 (2014). MR 3495795 Zbl 1377.11046 ArXiv 1310.7689 incollection

Let \( k \) be a field, let \( G \) be a reductive group, and let \( V \) be a linear representation of \( G \). Let \[ V \mathbin{\!/\!\!/\!} G = \operatorname{Spec}(\mathrm{Sym}^*(V^*))^G \] denote the geometric quotient and let \( \pi:V \to V \mathbin{\!/\!\!/\!} G \) denote the quotient map. Arithmetic invariant theory studies the map \( \pi \) on the level of \( k \)-rational points. In this article, which is a continuation of the results of our earlier paper “Arithmetic invariant theory”, we provide necessary and sufficient conditions for a rational element of \( V \mathbin{\!/\!\!/\!} G \) to lie in the image of \( \pi \), assuming that generic stabilizers are abelian. We illustrate the various scenarios that can occur with some recent examples of arithmetic interest.

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	AUTHOR = {Bhargava, Manjul and Gross, Benedict
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	TITLE = {Arithmetic invariant theory, {II}: {P}ure
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	BOOKTITLE = {Representations of reductive groups:
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	NUMBER = {312},
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B. K. Gross: “On the Langlands correspondence for symplectic motives,” Izv. Ross. Akad. Nauk Ser. Mat. 80 : 4 (2016), pp. 678–692. English translation of Russian original published in Izv. Ross. Akad. Nauk Ser. Mat. 80:4 (2016). MR 3535358 Zbl 1391.11075 article

R. Coulangeon, B. H. Gross, and G. Nebe: “Lattices and applications in number theory,” Oberwolfach Rep. 13 : 1 (2016), pp. 87–154. Abstracts from workshop held in Oberwolfach, Germany, 17–23 January 2016. MR 3586050 Zbl 1380.00058 article

This is a report on the workshop on Lattices and Applications in Number Theory held in Oberwolfach, from January 17 to January 23, 2016. The workshop brought together people working in various areas related to the field: classical geometry of numbers, packings, Diophantine approximation, Arakelov geometry, cohomology of arithmetic groups, algebraic modular forms and Hecke operators, algebraic topology. The meeting consisted of a few long talks which included an introductory part to each of the topics in the previous list, and a series of shorter talks mainly devoted to recent developments. The present report contains extended abstracts of all presentations.

M. Bhargava, B. H. Gross, and X. Wang: “A positive proportion of locally soluble hyperelliptic curves over \( \mathbb{Q} \) have no point over any odd degree extension,” J. Am. Math. Soc. 30 : 2 (2017), pp. 451–493. With an appendix by Tim Dokchitser and Vladimir Dokchitser. MR 3600041 Zbl 1385.11043 ArXiv 1310.7692 article

A hyperelliptic curve over \( \mathbb{Q} \) is called “locally soluble” if it has a point over every completion of \( \mathbb{Q} \). In this paper, we prove that a positive proportion of hyperelliptic curves over \( \mathbb{Q} \) of genus \( g\geq 1 \) are locally soluble but have no points over any odd degree extension of \( \mathbb{Q} \). We also obtain a number of related results. For example, we prove that for any fixed odd integer \( k > 0 \), the proportion of locally soluble hyperelliptic curves over \( \mathbb{Q} \) of genus \( g \) having no points over any odd degree extension of \( \mathbb{Q} \) of degree at most \( k \) tends to 1 as \( g \) tends to infinity. We also show that the failures of the Hasse principle in these cases are explained by the Brauer–Manin obstruction. Our methods involve a detailed study of the geometry of pencils of quadrics over a general field of characteristic not equal to 2, together with suitable arguments from the geometry of numbers.

B. H. Gross: “On Hecke’s decomposition of the regular differentials on the modular curve of prime level,” Res. Math. Sci. 5 : 1 (2018). Paper no. 1, 19 pages. MR 3749283 Zbl 1455.11083 article

B. Gross, J. Harris, and E. Riehl: Fat chance: Probability from 0 to 1. Cambridge University Press, 2019. MR 3931738 Zbl 1423.00005 book

B. Gross: “A short note on S.-T. Yau,” pp. 46 in Celebrating Shing-Tung Yau on his 70th birthday, published as ICCM Not. 7 : 1. Issue edited by S.-Y. Cheng, L. Ji, and H. X. Xiaokui Yang. International Press (Somerville, MA), 2019. MR 3960553 incollection

J. Achter, F. Andreatta, L. Bary-Soroker, B. Bhatt, P. Cassou-Noguès, T. Chinburg, B. Morin, M. J. Taylor, B. Conrad, C.-L. Chai, H. Esnault, A. Shiho, F. Oort, D. Gross, S. Harashita, D. Harbater, T. Katsura, R. Pries, and M. Zieve: “Appendix 3: Questions in arithmetic algebraic geometry,” pp. 295–331 in Open problems in arithmetic algebraic geometry. Edited by F. Oort. Advanced Lectures in Mathematics 46. International Press (Somerville, MA), 2019. MR 3971189 Zbl 1428.14005 incollection

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	AUTHOR = {Achter, Jeff and Andreatta, Fabrizio
		 and Bary-Soroker, Lior and Bhatt, Bhargav
		 and Cassou-Nogu\`es, P. and Chinburg,
		 T. and Morin, B. and Taylor, M. J. and
		 Conrad, Brian and Chai, Ching-Li and
		 Esnault, H\'el\`ene and Shiho, Atsushi
		 and Oort, Frans and Gross, Dick and
		 Harashita, Shushi and Harbater, David
		 and Katsura, Toshiyuki and Pries, Rachel
		 and Zieve, Michael},
	TITLE = {Appendix 3: {Q}uestions in arithmetic
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	PAGES = {295--331},
	NOTE = {MR:3971189. Zbl:1428.14005.},
	ISSN = {2379-3589},
	ISBN = {9781571463739},
}

B. H. Gross: Simple supercuspidals and the Langlands correspondence. Preprint, May 2020. ArXiv 2005.09078 techreport

W. T. Gan, B. H. Gross, and D. Prasad: “Branching laws for classical groups: The non-tempered case,” Compos. Math. 156 : 11 (2020), pp. 2298–2367. MR 4190046 Zbl 1470.11126 ArXiv 1911.02783 article

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic \( L \)-packets to Arthur packets, especially for the non-generic \( L \)-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for \( \mathrm{GL}_n \) and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical \( L \)-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

Wee Teck Gan	Related
Dipendra Prasad	Related

B. H. Gross: “On the periods of Abelian varieties,” ICCM Not. 8 : 2 (2020), pp. 10–18. MR 4245911 Zbl 1475.11121 ArXiv 2005.04194 article

K. Acquista, J. Buhler, D. Clausen, J. H. Coates, B. H. Gross, S. Lichtenbaum, J. Lubin, B. Mazur, V. K. Murty, B. Perrin-Riou, C. D. Popescu, K. A. Ribet, J. H. Silverman, K. Uhlenbeck, and J. F. Voloch: “Memorial article for John Tate.” Edited by B. Mazur and K. Ribet. Notices Am. Math. Soc. 68 : 5 (2021), pp. 768–781. MR 4249436 Zbl 1470.01012 article

Bernadette Perrin-Riou	Related
Karen Acquista	Related
Dustin Clausen	Related
Karen Uhlenbeck	Related	Volume
Barry C. Mazur	Related
Cristian Dumitru Popescu	Related
Joe Peter Buhler	Related
José Felipe Voloch	Related
John Torrence Tate, Jr.	Related
Joseph Hillel Silverman	Related
John Henry Coates	Related
Kenneth A. Ribet	Related
Stephen Lichtenbaum	Related
Jonathan Darby Lubin	Related
Vijaya Kumar Murty	Related

B. H. Gross and S. Garibaldi: “Minuscule embeddings,” pp. 987–1004 in Special issue to the memory of T. A. Springer, published as Indag. Math. (N.S.) 32 : 5. Issue edited by G. Cornelissen and E. Opdam. Elsevier (Amsterdam), September 2021. MR 4310010 Zbl 07394423 incollection

We study embeddings \( J\to G \) of simple linear algebraic groups with the following property: the simple components of the \( J \) module \( \mathrm{Lie}(G)/\mathrm{Lie}(J) \) are all minuscule representations of \( J \). One family of examples occurs when the group \( G \) has roots of two different lengths and \( J \) is the subgroup generated by the long roots. We classify all such embeddings when \( J = \mathrm{SL}_2 \) and \( J = \mathrm{SL}_3 \), show how each embedding implies the existence of exceptional algebraic structures on the graded components of \( \mathrm{Lie}(G) \), and relate properties of those structures to the existence of various twisted forms of \( G \) with certain relative root systems.

Y. K. Leong: “Benedict Gross: Elliptic curves, Millennium Problem,” pp. 131–140 in The art and practice of mathematics: Interviews at the Institute for Mathematical Sciences, National University of Singapore, 2010–2019. World Scientific (Singapore), 2021. Interview conducted on 19 March 2012. Originally published in Imprints 26 (2015). Zbl 1473.01034 incollection

B. H. Gross: “Incoherent definite spaces and Shimura varieties,” pp. 187–215 in Relative trace formulas (Schloss Elmau, Germany, 22–28 April 2018). Edited by W. Müller, S. W. Shin, and N. Templier. Simons Symposia. Springer (Cham, Switzerland), 2021. Zbl 1475.11124 ArXiv 2005.05188 incollection

W. T. Gan, B. H. Gross, and D. Prasad: Twisted GGP problems and conjectures. Preprint, April 2022. ArXiv 2204.10108 techreport

Benedict H. Gross

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