Introduction to the Mathematical Legacy of Hel Braun

2.1 Representations of integers

Braun’s first contributions concerned representations of integers as sums of squares. A familiar question of this sort is: How many ways can you write a given positive integer n as a sum of m squares? That is, what is the value of r_m(n), where

For example, r₁(7) = r₂(7) = r₃(7) = 0, but r₄(7) = 64. Lagrange proved in the eighteenth century that every positive integer can be written as a sum of four squares, an assertion that the ancient Greek mathematician Diophantus had already made around the third century. In the nineteenth century, work of Gauss, Jacobi, and Eisenstein led to formulas for r_m(n) for each positive integer m ≤ 8.

In the 1930s, when Braun came on the scene, this line of thought had expanded to include quadratic forms. This is where Braun’s dissertation results lie. Suppose you are handed a positive definite quadratic form Q in g variables with integral coefficients, i.e.,

with x denoting the g × 1 column vector

^tx its transpose, and T = T_Q a positive definite g × g matrix whose entries on the diagonal are integers and whose entries off the diagonal lie in 12Z How many ways can you write Q as a sum of m squares of linear terms with integral coefficients? Equivalently, what is the value of r_m(Q), where

(Here, M_m×g (𝕫) denotes m × g integer matrices.) When g = 1 and T_Q = n, r_m(Q) = r_m(n). In the 1930s, work of Mordell [34] and Ko [30] showed for m ≤ 8, that r_m(Q) > 0 whenever m ≥ g + 3.

In her dissertation, Braun gave a formula for r_m(Q) under these same conditions. Her formula recovers the aforementioned ones of Gauss, Jacobi, and Eisenstein as special cases. Her dissertation was published in Crelle in 1938 [13]. She graduated summa cum laude, and Siegel declared, „Nach dieser vorzüglichen ersten Arbeit kann die Wissenschaft noch weitere wertvolle Leistungen von Frl. Braun erwarten“ [42].

Soon, Braun and Siegel expanded their focus even further. Consider positive definite quadratic forms Q and Q˜ with integral coefficients, in g and m variables, respectively. How many A ∈ M_m×g (𝕫) are there so thatQ(x)=Q˜(Ax) ? When Q˜(x)=x12+⋯+xm2, or equivalently TQ˜ is the m×m identity matrix, we are simply asking for r_m(Q), which Braun computed in her dissertation. More generally, we are asking for A∈Mm×g(Z)tATQ˜A=TQ. Siegel had planned to write a book with Braun on quadratic forms [40, 41], but it never came to fruition. In 1940, the year after the war began, Siegel left for a long-term position at the Institute for Advanced Study in Princeton, while Braun remained at Göttingen. By the time she visited the IAS in 1947–48, she had turned her focus to Hermitian forms.

In her habilitation thesis, which was published in [16], Braun made significant advances for Hermitian forms. Now, instead of symmetric matrices, she considered Hermitian matrices T, i.e., tTˉ=T, and matrices A ∈ M_m×g (𝒪) for 𝒪 the ring of integers in a quadratic imaginary extension of ℚ. For example, 𝒪 could be 𝕫[i], but it could also be a ring with class number > 1, which can present new challenges. Given Hermitian matrices S and T under appropriate conditions, Braun considered |{A ∈ M_m×g (𝒪) | ^tĀSA = T }|. These problems led Braun to develop sophisticated tools whose applications extend far beyond what anyone likely could have foreseen at the time.

2.2 Siegel modular forms

This circle of ideas eventually led Siegel and Braun to make substantial advances for modular forms.^[1] Modular forms had been around since the 1800s, but Siegel and Braun’s foundational work extending them to higher dimensional settings would transform the landscape. One reason for interest in them stems from the fact generating functions for certain counting problems arise as the Fourier expansions of modular forms (for example, in Jacobi’s determination of r₄(n), as we shall see in Equation (2) below), but they also have turned out to be essential in an array of problems in number theory.

In the mid-1930s, Siegel introduced a class of functions that came to be known as Siegel modular forms [43, 44, 45, 46]. Braun would go on to make key contributions for Siegel modular forms, e.g., [14,15]. Siegel noted, “Einige beachtenswerte Beiträge auf diesem Gebiete verdankt man H. Braun” [46, p. 618]. Braun would also independently develop the foundations of related functions, Hermitian modular forms [17, 18, 19].

A Siegel modular form of degree g, weight k, and level 1 is a ℂ-valued holomorphic function f on the space

such that

for all Z ∈ ℋ_g and

(Here, Sym_g (ℝ) denotes the set of g×g symmetric real matrices.) This definition can be extended to handle other levels, which means replacing Sp_2g (𝕫) by certain subgroups. When g = 1, we also require that f is bounded as Y →∞. Siegel’s original definition included this boundedness condition for all g, but as Braun recounted in a letter to the mathematician Hans Maass [7], her former PhD student Max Koecher^[2] proved in the 1950s that the condition is automatically satisfied when g > 1 (now known as the Koecher principle).

The case g = 1 consists of modular forms, which had been studied since the 1800s. In 1838, Jacobi gave an explicit formula for the number r₄(n) of ways a positive integer n can be expressed as a sum of four squares. His proof that

relied on modular forms. His idea was as follows: Consider θm(q)=∑n∈Zrm(n)qn where q = e^2πiz. For each positive integer m,

Jacobi observed that θ^m is a modular form of weight m/2. To obtain a formula for r₄(n), he expressed θ⁴ as a linear combination of modular forms whose Fourier coefficients he knew explicitly. A similar idea works for other small values of m. The key point is that for small values of k, the space of weight k modular forms (of specified level) is generated by Eisenstein series whose Fourier coefficients can be written explicitly.

Eisenstein series are typically among the first examples of modular forms one encounters. The Eisenstein series of weight k and level 1 are the functions on the upper half plane defined by

Eisenstein series of other levels involve similar sums over other lattices, and there are also additional variants one can consider. Eisenstein series also play important roles in the space of degree g Siegel modular forms, where they arise as certain sums over lattices in spaces of g × 2g matrices.

Braun’s contributions to the foundations of Siegel modular forms include important work on Eisenstein series. Early on, she proved an identity relating certain Eisenstein series and with an analogue of θ^k. Jun-Ichi Igusa referred to this as the Braun identity [29], but the name seems not to have stuck. Later, Braun proved the convergence of Eisenstein series in the space of Siegel modular forms degree g > 1, including for sufficiently large low weights k [15]. Convergence of Eisenstein series of low weights is important, including for the above counting problems, but delicate to prove. Siegel made a point of crediting her [46], as he had only proved a weaker convergence result [43].

This relationship between θ and Eisenstein series is just one instance of a much broader phenomenon. Given a positive definite quadratic form Q with integral coefficients in m variables like above, we obtain Siegel modular forms

Hel Braun and her former PhD student Max Koecher in Hannover in 1974

Oberwolfach Photo Collection/Konrad Jacobs

where the sum is over all nonnegative definite symmetric half-integral (i.e., integers on the diagonal, half integers on the off-diagonal) g × g matrices S, q^S := e^{2πitrace(ZS)} with Z ∈ ℋ_g, and

When g=1 and Q(x)=x12+⋯+xm2,θQ(q)=θm(q). Siegel related weighted averages of the forms θ_Q to certain Eisenstein series in the space of Siegel modular forms, which led to the Siegel–Weil formula and currently ongoing research.

2.3 Hermitian modular forms

In the three aforementioned Annals papers [17, 18, 19], Braun developed a vast extension of this theory. She introduced Hermitian modular forms. These are an early instance of automorphic forms on unitary groups, which have turned out to be particularly useful objects, including in modern work on the Langlands program. Hermitian modular forms extend ideas introduced for Siegel modular forms to the Hermitian setting, i.e., the area of Braun’s habilitation thesis. Braun defined Hermitian modular forms similarly to Siegel modular forms, except for replacing ℋ_g by

(where Her_g (ℂ) is the set of conjugate symmetric complex matrices), replacing 𝕫 by the ring of integers 𝒪 in an imaginary quadratic extension of ℚ like in Braun’s habilitation, and replacing the symplectic group Sp_2g by the unitary group

Under the unofficial conventions of the field, Hermitian modular forms ought to be called Braun modular forms. They are examples of automorphic forms, a class of functions that includes Siegel modular forms, Hilbert modular forms, Picard modular forms, and so forth. (In the context of the above treatment, automorphic forms can be viewed as holomorphic functions on analogues of ℋ_g and 𝔥_g that satisfy analogues of the transformation property in Equation (1) for appropriate groups in place of the groups Sp_2g and U(g,g).) Certain families of automorphic forms also acquire names, such as Hida families and Coleman families. Some kinds of Eisenstein series also acquire names, like Siegel Eisenstein series and Klingen Eisenstein series, named for Siegel’s student Helmut Klingen. All these examples, except for Hermitian modular forms, are named for the mathematicians who developed their foundations. A significant amount of important work building on Braun’s foundations failed to cite or mention her.

In some sense, Braun’s naming her functions “Hermitian modular forms” was a service to mathematicians. They are associated to Hermitian forms. With this naming convention, Siegel modular forms ought to be called “skew-symmetric” or “symplectic” modular forms, since they are associated to symplectic bilinear forms. Maybe adopting that descriptive naming convention would be helpful, but that is not the one employed in this field.

With her name rarely mentioned in the context of these automorphic forms, Braun fell into relative obscurity among future generations building on her work. Many do not know to look for Braun’s contributions or perspective in this area. It is as if we are trying to view mathematics through a fog that we do not realize envelops us. We do not know what we are missing, but we are missing an important perspective nevertheless.

Last year, one of my PhD students experienced the consequences of this ignorance. He was hoping to find a paper he could consult and cite for a particular aspect of Hermitian modular forms. Instead, an expert whom I greatly respect asserted that as just one more instance of automorphic forms, Hermitian modular forms did not merit their own paper explicitly recording what my student wanted. Earlier this year, I discovered Braun had written a paper elegantly working out these details, and it was clearly warranted.

Braun’s Hermitian modular forms are important instances of automorphic forms on unitary groups of signature (n,n), which play a particularly powerful role among all automorphic forms. Their significance today extends beyond what anyone is likely to have foreseen when Braun started on the topic in the late 1940s. In the introduction to the first of her papers on Hermitian modular forms, she modestly wrote, “a detailed study . . . may be useful” [17]. Today, that looks like a huge understatement.

Unitary groups, groups preserving Hermitian forms, have turned out in recent years to be particularly important within the context of automorphic forms. On one hand, unitary groups have convenient geometric features. For instance, they have associated Shimura varieties, which can be powerful for establishing results in algebraic number theory but are not available for some other groups of central interest, in particular GL_n for n ≥ 3. (Certain quotients of 𝔥_g are the complex points of these Shimura varieties, which are moduli spaces parametrizing abelian varieties with additional structures.) On the other hand, they have close relationships with GL_n. In fact, some key results for GL_n that cannot be proved directly for GL_n can be established via sophisticated work in the setting of unitary groups U(n,n) of signature (n,n).

In the Langlands program, which arose two decades after Braun’s early work, unitary groups play an outsized role. Unitary groups, especially U(n,n), are particularly important. For example, Ana Caraiani and Peter Scholze exploit Shimura varieties associated to unitary groups to study Galois representations for GL_n, e.g., in [23], as recounted in the expository papers [22, Remark 10] and [24, Section 3.1.2]. Hermitian modular forms, together with structure of the spaces on which they are defined, also play important roles in other recent developments, including in many of my own papers.

Unsurprisingly, Braun developed an interest in the geometry of the spaces on which automorphic forms are defined. She understood particular aspects of ℋ_g and 𝔥_g, along with other so-called symmetric spaces, certain manifolds arising as the quotient of a group by a maximal compact subgroup. Mathematicians had been treating symmetric spaces in an ad hoc manner. In letters to Maass, Braun mused on the possibility for a broader framework in which to treat a large collection of symmetric spaces [8,9]. In the early 1940s, Siegel had introduced symplectic geometry [35,47], which made it natural to be asking questions about broader collections of symmetric spaces, as opposed to just specific instances. In the 1960s, Braun and her student Koecher would develop a unifying approach.

2.4 Jordan algebras

For the final three decades of her career, Braun focused on Jordan algebras, certain nonassociative algebras. At first glance, this topic looks distant from Braun’s earlier work, but she arrived there organically as she considered questions about symmetric spaces and how to treat them in a systematic way. Jordan algebras are nonassociative algebras A that are defined over a field and whose multiplication ∗ is commutative and also satisfies (x ∗ x) ∗ (x ∗ y) = x ∗ ((x ∗ x) ∗ y) for all x and y in A. In the context of an example that has already arisen in this article, namely the algebra A of symplectic or Hermitian matrices appearing in the symmetric spaces on which Siegel and Hermitian modular forms are defined, we get a Jordan algebra by defining multiplication ∗ on A by x∗y=xy+yx2. Jordan algebras turn out to be the key ingredient in Braun and Koecher’s development of a uniform treatment of symmetric spaces.

Her influence here is also substantial, but its full extent is hidden. She and Koecher wrote the first book on Jordan algebras [21]. A particularly telling mark of her influence is seen in the library card in the copy in the IAS’s library. If library cards had titles, this one could be called The Hidden Influence of Hel Braun. The library card reads like a who’s who of major players in her area. Right after the book was published in 1966, Walter Baily checked it out, followed by Sigurður Helgason, Ichino Satake, and other prominent mathematicians.

Soon after the book was released, Braun wrote to Maass that she had “gone very far into the algebraic side,” perhaps foreseeing the direction the field was moving. Of herself and Koecher, she wrote, “Wir sind auch beide sehr auf die algebraische Seite gegangen, und eigentlich unabhängig von einander. Aber wir haben halb das sichere Gefühl, dass dies in Moment die interessantere Siete ist, schon nur von Characteristik 0 wegzukommen, bis bei characteristik p > 0 durchzukommen” [10]. Indeed, despite their origins in characteristic 0, fundamental topics in Braun’s research would go on to see important developments in positive characteristic.

Braun’s influence stretched further through her personal contact with mathematicians at all career stages. In addition to her unofficial mentoring of students like Koecher, she officially supervised 18 PhD students during her years studying Jordan algebras. Her student Helmut Strade wrote, “Her support was really quite unusual, her everlasting confidence had been an extreme encouragement to me, and without her I would find myself at a different place” [48]. After Helmut Hasse wrote of one mathematician, “Seine Vortragsweise ist musterhaft klar und ueberzeugend, und er ist immer sehr um die Betreuung und Foerderung seiner Schueler bemueht, Hasse said that for Braun this “trifft in noch hoeherem Masse auf sie zu” [28]. An admirer of Braun’s mathematics and her lecturing, Hasse had written earlier, “Sie hält hier in Hamburg seit über 15 Jahren Vorlesungen auf allen möglichen Gebieten, mit bestem Erfolg, und ist bei Ihren Hörern, um die sie sich mit besonderer Fürsorge kümmert, sehr beliebt” [27]. Her direct impact extended well beyond Germany, including through lectures in India for a few months in 1970, at the end of which she lectured in Iran [1,12,38].

It seems likely that some of Braun’s vision for the use of Jordan algebras for automorphic forms has been obscured and remains yet to be (re)discovered. Originally, Braun and Koecher intended to write a book on Jordan algebras with Emil Artin, who had become interested in them through conversations with Braun [49, p. 129]. All copies of their original manuscript, which Koecher cites in [31,32], appear to have been lost.^[3]

Incidentally, you can find a book on a different subject by Artin and Braun. That book, Vorlesungen über algebraische Topologie, was based on lectures they gave in an introductory algebraic topology course at the University of Hamburg [2]. Armin Thedy, their student who wrote the article [49] cited in the previous paragraph, took the notes on which the book is based.

Braun also appears to have had broader ambitions for studying connections between automorphic forms and Jordan algebras, but her plans were thwarted. In 1963, she applied to return to the IAS to further these investigations [6]. In her application, she said she thought she “could study more efficiently if [she] had close contact with . . . A. Selberg.” Atle Selberg had done important work on automorphic forms, including on the eponymous Rankin–Selberg method. Selberg rejected her application [39], and he appears never to have pursued the study of Jordan algebras.

It is impossible to know for certain the details Braun had in mind to discuss with Selberg. Braun might have been onto something: Would she have been able to explain a recent observation of a “mysterious” connection between certain Rankin—Selberg integrals and a construction in terms of Jordan algebras [37, Appendix B]? At least for now, the underpinnings of that connection still remain to be discovered, or depending on what Braun knew, merely rediscovered.

Hel Braun in Hamburg in 1986

Oberwolfach Photo Collection/Dirk Ferus