In the 60s, Shimura studied certain algebraic curves as analogues of classical modular curves in order to construct class fields of totally real number fields. These curves were later coined "Shimura curves" and vastly generalized by Deligne. We will take a tour of the rich geometry and arithmetic of Shimura curves. Along the way, we may encounter tessellations of disks, quaternion algebras, abelian surfaces, elliptic curves with CM, Hurwitz curves ... and the answer to life, the universe and everything.
Briefly speaking, Shimura curves are simply one-dimensional Shimura varieties. I have accomplished my trivial notion task because I have told you a trivial notion. But obviously it does not help much if you do not know what the term Shimura varieties means. It only takes 5 chapters in Milne's notes in order to define them — not too bad — but initially Shimura invented them really because they are natural analogues of classical modular curves.
Let be the upper half plane. Then acts on by Mobius transformations. For each complex number , we can associate an elliptic curve . The endomorphism ring is given by , which is either or an order in some imaginary quadratic field . The latter case is rarer and is given the name complex multiplication (CM). Two such elliptic curves and are isomorphic if and only if lie in the same -orbit. Hence we have the following bijection.
can be identified with the fundamental set The elliptic point and have nontrivial stabilizer of order 2 and 3, which correspond to elliptic curves with complex multiplication by and and automorphisms groups of order and . can be viewed as the Riemann sphere with the north pole missing. By adding the cusp , we get the compactification . The cusp has the moduli interpretation as degenerate elliptic curves — nodal cubic curves.
Analogously, for the congruence subgroups , where
we get the compact Riemann surfaces after adding cusps to the quotient . These classical modular curves, which date back to Klein and Fricke in the 19th century, also play an important role in the modern proof of Fermat's last theorem. They are coverings of and are coarse moduli spaces of elliptic curves with additional torsion data:
Due to the moduli interpretation, and both have models over . has the function field , so there is a polynomial such that . A remarkable fact is that actually has integer coefficients. One can utilize this to show that for an elliptic curve with CM by , where is the ring of integers of , is actually an algebraic integer (Gross-Zagier have very explicit formula for these values). Moreover, the theory of complex multiplication shows:
This main theorem of complex multiplication implies an interesting result discovered by Ramanujan: is an "almost integer" as has class number 1!
One problem with is that its fundamental domain is not compact. As a consequence, every subgroup of commensurable with is not cocompact and we have to add cusps to obtain modular curves.
You have definitely seen many examples from the artwork of M. C. Escher. Using the Poincare disk model for , we can tessellate with infinitely many hyperbolic triangles with geodesic sides.
More generally, any triple satisfying gives us a triangle group. It is a cocompact subgroup of and has exactly three elliptic points of orders . From this point view, is simply the limiting case , where and are the order 2 and 3 elliptic points and the cusp emerges as the limit vertex.
By "arithmetically" we mean the way we obtained and other congruence subgroups by "taking -points" of a matrix group. More precisely,
Roughly speaking, after ignoring compact factors, an arithmetic subgroup of is simply a subgroup commensurable with provided . The congruence subgroups of are obtained by taking . But none of them are cocompact! In order get a cocompact arithmetic subgroup, we need to find some other algebraic group with . Instead of working with the matrix algebra and , we need some other -algebra structure. One example is given by the quaternion algebra.
By Wedderburn's theorem, every central simple algebra is a matrix algebra over a central division -algebra. All the central division -algebra are classified by the Brauer group . Quaternion algebras are characterized from division -algebras as those having a quadratic splitting field.
Now on let be a rational quaternion algebra split at . We can generalize the procedure of taking from by taking a maximal order (a -lattice of rank 4 which also a subring) and its norm 1 elements . Since , is an arithmetic subgroup of . As has no parabolic elements, we know that is cocompact. The resulting complex algebraic curve is the counterpart of the classical modular curve .
Now given any , we have a rank 4-lattice in , where we view . So we obtain a complex torus with an -action. It is actually an abelian surface via the Riemann form , where is chosen in the way that . A special case is when and , is just two copies of the elliptic curve . So the same logic should apply and we can check that
In this case we say is a QM-surface (quaternionic multiplication). For a fixed choice of , a theorem of Milne implies that if there is an embedding , then there is a unique principal polarization of such that the corresponding Rosati involution on coincides with the positive involution on .
Now assume is nonsplit and let be a QM-surface.
So we have proved the following
In the latter case, the corresponding point on the Shimura curve is called a CM point. Those are in some sense "degenerate" points on the moduli space of QM-surfaces. One can expect that these CM points could play an important role, since, unlike the modular curves case, we do not have truly degenerate cusps to work with.
Due to the moduli interpretation, has a canonical model over . More generally, one can define the order by imposing a congruence condition . The resulting Shimura curve has a similar moduli interpretation as QM-surfaces with extra level- structures, which I do not quite bother writing down here.
Instead of , one can also work more generally with a totally real number field of degree and a quaternion algebra over split only at one real place. Then we have an embedding from to the split factor of . Then norm one element in the maximal order will again form a cocompact arithmetic subgroup of and the quotient is a Shimura curve. These curves also have a moduli interpretation as abelian varieties of dimension with -actions, which is more complicated than the case. Using the moduli interpretation, Shimura proved that has a canonical model , where is a complete algebraic curve over , the maximal abelian extension of unramified at all finite primes. Shimura then constructed class fields for totally imaginary extension of totally real number fields:
Whatever the word "regular fixed point" means, it can be viewed as an analogue of the value of in the modular curves case. Moreover, the action of can be described explicitly by the Shimura reciprocity law. We shall not go into the excessive details here.
Now let us look at an interesting example of Shimura curves which relates to
In the mathematical context, you may have seen this magic number as a bound for the number of automorphisms of a complex algebraic curve .
You know the proof if you went to Anand's class and listened carefully. Those curves with equality are called Hurwitz curves. Now let me show you how quaternion algebras and Shimura curves could help us in finding Hurwitz curves. Let us get started by finding a volume formula for the fundamental domain of . Suppose is the half fundamental domain consisting of elliptic points of orders and quadruples of sides which are glued together in a way you all know. By the Gauss-Bonnet formula where is the Gaussian curvature of , is the geodesic curvature of . In our case, , and . Therefore, In other words, we recover the Riemann-Hurwitz formula, A simple calculation shows that the fundamental domain has the minimal volume when , and , which corresponds to . Suppose is a normal subgroup, then we know acts on as automorphisms and . If furthermore has no elliptic points, then , hence and is indeed a Hurwitz curve!
The remarkable thing is that is actually an arithmetic subgroup coming from a quaternion algebra. How can one possibly figure this out? The above volume formula is true for an arbitrary discrete subgroup . But since arithmetic subgroups are defined in an arithmetic way, so one could expect that the arithmetic properties of quaternion algebras would benefit us. This is the case and here is an amazing volume formula worked out by Shimizu  using only arithmetic properties.
Given a quaternion algebra, we can work out the volume of with Shimizu's formula in hand (and the zeta function is going to take over the world). Hence we can possibly solve and using the general volume formula. In fact, can also be determined by investigating arithmetics by the work of Eichler. So one can sit down and work hard with quaternion algebras with a hope of finding triangle groups, i.e. those with and . The complete list of arithmetic triangle groups are determined by Takeuchi , .
The above five are actually the first five smallest genus Hurwitz curves. The next one is of genus 17, which is non-arithmetic. The next arithmetic one is of genus 118 given by the inert prime above 3.
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